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Spacetime
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{{other uses}}{{Spacetime|cTopic=Types}}In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur.Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. However, in 1905, Albert Einstein based (wikisource:Translation:On the Electrodynamics of Moving Bodies|his seminal work on special relativity) on two postulates: (1) The laws of physics are invariant (i.e., identical) in all inertial systems (i.e., non-accelerating frames of reference); (2) The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.The logical consequence of taking these postulates together is the inseparable joining together of the four dimensions, hitherto assumed as independent, of space and time. Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source, the speed of light has the same speed regardless of the frame of reference in which it is measured; the distances and even temporal ordering of pairs of events change when measured in different inertial frames of reference (this is the relativity of simultaneity); and the linear additivity of velocities no longer holds true.Einstein framed his theory in terms of kinematics (the study of moving bodies). His theory was a breakthrough advance over Lorentz's 1904 theory of electromagnetic phenomena and PoincarÃ©'s electrodynamic theory. Although these theories included equations identical to those that Einstein introduced (i.e. the Lorentz transformation), they were essentially ad hoc models proposed to explain the results of various experimentsâ€”including the famous Michelsonâ€“Morley interferometer experimentâ€”that were extremely difficult to fit into existing paradigms.In 1908, Hermann Minkowskiâ€”once one of the math professors of a young Einstein in ZÃ¼richâ€”presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the formal definition of the spacetime interval. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded.Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy curve this flat spacetime to a Pseudo Riemannian manifold.{{anchor|Contents}}{{TOC limit|3}}- the content below is remote from Wikipedia
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Introduction
{{anchor|Introduction}}{{anchor|Definitions}}Definitions
Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external.WEB, Rynasiewicz, Robert, Newton's Views on Space, Time, and Motion,weblink Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, 24 March 2017, Furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense.BOOK, Davis, Philip J., Mathematics & Common Sense: A Case of Creative Tension, 2006, A.K. Peters, Wellesley, Massachusetts, 9781439864326, 86, In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, and z. A position in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). Spacetime is thus four dimensional. An event is something that happens instantaneously at a single point in spacetime, represented by a set of coordinates x, y, z and t.The word "event" used in relativity should not be confused with the use of the word "event" in normal conversation, where it might refer to an "event" as something such as a concert, sporting event, or a battle. These are not mathematical "events" in the way the word is used in relativity, because they have finite durations and extents. Unlike the analogies used to explain events, such as firecrackers or lightning bolts, mathematical events have zero duration and represent a single point in spacetime.The path of a particle through spacetime can be considered to be a succession of events. The series of events can be linked together to form a line which represents a particle's progress through spacetime. That line is called the particle's world line.{{rp|105}}Mathematically, spacetime is a manifold, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat.WEB, Rowland, Todd, Manifold,weblink Wolfram Mathworld, Wolfram Research, 24 March 2017, An extremely large scale factor, c (conventionally called the speed-of-light) relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly {{convert|300000|km|disp=or||}} in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe which is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelsonâ€“Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.BOOK, French, A.P., Special Relativity, 1968, CRC Press, Boca Raton, Florida, 0748764224, 35â€“60, {{anchor|Figure 1-1}}File:Observer in special relativity.svg|thumb|Figure 1-1. Each location in spacetime is marked by four numbers defined by a (frame of reference]]: the position in space, and the time (which can be visualized as the reading of a clock located at each position in space). The 'observer' synchronizes the clocks according to their own reference frame.)In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events are being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having a location. In Fig. 1‑1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the time and position of events taking place within the whole frame. The term observer refers to the entire ensemble of clocks associated with one inertial frame of reference.BOOK, Taylor, Edwin F., Wheeler, John Archibald, Spacetime Physics: Introduction to Special Relativity, 1966, Freeman, San Francisco, 071670336X, 1st,weblink 14 April 2017, {{rp|17â€“22}} In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, however, will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.Physicists distinguish between what one measures or observes (after one has factored out signal propagation delays), versus what one visually sees without such corrections. Failure to understand the difference between what one measures/observes versus what one sees is the source of much error among beginning students of relativity.JOURNAL, Scherr, Rachel E., Shaffer, Peter S., Vokos, Stamatis, Student understanding of time in special relativity: Simultaneity and reference frames, American Journal of Physics, July 2001, 69, S1, S24â€“S35, 10.1119/1.1371254,weblink 11 April 2017, 2001AmJPh..69S..24S, physics/0207109,History
{{anchor|History}}{{multiple image
| direction = vertical
| width = 220
| image1 = Michelson-Morley experiment conducted with white light.png
| image2 = MichelsonMorleyAnimationDE.gif
| caption2 = Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration.
}}By the mid-1800s, various experiments such as the observation of the Arago spot (a bright point at the center of a circular object's shadow due to diffraction) and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to a corpuscular theory.BOOK, Hughes, Stefan, Catchers of the Light: Catching Space: Origins, Lunar, Solar, Solar System and Deep Space, 2013, ArtDeCiel Publishing, Paphos, Cyprus, 9781467579926, 202â€“233,weblink 7 April 2017, Propagation of waves was then assumed to require the existence of a medium which waved: in the case of light waves, this was considered to be a hypothetical luminiferous aether.luminiferous from the Latin lumen, light, + ferens, carrying; aether from the Greek Î±á¼°Î¸Î®Ï (aithÄ“r), pure air, clear sky However, the various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction. Among other issues, the dependence of the partial aether-dragging implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether simultaneously flows at different speeds for different colors of light.BOOK, Stachel, John, Kox, A. J., Eisenstaedt, Jean, The Universe of General Relativity, BirkhÃ¤user, Boston, 2005, 1â€“13, Fresnelâ€™s (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies., 081764380X,weblinkweblink 13 April 2017, yes, dmy-all, The famous Michelsonâ€“Morley experiment of 1887 (Fig. 1‑2) showed no differential influence of Earth's motions through the hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of stellar aberration.George Francis FitzGerald in 1889 and Hendrik Lorentz in 1892 independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson-Morley experiment. (No length changes occur in directions transverse to the direction of motion.)By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein were to derive later (i.e. the Lorentz transform), but with a fundamentally different interpretation. As a theory of dynamics (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter.BOOK, Pais, Abraham, ""Subtle is the Lord-- ": The Science and the Life of Albert Einstein, 1982, Oxford University Press, Oxford, 019853907X, 11th, {{rp|163â€“174}} Lorentz's equations predicted a quantity that he called local time, with which he could explain the aberration of light, the Fizeau experiment and other phenomena. However, Lorentz considered local time to be only an auxiliary mathematical tool, a trick as it were, to simplify the transformation from one system into another.Other physicists and mathematicians at the turn of the century came close to arriving at what is currently known as spacetime. Einstein himself noted, that with so many people unraveling separate pieces of the puzzle, "the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905."BOOK, Born, Max, Physics in My Generation, 1956, Pergamon Press, London & New York, 194,weblink 10 July 2017, {{multiple image|perrow = 2|total_width=300< 0, and so in this measurement the moving clock W passes W'2 in the event C.In the upper picture the ct-coordinate At of the event A (the reading of W2) is labeled B, thus giving the elapsed time between the two events, measured with W1 and W2, as OB. For a comparison, the length of the time interval OA, measured with W', must be transformed to the scale of the ct-axis. This is done by the invariant hyperbola (see also Fig. 2-8) through A, connecting all events with the same spacetime interval from the origin as A. This yields the event C on the ct-axis, and obviously: OC < OB, the "moving" clock W' runs slower.To show the mutual time dilation immediately in the upper picture, the event D may be constructed as the event at xâ€² = 0 (the location of W' in S'), that is simultaneous to C (OC has equal spacetime interval as OA) in S'. This shows that the time interval OD is longer than OA, again, the "moving" clock, now W, runs slower.{{rp|124}}In the lower picture the frame S is moving with velocity -v in the frame S' at rest. The worldline of W is the ct-axis, slanted to the left, the worldline of W'1 is the vertical ctâ€²-axis and the worldline of W'2 is the vertical through event C, with ctâ€²-coordinate D. The invariant parabola through event C scales the time interval OC to OA, which is shorter than OD; also, B is constructed (similar to D in the upper pictures) as simultaneous to A in S, at x = 0. The result OB > OC corresponds again to above.Please note the importance of the word "measure". In classical physics an observer cannot affect an observed object, but the objects state of motion can affect the observer's observations of the object.| width = 220
| image1 = Michelson-Morley experiment conducted with white light.png
| image2 = MichelsonMorleyAnimationDE.gif
| caption2 = Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration.
Twin paradox
Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light. The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.BOOK, Schutz, Bernard F., A first course in general relativity, 1985, Cambridge University Press, Cambridge, UK, 0521277035, 26, The twin paradox is a thought experiment involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock.{{rp|207}} Nevertheless, the twin paradox is not a true paradox because it is easily understood within the context of special relativity.The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not.(File:Introductory Physics fig 4.9.png|thumb|Figure 2-11. Spacetime explanation of the twin paradox)Deeper analysis is needed before we can understand why these distinctions should result in a difference in the twins' ages. Consider the spacetime diagram of Fig. 2‑11. This presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the path integral) to calculate the total amount of proper time experienced by the traveling twin.WEB, Weiss, Michael, The Twin Paradox,weblink The Physics and Relativity FAQ, 10 April 2017, Complications arise if the twin paradox is analyzed from the traveling twin's point of view.For the rest of this discussion, we adopt Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella.We had previously noted that Stella is not in an inertial frame. Given this fact, it is sometimes stated that full resolution of the twin paradox requires general relativity. This is not true.A pure SR analysis would be as follows: Analyzed in Stella's rest frame, she is motionless for the entire trip. When she fires her rockets for the turnaround, she experiences a pseudo force which resembles a gravitational force. Figs. 2‑6 and 2‑11 illustrate the concept of lines (planes) of simultaneity: Lines parallel to the observer's x-axis (xy-plane) represent sets of events that are simultaneous in the observer frame. In Fig. 2‑11, the blue lines connect events on Terence's world line which, from Stella's point of view, are simultaneous with events on her world line. (Terence, in turn, would observe a set of horizontal lines of simultaneity.) Throughout both the outbound and the inbound legs of Stella's journey, she measures Terence's clocks as running slower than her own. But during the turnaround (i.e. between the bold blue lines in the figure), a shift takes place in the angle of her lines of simultaneity, corresponding to a rapid skip-over of the events in Terence's world line that Stella considers to be simultaneous with her own. Therefore, at the end of her trip, Stella finds that Terence has aged more than she has.Although general relativity is not required to analyze the twin paradox, application of the Equivalence Principle of general relativity does provide some additional insight into the subject. We had previously noted that Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of gravitational time dilation, his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together. As will be discussed in the forthcoming section Curvature of time, the theoretical arguments predicting gravitational time dilation are not exclusive to general relativity. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory.{{rp|16}}{{anchor|Gravitation}}Gravitation
This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena.{{rp|221}} A few of these phenomena are described in the later sections of this article.Basic mathematics of spacetime
{{anchor|Galilean transformations}}Galilean transformations
A basic goal is to be able to compare measurements made by observers in relative motion. Say we have an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks {{nowrap|1=(x, y, z, t)}} (see Fig. 1‑1). A second observer Oâ€² in a different frame Sâ€² measures the same event in her coordinate system and her lattice of synchronized clocks {{nowrap|1=({{â€²|x}}, {{â€²|y}}, {{â€²|z}}, {{â€²|t}})}}. Since we are dealing with inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates {{nowrap|1=(x, y, z, t)}} to {{nowrap|1=({{â€²|x}}, {{â€²|y}}, {{â€²|z}}, {{â€²|t}})}}. Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel {{nowrap|1=(x, y, z)}} coordinates and that {{nowrap|1=t = 0}} when {{nowrap|1={{â€²|t}} = 0}}, the coordinate transformation is as follows:BOOK, Mould, Richard A., Basic Relativity, 1994, Springer, 9780387952109, 42, 1st, 22 April 2017,weblink BOOK, Lerner, Lawrence S., Physics for Scientists and Engineers, Volume 2, 1997, Jones & Bartlett Pub, 9780763704605, 1047, 1st, 22 April 2017,weblink
x' = x - v t
y' = y
z' = z
t' = t .
(File:Galilean Spacetime and composition of velocities.svg|thumb|Figure 3-1. Galilean Spacetime and composition of velocities)Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light.{{rp|36â€“37}} Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.More generally, assume that frame Sâ€² is moving at velocity v with respect to frame S. Within frame Sâ€², observer Oâ€² measures an object moving with velocity {{â€²|u}}. What is its velocity u with respect to frame S? Since {{nowrap|1=x = ut}}, {{nowrap|1={{â€²|x}} = x âˆ’ vt}}, and {{nowrap|1=t = {{â€²|t}}}}, we can write {{nowrap|1={{â€²|x}} = ut âˆ’ vt}} = {{nowrap|1=(u âˆ’ v)t}} = {{nowrap|1=(u âˆ’ v){{â€²|t}}}}. This leads to {{nowrap|1={{â€²|u}} = {{â€²|x}}/{{â€²|t}}}} and ultimately
u' = u - v or u = u' + v ,
which is the common-sense Galilean law for the addition of velocities.{{anchor|Relativistic composition of velocities}}Relativistic composition of velocities
(File:Relativistic composition of velocities.svg|thumb|330px|Figure 3-2. Relativistic composition of velocities)The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,
beta = v/c
Fig. 3-2a illustrates a red train that is moving forward at a speed given by {{nowrap|1=v/c = β = s/a}}. From the primed frame of the train, a passenger shoots a bullet with a speed given by {{nowrap|1={{â€²|u}}/c = {{â€²|β}} = n/m}}, where the distance is measured along a line parallel to the red {{â€²|x}} axis rather than parallel to the black x axis. What is the composite velocity u of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3‑2b: - From the platform, the composite speed of the bullet is given by {{nowrap|1=u = c(s + r)/(a + b)}}.
- The two yellow triangles are similar because they are right triangles that share a common angle α. In the large yellow triangle, the ratio {{nowrap|1=s/a = v/c = β}}.
- The ratios of corresponding sides of the two yellow triangles are constant, so that {{nowrap|1=r/a = b/s}} = {{nowrap|1=n/m = {{â€²|β}}}}. So {{nowrap|1=b = {{â€²|u}}s/c}} and {{nowrap|1=r = {{â€²|u}}a/c}}.
- Substitute the expressions for b and r into the expression for u in step 1 to yield Einstein's formula for the addition of velocities:BOOK, Bais, Sander, Very Special Relativity: An Illustrated Guide, 2007, Harvard University Press, Cambridge, Massachusetts, 067402611X, {{rp|42â€“48}}
u = {v+u'over 1+(vu'/c^2)} .
- If {{â€²|u}} and v are both very small compared with the speed of light, then the product {{â€²|vu}}/c2 becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: u = {{â€²|u}} + v. The Galilean formula is a special case of the relativistic formula applicable to low velocities.
- If {{â€²|u}} is set equal to c, then the formula yields u = c regardless of the starting value of v. The velocity of light is the same for all observers regardless their motions relative to the emitting source.{{rp|49}}
Time dilation and length contraction revisited
(File:Spacetime Diagrams Illustrating Time Dilation and Length Contraction.png|thumb|330px|Figure 3-3. Spacetime diagrams illustrating time dilation and length contraction)We had previously discussed, in qualitative terms, time dilation and length contraction. It is straightforward to obtain quantitative expressions for these effects. Fig. 3‑3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.To reduce the complexity of the equations slightly, we see in the literature a variety of different shorthand notations for ct :
Tau = ct and w = ct are common.
One also sees very frequently the use of the convention c = 1.
(File:Lorentz factor.svg|thumb|Figure 3-4. Lorentz factor as a function of velocity)In Fig. 3-3a, segments OA and OK represent equal spacetime intervals. Time dilation is represented by the ratio OB/OK. The invariant hyperbola has the equation {{nowrap|1={{math|w {{=}} {{radical|x2 + k2}}}}}} where k = OK, and the red line representing the world line of a particle in motion has the equation w = x/β = xc/v. A bit of algebraic manipulation yields OB = OK / sqrt{1 - v^2/c^2} .The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma gamma:BOOK, Forshaw, Jeffrey, Smith, Gavin, Dynamics and Relativity, 2014, John Wiley & Sons, 9781118933299, 118,weblink 24 April 2017, en,
gamma = frac{1}{sqrt{1 - v^2/c^2}} = frac{1}{sqrt{1 - beta^2}}
We note that if v is greater than or equal to c, the expression for gamma becomes physically meaningless, implying that c is the maximum possible speed in nature. Next, we note that for any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.In Fig. 3-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB/OK. The invariant hyperbola has the equation {{nowrap|1={{math|x {{=}} {{radical|w2 + k2}}}}}}, where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/β = c/v. Event A has coordinates (x, w) = (γk, γβk). Since the tangent line through A and B has the equation w = (x âˆ’ OB)/β, we have γβk = (γk âˆ’ OB)/β and
OB/OK = gamma (1 - beta ^ 2) = frac{1}{gamma}
{{anchor|Lorentz transformations}}Lorentz transformations
The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.To transform the coordinates of an event from one frame to another in special relativity, we use the Lorentz transformations.The Lorentz factor appears in the Lorentz transformations:
begin{align}
t' &= gamma left( t - frac{v x}{c^2} right)
x' &= gamma left( x - v t right)
y' &= y
z' &= z
end{align}The inverse Lorentz transformations are:
x' &= gamma left( x - v t right)
y' &= y
z' &= z
begin{align}
t &= gamma left( t' + frac{v x'}{c^2} right)
x &= gamma left( x' + v t' right)
y &= y'
z &= z'
end{align}When v â‰ª c and x is small enough, the v2/c2 and vx/c2 terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.As noted before, when we write t' = gamma ( t - v x/c^2), x' = gamma( x - v t) and so forth, we most often really mean Delta t' = gamma (Delta t - v Delta x/c^2), Delta x' = gamma(Delta x - v Delta t) and so forth. Although, for brevity, we write the Lorentz transformation equations without deltas, it should be understood that x means Δx, etc. We are, in general, always concerned with the space and time differences between events.Note on nomenclature: Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the S frame can only be moving forwards or reverse with respect to {{â€²|S}}. So inverting the equations simply entails switching the primed and unprimed variables and replacing v with âˆ’v.{{rp|71â€“79}}Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time {{nowrap|1=t = {{â€²|t}} = 0}}, Stella's spaceship accelerates instantaneously to a speed of 0.5 c. The distance from Earth to Mars is 300 light-seconds (about {{val|90.0|e=6|u=km}}). Terence observes Stella crossing the finish-line clock at t = 600.00 s. But Stella observes the time on her ship chronometer to be {{nowrap|1={{â€²|t}} =}} {{nowrap|1=gamma(t âˆ’ vx/c2)}} = 519.62 s as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about {{val|77.9|e=6|u=km}}).1).x &= gamma left( x' + v t' right)
y &= y'
z &= z'
Deriving the Lorentz transformations
(File:Derivation of Lorentz Transformation.svg|thumb|Figure 3-5. Derivation of Lorentz Transformation)There have been many dozens of derivations of the Lorentz transformations since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality, which states that the influence that one particle exerts on another can not be transmitted instantaneously.BOOK, Landau, L. D., Lifshitz, E. M., The Classical Theory of Fields, Course of Theoretical Physics, Volume 2, 2006, Elsevier, Amsterdam, 9780750627689, 1â€“24, 4th, The derivation given here and illustrated in Fig. 3‑5 is based on one presented by Bais{{rp|64â€“66}} and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event P has coordinates (w, x) in the black "rest system" and coordinates ({{â€²|w}}, {{â€²|x}}) in the red frame that is moving with velocity parameter β = v/c. How do we determine {{â€²|w}} and {{â€²|x}} in terms of w and x? (Or the other way around, of course.)It is easier at first to derive the inverse Lorentz transformation.- We start by noting that there can be no such thing as length expansion/contraction in the transverse directions. y{{'}} must equal y and {{â€²|z}} must equal z, otherwise whether a fast moving 1 m ball could fit through a 1 m circular hole would depend on the observer. The first postulate of relativity states that all inertial frames are equivalent, and transverse expansion/contraction would violate this law.BOOK, Morin, David, Special Relativity for the Enthusiastic Beginner, 2017, CreateSpace Independent Publishing Platform, 9781542323512, {{rp|27â€“28}}
- From the drawing, w = a + b and x = r + s
- From previous results using similar triangles, we know that s/a = b/r = v/c = Î².
- We know that because of time dilation, a = γ{{prime|w}}
- Substituting equation (4) into s/a = Î² yields s = γ{{prime|w}}Î².
- Length contraction and similar triangles give us r = γ{{prime|x}} and b = βr = βγ{{prime|x}}
- Substituting the expressions for s, a, r and b into the equations in Step 2 immediately yield
- :w = gamma w' + beta gamma x'
- :x = gamma x' + beta gamma w'
Linearity of the Lorentz transformations
The Lorentz transformations have a mathematical property called linearity, since x{{'}} and t{{'}} are obtained as linear combinations of x and t, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that we tacitly assumed while performing the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere.{{rp|67}} All inertial observers will agree on what constitutes accelerating and non-accelerating motion.{{rp|72â€“73}} Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well.{{rp|190}}A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.Example: Terence observes Stella speeding away from him at 0.500 c, and he can use the Lorentz transformations with β = 0.500 to relate Stella's measurements to his own. Stella, in her frame, observes Ursula traveling away from her at 0.250 c, and she can use the Lorentz transformations with β = 0.250 to relate Ursula's measurements with her own. Because of the linearity of the transformations and the relativistic composition of velocities, Terence can use the Lorentz transformations with β = 0.666 to relate Ursula's measurements with his own.{{anchor|Doppler effect}}Doppler effect
The Doppler effect is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line (transverse Doppler effect). We are ignoring scenarios where they move along intermediate angles.Longitudinal Doppler effect
The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of vs for a velocity parameter of βs, the wavelength is increased, and the observed frequency f is given by
f = frac{1}{1+beta _s}f_0
On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of vr for a velocity parameter of βr, the wavelength is not changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency f is given by
f = (1-beta _r)f_0
(File:Spacetime Diagram of Relativistic Doppler Effect.svg|thumb|Figure 3-6. Spacetime diagram of relativistic Doppler effect)Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig. 3‑6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter β, so that the separation between source and receiver at time w is βw. Because of time dilation, w = γw{{'}}. Since the slope of the green light ray is âˆ’1, T = w+βw = γw{{'}}(1+β). Hence, the relativistic Doppler effect is given by{{rp|58â€“59}}
f = sqrt{frac{1 - beta}{1 + beta}},f_0.
Transverse Doppler effect
(File:Transverse Doppler effect scenarios 2.svg|thumb|300px|Figure 3-7. Transverse Doppler effect scenarios)Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:BOOK, Morin, David, Introduction to Classical Mechanics: With Problems and Solutions, 2008, Cambridge University Press, 978-0-521-87622-3, {{rp|541â€“543}}{{plainlist|- Fig. 3-7a. What is the frequency measurement when the receiver is geometrically at its closest approach to the source? This scenario is most easily analyzed from the frame S' of the source.
- Fig. 3-7b. What is the frequency measurement when the receiver sees the source as being closest to it? This scenario is most easily analyzed from the frame S of the receiver.
- Fig. 3-7c. If the receiver is moving in a circle around the source, what frequency does the receiver measure?
- Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?
f = f' gamma = f' / sqrt { 1 - beta ^2 }
In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is redshifted with frequency
f = f' / gamma = f' sqrt { 1 - beta ^2 }
Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of gamma, and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.){{rp|541â€“543}} Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).Not all experiments characterize the effect in terms of a redshift. For example, the KÃ¼ndig experiment was set up to measure transverse blueshift using a MÃ¶ssbauer source setup at the center of a centrifuge rotor and an absorber at the rim.{{anchor|Energy and momentum}}Energy and momentum
Extending momentum to four dimensions
(File:Relativistic spacetime momentum vector.svg|thumb|330px|Figure 3-8. Relativistic spacetime momentum vector)In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. Linear momentum, the product of a particle's mass and velocity, is a vector quantity, possessing the same direction as the velocity: p = mv. It is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector (x, t). In exploring the properties of the spacetime momentum, we start, in Fig. 3‑8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. p = 0, but the time component equals mc.We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that (mc){{'}} = γmc and p{{'}} = âˆ’βγmc, since the red axes are rescaled by gamma. Fig. 3‑8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches c.{{rp|84â€“87}}We will use this information shortly to obtain an expression for the four-momentum.Momentum of light
(File:Calculating the energy of light in different inertial frames.svg|thumb|Figure 3-9. Energy and momentum of light in different inertial frames)Light particles, or photons, travel at the speed of c, the constant that is conventionally known as the speed of light. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a light-like world line and, in appropriate units, have equal space and time components for every observer.A consequence of Maxwell's theory of electromagnetism is that light carries energy and momentum, and that their ratio is a constant: E/p = c. Rearranging, E/c = p, and since for photons, the space and time components are equal, E/c must therefore be equated with the time component of the spacetime momentum vector.Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in γmc must be zero, meaning that photons are massless particles. Infinity times zero is an ill-defined quantity, but E/c is well-defined.By this analysis, if the energy of a photon equals E in the rest frame, it equals {{nowrap|1=E{{'}} = (1 âˆ’ β)γE}} in a moving frame. This result can be derived by inspection of Fig. 3‑9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.{{rp|88}}Mass-energy relationship
Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several famous conclusions.- In the low speed limit as β = v/c approaches zero, gamma approaches 1, so the spatial component of the relativistic momentum {{nowrap|1=βγmc = γmv}} approaches mv, the classical term for momentum. Following this perspective, γm can be interpreted as a relativistic generalization of m. Einstein proposed that the relativistic mass of an object increases with velocity according to the formula {{nowrap|1=mrel = γm}}.
- Likewise, comparing the time component of the relativistic momentum with that of the photon, {{nowrap|1=γmc = mrelc = E/c}}, so that Einstein arrived at the relationship {{nowrap|1=E = mrelc2}}. Simplified to the case of zero velocity, this is Einstein's famous equation relating energy and mass.
E = gamma m c^2 =frac{m c^2}{sqrt{1 - beta ^ 2}} approx m c^2 + frac{1}{2} m v^2 ...
The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.{{rp|90â€“92}}{{rp|129â€“130,180}}The concept of relativistic mass that Einstein introduced in 1905, mrel, although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes,JOURNAL, Rose, H. H., Optics of high-performance electron microscopes, Science and Technology of Advanced Materials, 21 April 2008, 9, 1, 014107, 10.1088/0031-8949/9/1/014107,weblink 4 July 2017, 2008STAdM...9a4107R, bot: unknown,weblink 3 July 2017, dmy-all, old-fashioned color television sets, etc.), has nevertheless not proven to be a fruitful concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy.BOOK, Griffiths, David J., Revolutions in Twentieth-Century Physics, 2013, Cambridge University Press, Cambridge, 9781107602175, 60,weblink 24 May 2017, en, "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or invariant mass, and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula,
E^2 - p^2c^2 = m^2 c^4
This formula applies to all particles, massless as well as massive. For massless photons, it yields the same relationship that we had earlier established, {{nowrap|1=E = Â±pc}}.{{rp|90â€“92}}Four-momentum
Because of the close relationship between mass and energy, the four-momentum (also called 4‑momentum) is also called the energy-momentum 4‑vector. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as
P equiv (E/c, vec{p}) = (E/c, p_x, p_y, p_z) or alternatively,
P equiv (E, vec{p}) = (E, p_x, p_y, p_z) using the convention that c = 1 .{{rp|129â€“130,180}}
{{anchor|Conservation laws}}Conservation laws
In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature.ARXIV, Byers, Nina, E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws, physics/9807044, 1998, The fact that physical processes don't care where in space they take place (space translation symmetry) yields conservation of momentum, the fact that such processes don't care when they take place (time translation symmetry) yields conservation of energy, and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.Total momentum
(File:Relativistic conservation of momentum.png|thumb|Figure 3-10. Relativistic conservation of momentum)To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity: (1) The two bodies rebound from each other in a completely elastic collision. (2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision. For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.In case (2), two masses with momentums {{nowrap|1=p1 = m1v1}} and {{nowrap|1=p2 = m2v2}} collide to produce a single particle of conserved mass {{nowrap|1=m = m1 + m2}} traveling at the center of mass velocity of the original system, {{nowrap|1=vcm = (m1v1 + m2v2)/(m1 + m2)}}. The total momentum {{nowrap|1=p = p1 + p2}} is conserved.Fig. 3‑10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components E1/c and E2/c add up to total E/c of the resultant vector, meaning that energy is conserved. Likewise, the space components p1 and p2 add up to form p of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: {{nowrap|1=m > m1 + m2}}.{{rp|94â€“97}}Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.{{rp|134â€“138}}Choice of reference frames
{{multiple image|align=right|image1=2-body Particle Decay-Lab.svg|width1=115|image2=2-body Particle Decay-CoM.svg|width2=105|caption1=Figure 3-11. (above) Lab Frame.(right) Center of Momentum Frame.| }}The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "center-of-momentum frame" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig. 3‑11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same.Return to IntroductionEnergy and momentum conservation
In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since {{nowrap|1=v{{'}} = v âˆ’ u}}, the momentum {{nowrap|1=p{{'}} = p âˆ’ mu}}. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame.{{rp|241â€“245}}Conservation of momentum in the COM frame amounts to the requirement that p = 0 both before and after collision. In the Newtonian analysis, conservation of mass dictates that {{nowrap|1=m = m1 + m2}}. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero.{{rp|241â€“245}}Newtonian momenta, calculated as {{nowrap|1=p = mv}}, fail to behave properly under Lorentzian transformation. The linear transformation of velocities {{nowrap|1=v{{'}} = v âˆ’ u}} is replaced by the highly nonlinear{{nowrap|1=v{{'}} = (v âˆ’ u)/(1 âˆ’ vu/c2),}} so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. As we have discussed in the previous section on four-momentum, this second option was what he chose.{{rp|104}}{{multiple image
| align = right
| direction = vertical
| width = 220
| direction = vertical
| width = 220
| image1 = Energy-momentum diagram for pion decay (A).png
| width1 =
| alt1 =
| caption1 = Figure 3-12a. Energy-momentum diagram for decay of a charged pion.
| width1 =
| alt1 =
| caption1 = Figure 3-12a. Energy-momentum diagram for decay of a charged pion.
| image2 = Energy-momentum diagram for pion decay (B).png
| width2 =
| alt2 =
| caption2 = Figure 3-12b. Graphing calculator analysis of charged pion decay.
}}The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.{{rp|127}}Example: Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where {{nowrap|1=1 MeV = 1Ã—106}} electron volts. A charged pion is a particle of mass 139.57 MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66 MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91 MeV.
| width2 =
| alt2 =
| caption2 = Figure 3-12b. Graphing calculator analysis of charged pion decay.
{{SubatomicParticle|Pion-}} â†’ {{SubatomicParticle|link=yes|Muon-}} + {{SubatomicParticle|link=yes|Muon antineutrino}}
Fig. 3‑12a illustrates the energy-momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is {{nowrap|1=Eν = pc,}} which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.Algebraic analyses of the energetics of this decay reaction are available online,WEB, Nave, R., Energetics of Charged Pion Decay,weblink Hyperphysics, Department of Physics and Astronomy, Georgia State University, 27 May 2017, so Fig. 3‑12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is {{nowrap|1=33.91 âˆ’ 29.79 = 4.12 MeV.}} Most of the energy is carried off by the near-zero-mass neutrino. Beyond the basics
{{anchor|Rapidity}}The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime.Rapidity
{{multiple image
| direction = horizontal
| width1 = 135
| image1 = Trig functions (sine and cosine).svg
| caption1 = Figure 4-1a. A ray through the unit circle {{nowrap|1=x2 + y2 = 1}} in the point {{nowrap|1=(cos a, sin a)}}, where a is twice the area between the ray, the circle, and the x-axis.
| width2 = 190
| image2 = Hyperbolic functions-2.svg
| caption2 = Figure 4-1b. A ray through the unit hyperbola {{nowrap|1=x2 - y2 = 1}} in the point {{nowrap|1=(cosh a, sinh a)}}, where a is twice the area between the ray, the hyperbola, and the x-axis.
}}File:Sinh+cosh+tanh.svg|thumb|180px|Figure 4-2. Plot of the three basic (:en:Hyperbolic function|Hyperbolic functions): hyperbolic sine ((:Image:Hyperbolic Sine.svg|sinh)), hyperbolic cosine ((:Image:Hyperbolic Cosine.svg|cosh)) and hyperbolic tangent ((:Image:Hyperbolic Tangent.svg|tanh)). Sinh is red, cosh is blue and tanh is green.]]Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.This nonlinearity is an artifact of our choice of parameters.{{rp|47â€“59}} We have previously noted that in an {{nowrap|1=xâ€“ct}} spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. Fig. 4‑1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the {{nowrap|1=x-axis}}, but as twice the area of the sector swept out by the ray from the {{nowrap|1=x-axis}}. (Numerically, the angle and {{nowrap|1=2 Ã— area}} measures for the unit circle are identical.) Fig. 4‑1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.BOOK, Thomas, George B., Weir, Maurice D., Hass, Joel, Giordano, Frank R., Thomas' Calculus: Early Transcendentals, 2008, Pearson Education, Inc., Boston, 0321495756, 533, Eleventh, Fig. 4‑2 presents plots of the sinh, cosh, and tanh functions.For the unit circle, the slope of the ray is given by
| width1 = 135
| image1 = Trig functions (sine and cosine).svg
| caption1 = Figure 4-1a. A ray through the unit circle {{nowrap|1=x2 + y2 = 1}} in the point {{nowrap|1=(cos a, sin a)}}, where a is twice the area between the ray, the circle, and the x-axis.
| width2 = 190
| image2 = Hyperbolic functions-2.svg
| caption2 = Figure 4-1b. A ray through the unit hyperbola {{nowrap|1=x2 - y2 = 1}} in the point {{nowrap|1=(cosh a, sinh a)}}, where a is twice the area between the ray, the hyperbola, and the x-axis.
text{slope} = tan a = frac{sin a }{cos a }.
In the Cartesian plane, rotation of point {{nowrap|1=(x, y)}} into point {{nowrap|1=(x{{'}}, y{{'}})}} by angle θ is given by
beta equiv tanh phi equiv frac{v}{c},
where
tanh phi = frac{sinh phi}{cosh phi} = frac{e^phi-e^{-phi}}{e^phi+e^{-phi}}.
The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;{{rp|47â€“59}}
beta = frac{beta_1 + beta_2}{1 + beta_1 beta_2} = frac{tanh phi_1 + tanh phi_2}{1 + tanh phi_1 tanh phi_2} = tanh(phi_1 + phi_2),
or in other words, phi = phi_1 + phi_2.The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as
gamma = frac{1}{sqrt{1 - beta^2}} = frac{1}{sqrt{1 - tanh^2 phi}} = cosh phi,
gamma beta = frac{beta}{sqrt{1 - beta^2}} = frac{tanh phi}{sqrt{1 - tanh^2 phi}} = sinh phi.
Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts.Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the {{nowrap|1=x direction}} may be written as
begin{pmatrix}
c t'
x'
end{pmatrix}
=
begin{pmatrix}
cosh phi & -sinh phi
-sinh phi & cosh phi
end{pmatrix}
begin{pmatrix}
ct
x
end{pmatrix},
and the inverse Lorentz boost in the {{nowrap|1=x direction}} may be written as
c t'
x'
end{pmatrix}
=
begin{pmatrix}
cosh phi & -sinh phi
-sinh phi & cosh phi
end{pmatrix}
begin{pmatrix}
ct
x
end{pmatrix},
begin{pmatrix}
c t
x
end{pmatrix}
=
begin{pmatrix}
cosh phi & sinh phi
sinh phi & cosh phi
end{pmatrix}
begin{pmatrix}
c t'
x'
end{pmatrix}.
In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime.{{rp|96â€“99}}The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.BOOK, Taylor, Edwin F., Wheeler, John Archibald, Spacetime Physics, 1992, W. H. Freeman, 0716723271, 2nd, Rapidity arises naturally as a coordinates on the pure boost generators inside the Lie algebra algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo {{nowrap|2{{pi}}}}) on the pure rotation generators in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.{{anchor|4â€‘vectors}}c t
x
end{pmatrix}
=
begin{pmatrix}
cosh phi & sinh phi
sinh phi & cosh phi
end{pmatrix}
begin{pmatrix}
c t'
x'
end{pmatrix}.
4â€‘vectors
Fourâ€‘vectors have been mentioned above in context of the energy-momentum {{nowrap|1=4â€‘vector}}, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, {{nowrap|1=4â€‘vectors}}, and more generally tensors, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation (really no more than an observation) using the field strength tensor formulation. On the other hand, general relativity, from the outset, relies heavily on {{nowrap|1=4â€‘vectors}}, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such {{nowrap|1=4â€‘vectors}} even within a curved spacetime, and not just within a flat one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.Definition of 4-vectors
A 4-tuple, {{nowrap|1=A = (A0, A1, A2, A3)}} is a "4-vector" if its component A i transform between frames according to the Lorentz transformation.If using {{nowrap|1=(ct, x, y, z)}} coordinates, A is a {{nowrap|1=4â€“vector}} if it transforms (in the {{nowrap|1=x-direction}}) according to
begin{align}
A_0' &= gamma left( A_0 - (v/c) A_1 right)
A_1' &= gamma left( A_1 - (v/c) A_0 right)
A_2' &= A_2
A_3' &= A_3
end{align}which comes from simply replacing ct with A0 and x with A1 in the earlier presentation of the Lorentz transformation.As usual, when we write x, t, etc. we generally mean Δx, Δt etc.The last three components of a {{nowrap|1=4â€“vector}} must be a standard vector in three-dimensional space. Therefore, a {{nowrap|1=4â€“vector}} must transform like {{nowrap|1=(c Δt, Δx, Δy, Δz)}} under Lorentz transformations as well as rotations.{{rp|36â€“59}}A_1' &= gamma left( A_1 - (v/c) A_0 right)
A_2' &= A_2
A_3' &= A_3
Properties of 4-vectors
- Closure under linear combination: If A and B are {{nowrap|1=4-vectors}}, then {{nowrap|1=C = aA + aB}} is also a {{nowrap|1=4-vector}}.
- Inner-product invariance: If A and B are {{nowrap|1=4-vectors}}, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a {{nowrap|1=3-vector}}. In the following, vec{A} and vec{B} are {{nowrap|1=3-vectors}}:
A cdot B equiv A_0 B_0 - A_1 B_1 - A_2 B_2 - A_3 B_3 equiv A_0 B_0 - vec{A} cdot vec{B}
In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in {{nowrap|1=3-space}}.
Two vectors are said to be orthogonal if A cdot B = 0. Unlike the case with {{nowrap|1=3-vectors,}} orthogonal {{nowrap|1=4-vectors}} are not necessarily at right angles with each other. The rule is that two {{nowrap|1=4-vectors}} are orthogonal if they are offset by equal and opposite angles from the 45Â° line which is the world line of a light ray. This implies that a lightlike {{nowrap|1=4-vector}} is orthogonal with itself.
- Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a {{nowrap|1=4-vector}} with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which A cdot A = 0 , while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval c^2 t^2 - x^2 and the invariant length of the relativistic momentum vector E^2 - p^2 c^2 .{{rp|178â€“181}}{{rp|36â€“59}}
Examples of 4-vectors
- Displacement 4-vector: Otherwise known as the spacetime separation, this is {{nowrap|1=(Δt, Δx, Δy, Δz),}} or for infinitesimal separations, {{nowrap|1=(dt, dx, dy, dz)}}.
dS equiv (dt, dx, dy, dz)
- Velocity 4-vector: This results when the displacement {{nowrap|1=4-vector}} is divided by d tau, where d tau is the proper time between the two events that yield dt, dx, dy, and dz.
V equiv frac{dS}{d tau} = frac{(dt, dx, dy, dz)}{dt/gamma} = gamma left(1, frac{dx}{dt}, frac{dy}{dt}, frac{dz}{dt} right) = (gamma, gamma vec{v} )
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| caption1 = Figure 4-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.
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| caption2 = Figure 4-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).
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| caption1 = Figure 4-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.
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| caption2 = Figure 4-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).
The {{nowrap|1=4-velocity}} is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.
An accelerated particle does not have an inertial frame in which it is always at rest. However, as stated before in the earlier discussion of the transverse Doppler effect, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles.
Since photons move on null lines, d tau = 0 for a photon, and a {{nowrap|1=4-velocity}} cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.
- Energy-momentum 4-vector: As discussed in the section on Energy and momentum,
P equiv (E/c, vec{p}) = (E/c, p_x, p_y, p_z)
As indicated before, there are varying treatments for the energy-momentum {{nowrap|1=4-vector}} so that one may also see it expressed as (E, vec{p}) or (E, vec{p}c) . The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy-momentum {{nowrap|1=4-vector}} is a conserved quantity.
- Acceleration 4-vector: This results from taking the derivative of the velocity {{nowrap|1=4-vector}} with respect to tau .
A equiv frac{dV}{d tau} = frac{d}{d tau} (gamma, gamma vec{v}) = gamma left( frac{d gamma}{dt}, frac{d(gamma vec{v})}{dt} right)
- Force 4-vector: This is the derivative of the momentum {{nowrap|1=4-vector}} with respect to tau .
F equiv frac{dP}{d tau} = gamma left(frac{dE}{dt}, frac{d vec{p}}{dt} right) = gamma left( frac{dE}{dt},vec{f} right)
4-vectors and physical law
The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. As noted in the previous discussion of energy and momentum conservation, Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving {{nowrap|1=4-vectors}} rather than give up on conservation of momentum.Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving {{nowrap|1=4-vectors}} require the use of tensors with appropriate rank, which themselves can be thought of as being built up from {{nowrap|1=4-vectors}}.{{rp|186}}{{anchor|Acceleration}}Acceleration
{{See|Acceleration (special relativity)}}It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.WEB, Gibbs, Philip, Can Special Relativity Handle Acceleration?,weblink The Physics and Relativity FAQ, math.ucr.edu, 28 May 2017, Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.In this section, we analyze several scenarios involving accelerated reference frames.{{anchor|Dewanâ€“Beranâ€“Bell spaceship paradox}}Dewanâ€“Beranâ€“Bell spaceship paradox
The Dewanâ€“Beranâ€“Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.(File:Bell's spaceship paradox - two spaceships - initial setup.png|thumb|Figure 4-4. Dewanâ€“Beranâ€“Bell spaceship paradox)In Fig. 4‑4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Will the string break?The main article for this section recounts how, when the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.{{rp|106,120â€“122}}- To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance {{nowrap|1=L{{'}} = γL}} in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break.
- Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.{{rp|106,120â€“122}}
begin{align}
x'_{A}& = gammaleft(x_{A}-vtright)x'_{B}& = gammaleft(x_{A}+L-vtright)L'& = x'_{B}-x'_{A} =gamma Lend{align}The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame S. As shown in Fig. 4‑5, Bell's example asserts the moving lengths AB and A'B' measured in frame S to be fixed, thereby forcing the rest frame length A'B'' in frame S' to increase.{{anchor|Accelerated observer with horizon}}Accelerated observer with horizon
Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. In the text accompanying Fig. 2‑7, we had noted that the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.File:Accelerated relativistic observer with horizon.png|thumb|Figure 4-6. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed (:File:ConstantAcceleration02.jpg|here). ]]Fig. 4‑6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter beta approaches a limit of one as ct increases. Likewise, gamma approaches infinity.The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:- We remember that beta = ct/x.
- Since c^2 t^2 - x^2 = s^2, we conclude that beta (ct) = ct/ sqrt{c^2 t^2 - s^2}.
- gamma = 1/sqrt{1 - beta ^2} = sqrt{c^2 t^2 - s^2}/s
- From the relativistic force law, F = dp/dt = dpc/d(ct) = d(beta gamma m c^2)/d(ct).
- Substituting beta(ct) from step 2 and the expression for gamma from step 3 yields F = mc^2 / s , which is a constant expression.{{rp|110â€“113}}
Introduction to curved spacetime
{{anchor|Basic propositions}}Basic propositions
Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space.Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it." In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.{{rp|175â€“190}}File:Principle of the tidal force.svg|thumb|Figure 5-1. Tidal effects.Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles {{nowrap|1=2 and 3}} but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter's naivetÃ©. This second reporter knows that in reality, the apparent forces between particles {{nowrap|1=2 and 3}} really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesicsgeodesicsIn spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic. No evidence of gravitation can be discovered following alongside the motions of a single particle.{{rp|175â€“190}}In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. In Fig. 5‑1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth.{{rp|175â€“190}}Two central propositions underlie general relativity.- The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."BOOK, Lorentz, H. A., Einstein, A., Minkowski, H., Weyl, H., The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, 1952, Dover Publications, 0486600815, {{rp|113}} This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.BOOK, Mook, Delo E., Vargish, Thoma s, Inside Relativity, 1987, Princeton University Press, Princeton, New Jersey, 0691084726, {{rp|137â€“149}}
- The equivalence principle states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration.
In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of g. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be Lilliputians), there are no experiments that A and B can perform which will enable them to tell which setting they are in.{{rp|141â€“149}}
An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, {{nowrap|1=F = GMmg /r2 = }} mgg and in Newton's second law, {{nowrap|1=F = m ia,}} there is no a priori reason why the gravitational mass mg should be equal to the inertial mass m i. The equivalence principle states that these two masses are identical.{{rp|141â€“149}}
To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations.Rather than this section attempting to offer a (yet another) relatively non-mathematical presentation about general relativity, the reader is referred to the featured Wikipedia articles Introduction to general relativity and General relativity.Instead, the focus in this section will be to explore a handful of elementary scenarios that serve to give somewhat of the flavor of general relativity.{{anchor|Curvature of time}}Curvature of time
(File:Einstein's argument suggesting gravitational redshift.svg|thumb|Figure 5-3. Einstein's argument suggesting gravitational redshift)In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame. In small enough regions of spacetime, local inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.{{rp|118â€“126}}Shortly after the publication of the general theory in 1916, a number of scientists pointed out that general relativity predicts the existence of gravitational redshift. Einstein himself suggested the following thought experiment: (i) Assume that a tower of height h (Fig. 5‑3) has been constructed. (ii) Drop a particle of rest mass m from the top of the tower. It falls freely with acceleration g, reaching the ground with velocity {{nowrap|1=v = (2gh)1/2}}, so that its total energy E, as measured by an observer on the ground, is {{nowrap|1=m + Â½mv2/c2 =}} {{nowrap|m + mgh/c2.}} (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon E{{'}} back into a particle of rest mass m{{'}}.{{rp|118â€“126}}It must be that {{nowrap|1=m = m{{'}}}}, since otherwise one would be able to construct a perpetual motion device. We therefore predict that {{nowrap|1=E{{'}} = m}}, so that
frac{E'}{E} = frac{h nu , '}{ h nu} = frac{m}{m + mgh/c^2} = 1 - frac{gh}{c^2}
A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by Pound & Rebka (1959) and later by Pound & Snider (1964).WEB, Mester, John, Experimental Tests of General Relativity,weblink Laboratoire Univers et ThÃ©ories, 9 June 2017,weblink 9 June 2017, yes, dmy-all, Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side-by-side with the ground clock.{{rp|16â€“18}} For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.Clocks in a gravitational field do not all run at the same rate. Experiments such as the Poundâ€“Rebka experiment have firmly established curvature of the time component of spacetime. The Poundâ€“Rebka experiment says nothing about curvature of the space component of spacetime. But note that the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence.{{rp|16}} This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the "Newtonian limit" (i.e. the particles are moving slowly, the gravitational field is weak, and the field is static), curvature of time alone is sufficient to derive Newton's law of gravity.ARXIV, Carroll, Sean M., Lecture Notes on General Relativity, 2 December 1997, gr-qc/9712019, {{rp|101â€“106}}Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:{{rp|229â€“232}}
Delta s^2 = left( 1 - frac{2GM}{c^2 r} right) (c Delta t)^2- , (Delta x)^2 - (Delta y)^2 - (Delta z)^2
{{anchor|Curvature of space}}Curvature of space
The (1 - 2GM/(c^2 r) ) coefficient in front of (c Delta t)^2 describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to G and M, and because of the r in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the (c Delta t)^2 term dwarfs the spatial terms.{{rp|234â€“238}}Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets.JOURNAL, Le Verrier, Urbain, Lettre de M. Le Verrier Ã M. Faye sur la thÃ©orie de Mercure et sur le mouvement du pÃ©rihÃ©lie de cette planÃ¨te, Comptes rendus hebdomadaires des sÃ©ances de l'AcadÃ©mie des Sciences, 1859, 49, 379â€“383,weblink The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.File:General relativity time and space distortion frame 1.png|thumb|Figure 5-4. General relativity is a theory of curved time and curved space. (:File:General relativity time and space distortion extract.gif|Click here to animate.) ]]As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.WEB, Worrall, Simon, The Hunt for Vulcan, the Planet That Wasnâ€™t There,weblink National Geographic, National Geographic, 12 June 2017, In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of Â±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.{{rp|234â€“238}}
Delta s^2 = left( 1 - frac{2GM}{c^2 r} right) (c Delta t)^2- , left( 1 + frac{2GM}{c^2 r} right) left[ (Delta x)^2 + (Delta y)^2 + (Delta z)^2 right]
In Newton's gravitation, the (1 - 2GM/(c^2 r) ) coefficient in front of (c Delta t)^2 predicts bending of light around a star. In general relativity, the (1 + 2GM/(c^2 r) ) coefficient in front of left[ (Delta x)^2 + (Delta y)^2 + (Delta z)^2 right] predicts a doubling of the total bending.{{rp|234â€“238}}The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.WEB, Levine, Alaina G., May 29, 1919: Eddington Observes Solar Eclipse to Test General Relativity,weblink APS News: This Month in Physics History, American Physical Society, 12 June 2017, {{anchor|Sources of spacetime curvature}}Sources of spacetime curvature
(File:StressEnergyTensor contravariant.svg|thumb|250px|Figure 5-5. Contravariant components of the stressâ€“energy tensor)In Newton's theory of gravitation, the only source of gravitational force is mass.In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations, the sources of gravity are presented on the right-hand side in T_{mu nu}, the stressâ€“energy tensor.Fig. 5‑5 classifies the various sources of gravity in the stressâ€“energy tensor:- T^{00} (red): The total mass-energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions.
- T^{0i} and T^{i0} (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum.
- T^{ij} are the rates of flow of the {{nowrap|1=i-component}} of momentum per unit area in the {{nowrap|1=j-direction}}. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the {{nowrap|1=i = j}} terms (green) represent isotropic pressure, and the {{nowrap|1=i â‰ j}} terms (blue) represent shear stresses.BOOK, Hobson, M. P., Efstathiou, G., Lasenby, A. N., General Relativity, 2006, Cambridge University Press, Cambridge, 9780521829519, 176â€“179,
Energy-momentum
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| footer = Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass-energy distributions with angular momentum J generate gravitomagnetic fields H.}}
In special relativity, mass-energy is closely connected to momentum. As we have discussed earlier in the section on Energy and momentum, just as space and time are different aspects of a more comprehensive entity called spacetime, mass-energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass-energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.BOOK, Thorne, Kip S., Fairbank, J. D., Deaver Jr., B. S., Everitt, W. F., Michelson, P. F., Near zero: New Frontiers of Physics, 1988, W. H. Freeman and Company, 573â€“586,weblinkweblink 30 June 2017, yes, dmy-all, (File:Special relativistic explanation of gravitomagnetism.svg|250px|thumb|Figure 5-7. Origin of gravitomagnetism)It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, {{nowrap|1=chapter 13â€“6}} of his Lectures on Physics, available online.BOOK, Feynman, R. P., Leighton, R. B., Sands, M., The Feynman Lectures on Physics, vol. 2, 1964, Basic Books, 9780465024162, 13â€“6 to 13â€“11, New Millenium,weblink 1 July 2017, ) Analogous logic can be used to demonstrate the origin of gravitomagnetism. In Fig. 5‑7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities âˆ’v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume {{nowrap|1=v| footer_align =
| footer = Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass-energy distributions with angular momentum J generate gravitomagnetic fields H.}}
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width1=280|height1=396| caption1 = Hendrik Lorentz | width2=371|height2=500| caption2 = Henri PoincarÃ© | width3=280|height3=396| caption3 = Albert Einstein | width4=813|height4=1093| caption4 = Hermann Minkowski | | footer = Figure 1-3.}}An important example is Henri PoincarÃ©,{{Citation|author=Darrigol, O. |title=The Genesis of the theory of relativity |year=2005 |journal=SÃ©minaire PoincarÃ©|volume=1|pages=1â€“22|url=http://www.bourbaphy.fr/darrigol2.pdf|format=PDF |doi=10.1007/3-7643-7436-5_1|isbn=978-3-7643-7435-8 |bibcode=2006eins.book....1D }}BOOK, Miller, Arthur I., Albert Einstein's Special Theory of Relativity, 1998, Springer-Verlag, New York, 0387948708, {{rp|73â€“80,93â€“95}} who in 1898 argued that the simultaneity of two events is a matter of convention.BOOK, Galison, Peter, Einstein's Clocks, PoincarÃ©'s Maps: Empires of Time, 2003, W. W. Norton & Company, Inc., New York, 0393020010, 13â€“47, {{refn|group=note|By stating that simultaneity is a matter of convention, PoincarÃ© meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher Henri Bergson, who argued that time, simultaneity, and duration were matters of intuitive understanding.}} In 1900, he recognized that Lorentz's "local time" is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization assuming constant light speed.{{refn|group=note|The operational procedure adopted by PoincarÃ© was essentially identical to what is known as Einstein synchronization, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.}} In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity, and in 1905/1906JOURNAL, Poincare, Henri, On the Dynamics of the Electron (Sur la dynamique de lâ€™Ã©lectron), Rendiconti del Circolo matematico di Palermo, 1906, 21, 129â€“176,weblink 15 July 2017, 10.1007/bf03013466, 1906RCMP...21..129P, he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity. While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional space-time by defining various four vectors, namely four-position, four-velocity, and four-force.{{Citation|author=Zahar, Elie|year=1989|orig-year=1983|title=Einstein's Revolution: A Study in Heuristic|chapter=PoincarÃ©'s Independent Discovery of the relativity principle |publisher=Open Court Publishing Company|location=Chicago|isbn=0-8126-9067-2}} He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world".BOOK, Walter, Scott A., Renn, JÃ¼rgen, Schemmel, Matthias, The Genesis of General Relativity, Volume 3, Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905â€“1910, 2007, Springer, Berlin, 193â€“252,weblink 15 July 2017,weblink 15 July 2017, yes, dmy-all, Furthermore, even as late as 1909, PoincarÃ© continued to believe in the dynamical interpretation of the Lorentz transform.{{rp|163â€“174}} For these and other reasons, most historians of science argue that PoincarÃ© did not invent what is now called special relativity.In 1905, Einstein introduced special relativity (even though without using the techniques of the spacetime formalism) in its modern understanding as a theory of space and time. While his results are mathematically equivalent to those of Lorentz and PoincarÃ©, it was Einstein who showed that the Lorentz transformations are not the result of interactions between matter and aether, but rather concern the nature of space and time itself. He obtained all of his results by recognizing that the entire theory can be built upon two postulates: The principle of relativity and the principle of the constancy of light speed.Einstein performed his analyses in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His seminal work introducing the subject was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.JOURNAL, Einstein, Albert, On the Electrodynamics of Moving Bodies ( Zur Elektrodynamik bewegter KÃ¶rper), Annalen der Physik, 1905, 322, 10, 891â€“921,weblink 7 April 2018, 1905AnP...322..891E, 10.1002/andp.19053221004, {{refn|group=note|A hallmark of Einstein's career, in fact, was his use of visualized thought experiments (Gedankenâ€“Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great Solvay Debates with Bohr on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the Heisenberg uncertainty principle might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement. BOOK, Isaacson, Walter, Einstein: His Life and Universe, 2007, Simon & Schuster, 978-0-7432-6473-0, {{rp|26â€“27;122â€“127;145â€“146;345â€“349;448â€“460}} }}In addition, Einstein in 1905 superseded previous attempts of an electromagnetic mass-energy relation by introducing the general equivalence of mass and energy, which was instrumental for his subsequent formulation of the equivalence principle in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass-energy equivalence, Einstein showed, in addition, that the gravitational mass of a body is proportional to its energy content, which was one of early results in developing general relativity. While it would appear that he did not at first think geometrically about spacetime,BOOK, Schutz, Bernard, Gravity from the Ground Up: An Introductory Guide to Gravity and General Relativity, 2004, Cambridge University Press, Cambridge, 0521455065, Reprint,weblink 24 May 2017, en, {{rp|219}} in the further development of general relativity Einstein fully incorporated the spacetime formalism.When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski, had also arrived at most of the basic elements of special relativity. Max Born recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:ARXIV, Weinstein, Galina, Max Born, Albert Einstein and Hermann Minkowski's Space-Time Formalism of Special Relativity, 1210.6929, physics.hist-ph, 2012, {{cquote|I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. [â€¦] He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery.}}Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, PoincarÃ© et al. However, it is not at all clear when Minkowski began to formulate the geometric formulation of special relativity that was to bear his name, or to which extent he was influenced by PoincarÃ©'s four-dimensional interpretation of the Lorentz transformation. Nor is it clear if he ever fully appreciated Einstein's critical contribution to the understanding of the Lorentz transformations, thinking of Einstein's work as being an extension of Lorentz's work.JOURNAL, Galison, Peter Louis, Minkowski's space-time: From visual thinking to the absolute world, Historical Studies in the Physical Sciences, 1979, 10, 85â€“121, 10.2307/27757388, 27757388, (File:Minkowski Diagram from 1908 'Raum und Zeit' lecture.jpg|thumb|330px|Figure 1-4. Hand-colored transparency presented by Minkowski in his 1908 Raum und Zeit lecture)On November 5, 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the GÃ¶ttingen Mathematical society with the title, The Relativity Principle (Das RelativitÃ¤tsprinzip).{{refn|group=note|In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the Annals of Physics (Annalen der Physik) was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.}} On September 21, 1908, Minkowski presented his famous talk, Space and Time (Raum und Zeit),JOURNAL, Minkowski, Hermann, Raum und Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung, 1909, 1â€“14,weblink Space and Time, B.G. Teubner, to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's famous statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." Space and Time included the first public presentation of spacetime diagrams (Fig. 1‑4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.{{refn|group=note|(In the following, the group Gâˆž is the Galilean group and the group Gc the Lorentz group.) "With respect to this it is clear that the group Gc in the limit for {{nowrap|1=c = ∞}}, i.e. as group Gâˆž, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since Gc is mathematically more intelligible than Gâˆž, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group Gâˆž, but rather for a group Gc, where c is definitely finite, and only exceedingly large using the ordinary measuring units."}}The spacetime concept and the Lorentz group are closely connected to certain types of sphere, hyperbolic, or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used.{{refn|group=note|For instance, the Lorentz group is a subgroup of the conformal group in four dimensions.JOURNAL, Cartan, Ã‰.; Fano, G., 1955, 1915, EncyclopÃ©die des sciences mathÃ©matiques pures et appliquÃ©es, 3.1, La thÃ©orie des groupes continus et la gÃ©omÃ©trie, 39â€“43,weblink (Only pages 1â€“21 were published in 1915, the entire article including pp. 39â€“43 concerning the groups of Laguerre and Lorentz was posthumously published in 1955 in Cartan's collected papers, and was reprinted in the EncyclopÃ©die in 1991.){{rp|41â€“42}}The Lorentz group is isomorphic to the Laguerre group transforming planes into planes,{{rp|39â€“42}}it is isomorphic to the MÃ¶bius group of the plane,JOURNAL, Kastrup, H. A., On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics, Annalen der Physik, 520, 9â€“10, 2008, 631â€“690, 0808.2730, 10.1002/andp.200810324, 2008AnP...520..631K, {{rp|22}}and is isomorphic to the group of isometries in hyperbolic space which is often expressed in terms of the hyperboloid model.BOOK, Ratcliffe, J. G., 1994, Foundations of Hyperbolic Manifolds, Hyperbolic geometry, 56â€“104, New York, 038794348X, {{rp|3.2.3}} }}Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as Ã¼berflÃ¼ssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital, and in 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.{{rp|151â€“152}} Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as Minkowski spacetime.Spacetime in special relativity{{anchor|Spacetime interval}}Spacetime intervalIn three-dimensions, the distance Delta{d} between two points can be defined using the Pythagorean theorem:
{left(Delta{d}right)}^2 = {left(Delta{x}right)}^2 + {left(Delta{y}right)}^2 + {left(Delta{z}right)}^2
Although two viewers may measure the x,y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both (assuming that they are measuring using the same units). The distance is "invariant".In special relativity, however, the distance between two points is no longer the same if measured by two different observers when one of the observers is moving, because of Lorentz contraction. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because (from their point of view) they are stationary, and the position of the event is receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events.In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure time and distance carefully will find the same spacetime interval between any two events. Suppose an observer measures two events as being separated in time by Delta t and a spatial distance Delta x. Then the spacetime interval {left(Delta{s}right)}^2 between the two events that are separated by a distance Delta{x} in space and by Delta{ct}= cDelta t in the ct-coordinate is:
(Delta s)^2 = (Delta ct)^2 - (Delta x)^2 , or for three space dimensions, (Delta s)^2 = (Delta ct)^2 - (Delta x)^2 - (Delta y)^2 - (Delta z)^2BOOK, Differential Manifolds and Theoretical Physics, W. D., Curtis, F. R., Miller, Academic Press, 1985, 978-0-08-087435-7, 223,weblink Extract of page 223
The constant textrm{c}, the speed of light, converts the units used to measure time (seconds) into units used to measure distance (meters).Note on nomenclature: Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, x means Delta{x}, etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.(File:Spacetime Diagram of Two Photons and a Slower than Light Object.png|thumb|Figure 2-1. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C) The equation above is similar to the Pythagorean theorem, except with a minus sign between the (textrm{c} , t)^2 and the x^2 terms. Note also that the spacetime interval is the quantity s^2, not s itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard s^2 as a distinct symbol in itself, rather than the square of something.{{rp|217}}Because of the minus sign, the spacetime interval between two distinct events can be zero. If s^2 is positive, the spacetime interval is timelike, meaning that two events are separated by more time than space. If s^2 is negative, the spacetime interval is spacelike, meaning that two events are separated by more space than time. Spacetime intervals are zero when x = pm textrm{c} , t. In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed lightlike or null. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage.A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2‑1 presents a spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by textrm{c} so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of Â±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.Note on nomenclature: There are two sign conventions in use in the relativity literature:
s^2 = (textrm{c} t)^2 - x^2 - y^2 - z^2
and
s^2 = - (textrm{c} t)^2 + x^2 + y^2 + z^2
These sign conventions are associated with the metric signatures {{nowrap|(+ âˆ’ âˆ’ âˆ’)}} and {{nowrap|(âˆ’ + + +).}} A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.{{anchor|Reference frames}}Reference frames(File:Standard configuration of coordinate systems.svg|thumb|Figure 2-2. Galilean diagram of two frames of reference in standard configuration)(File:Galilean and Spacetime coordinate transformations.png|thumb|330px|Figure 2-3. (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected light pulse)To gain insight in how spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration. With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2‑2, two Galilean reference frames (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame Sâ€² (pronounced "S prime") belongs to a second observer Oâ€².
tan theta = v/c .
The xâ€² axis is also tilted with respect to the x axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always Â±1. Fig. 2‑3c presents a spacetime diagram from the viewpoint of observer Oâ€². Event P represents the emission of a light pulse at xâ€² = 0, ctâ€² = âˆ’a. The pulse is reflected from a mirror situated a distance a from the light source (event Q), and returns to the light source at xâ€² = 0, ctâ€² = a (event R).The same events P, Q, R are plotted in Fig. 2‑3b in the frame of observer O. The light paths have slopes = 1 and âˆ’1 so that â–³PQR forms a right triangle. Since OP = OQ = OR, the angle between xâ€² and x must also be Î¸.{{rp|113â€“118}}While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a Cartesian plane, and should be considered no stranger than the manner in which, on a Mercator projection of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.{{anchor|Light cone}}Light cone{{anchor|Figure 2-4}}(File:ModernPhysicsSpaceTimeA.png|thumb|Figure 2-4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere")In Fig. 2-4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the light cone of the event O, since adding a second spatial dimension (Fig. 2‑5) makes the appearance that of two right circular cones meeting with their apices at O. One cone extends into the future (t>0), the other into the past (t 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the ctâ€²-axis) in the event labelled A, where "two clocks are simultaneously at one place". In the lower picture the place for W'2 is taken to be Cxâ€² |