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Special relativity
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{{short description|Theory of interwoven space and time by Albert Einstein}}{{for|history and motivation|History of special relativity}}{{Special relativity sidebar}}In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original pedagogical treatment, it is based on two postulates: - the content below is remote from Wikipedia
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- the laws of physics are invariant (i.e. identical) in all inertial systems (i.e. non-accelerating frames of reference); and
- the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
{{anchor|Postulates}} Traditional "two postulates" approach to special relativity
{{rquote|right|Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results... How, then, could such a universal principle be found?|Albert Einstein: Autobiographical NotesEinstein, Autobiographical Notes, 1949.}}Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in a vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:- The Principle of Relativity â€“ the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.
- The Principle of Invariant Light Speed â€“ "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface). That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.
Principle of relativity
Reference frames and relative motion
File:Frames of reference in relative motion.svg|thumb|right|300px|Figure 2-1. The primed system is in motion relative to the unprimed system with constant velocity v only along the x-axis, from the perspective of an observer stationary in the unprimed system. By the principle of relativityprinciple of relativityReference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space which is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).An event is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in spacetime. Since the speed of light is constant in relativity irrespective of reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called transformation equations.Standard configuration
To gain insight in how the 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.BOOK, A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity, Collier, Peter, Incomprehensible Books, 2017, 9780957389465, 3rd, {{rp|107}} With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2‑1, 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" or "S dash") belongs to a second observer Oâ€².- The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame Sâ€².
- Frame Sâ€² moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.
- The origins of frames S and Sâ€² are coincident when time t = 0 for frame S and tâ€² = 0 for frame Sâ€².
Lack of an absolute reference frame
The principle of relativity, which states that physical laws have the same form in each inertial reference frame, dates back to Galileo, and was incorporated into Newtonian physics. However, in the late 19th century, the existence of electromagnetic waves led physicists to suggest that the universe was filled with a substance that they called "aether", which, they postulated, would act as the medium through which these waves, or vibrations, propagated. The aether was thought to be an absolute reference frame against which all speeds could be measured, and could be considered fixed and motionless. Aether supposedly possessed some wonderful properties: it was sufficiently elastic to support electromagnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the Michelsonâ€“Morley experiment in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist.Staley, Richard (2009), "Albert Michelson, the Velocity of Light, and the Ether Drift", Einstein's generation. The origins of the relativity revolution, Chicago: University of Chicago Press, {{isbn|0-226-77057-5}} Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.Relativity without the second postulate
From the principle of relativity alone without assuming the constancy of the speed of light (i.e. using the isotropy of space and the symmetry implied by the principle of special relativity) it can be shown that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.BOOK, Yaakov Friedman, Physical Applications of Homogeneous Balls, Progress in Mathematical Physics, 40, 2004, 1â€“21, 978-0-8176-3339-4, David Morin (2007) Introduction to Classical Mechanics, Cambridge University Press, Cambridge, chapter 11, Appendix I, {{isbn|1-139-46837-5}}.{{anchor|Lorentz transformation}} Lorentz invariance as the essential core of special relativity
Alternative approaches to special relativity
Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.Das, A. (1993) The Special Theory of Relativity, A Mathematical Exposition, Springer, {{isbn|0-387-94042-1}}.Schutz, J. (1997) Independent Axioms for Minkowski Spacetime, Addison Wesley Longman Limited, {{isbn|0-582-31760-6}}.Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations.JOURNAL, Miller, D. J., A constructive approach to the special theory of relativity, American Journal of Physics, 78, 6, 633â€“638, 0907.0902, 10.1119/1.3298908, 2010, 2010AmJPh..78..633M, Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and WheelerBOOK, Taylor, Edwin, Wheeler, John Archibald, Spacetime Physics, 1992, W.H. Freeman & Co., 978-0-7167-2327-1, 2nd, and by Callahan.BOOK, Callahan, James J., The Geometry of Spacetime: An Introduction to Special and General Relativity, 2011, Springer, New York, 9781441931429, This is also the approach followed by the Wikipedia articles Spacetime and Minkowski diagram.Lorentz transformation and its inverse
Define an event to have spacetime coordinates {{nowrap|(t,x,y,z)}} in system S and {{nowrap|(tâ€²,xâ€²,yâ€²,zâ€²)}} in a reference frame moving at a velocity v with respect to that frame, Sâ€². Then the Lorentz transformation specifies that these coordinates are related in the following way:
begin{align}
t' &= gamma (t - vx/c^2) x' &= gamma (x - v t) y' &= y z' &= z ,end{align}where
gamma = frac{1}{sqrt{1 - frac{v^2}{c^2}}}
is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of Sâ€² is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.Solving the above four transformation equations for the unprimed coordinates yields the inverse Lorentz transformation:
begin{align}
t &= gamma ( t' + v x'/c^2)
x &= gamma ( x' + v t')
y &= y'
z &= z'.
end{align}Enforcing this inverse Lorentz transformation to coincide with the Lorentz transformation from the primed to the unprimed system, shows the unprimed frame as moving with the velocity vâ€² = âˆ’v, as measured in the primed frame.There is nothing special about the x-axis. The transformation can apply to the y- or z-axis, or indeed in any direction, which can be done by directions parallel to the motion (which are warped by the Î³ factor) and perpendicular; see the article Lorentz transformation for details.A quantity invariant under Lorentz transformations is known as a Lorentz scalar.Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates {{nowrap|(x1, t1)}} and {{nowrap|(xâ€²1, tâ€²1)}}, another event has coordinates {{nowrap|(x2, t2) }}and {{nowrap|(xâ€²2, tâ€²2)}}, and the differences are defined as
x &= gamma ( x' + v t')
y &= y'
z &= z'.
{{EquationRef|1|Eq. 1:}} Delta x' = x'_2-x'_1 , Delta t' = t'_2-t'_1 .
{{EquationRef|2|Eq. 2:}} Delta x = x_2-x_1 , Delta t = t_2-t_1 .
we get
{{EquationRef|3|Eq. 3:}} Delta x' = gamma (Delta x - v ,Delta t) , Delta t' = gamma left(Delta t - v Delta x / c^{2} right) .
{{EquationRef|4|Eq. 4:}} Delta x = gamma (Delta x' + v ,Delta t') , Delta t = gamma left(Delta t' + v Delta x' / c^{2} right) .
If we take differentials instead of taking differences, we get
{{EquationRef|5|Eq. 5:}} dx' = gamma (dx - v , dt) , dt' = gamma left( dt - v dx / c^{2} right) .
{{EquationRef|6|Eq. 6:}} dx = gamma (dx' + v ,dt') , dt = gamma left(dt' + v dx' / c^{2} right) .
Graphical representation of the Lorentz transformation
{{multiple image|perrow = 2|total_width=400width1=535|height1=535width2=535|height2=535width3=535|height3=535width4=535|height4=535| footer = Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.}}Spacetime diagrams (Minkowski diagrams) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2‑1.BOOK, Mermin, N. David, Space and Time in Special Relativity, 1968, McGraw-Hill, 978-0881334203, {{rp|155â€“199}} Fig. 3‑1a. Draw the x and t axes of frame S. The x axis is horizontal and the t (actually ct) axis is vertical, which is the opposite of the usual convention in kinematics. The ct axis is scaled by a factor of c so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45Â° diagonal lines represent the worldlines of two photons passing through the origin at time t = 0. The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, text{A} and text{B}, have been plotted on this graph so that their coordinates may be compared in the S and S' frames. Fig. 3‑1b. Draw the x' and ct' axes of frame S'. The ct' axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, v = c/2. Both the ct' and x' axes are tilted from the unprimed axes by an angle alpha = tan^{-1}(beta), where beta = v/c. The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that t=0 when t'=0.Fig. 3‑1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that (x', ct') coordinates of (0, 1) in the primed coordinate system transform to (beta gamma, gamma) in the unprimed coordinate system. Likewise, (x', ct') coordinates of (1, 0) in the primed coordinate system transform to (gamma, beta gamma) in the unprimed system. Draw gridlines parallel with the ct' axis through points (k gamma, k beta gamma) as measured in the unprimed frame, where k is an integer. Likewise, draw gridlines parallel with the x' axis through (k beta gamma, k gamma) as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between ct' units equals sqrt{(1 + beta ^2)/(1 - beta ^2)} times the spacing between ct units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as beta rightarrow 1.Fig. 3‑1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time t' = 0 still plot as 45Â° diagonal lines. The primed coordinates of text{A} and text{B} are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space.While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane. By analogy, planar maps of the world are unavoidably distorted, but with experience and intuition, one learns to mentally account for these distortions.Consequences derived from the Lorentz transformation
{{See also|Twin paradox|Relativistic mechanics}}The consequences of special relativity can be derived from the Lorentz transformation equations.BOOK, Introduction to special relativity, Robert Resnick, Wiley, 1968, 62â€“63,weblink These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.Invariant interval
In Galilean relativity, length (Delta r){{refn|group=note|In a spacetime setting, the length of a rigid object is the spatial distance between the ends of the object measured at the same time.}} and temporal separation between two events (Delta t) are independent invariants, the values of which do not change when observed from different frames of reference.{{refn|group=note|The results of the Michelsonâ€“Morley experiment led George Francis FitzGerald and Hendrik Lorentz independently to propose the phenomenon of length contraction. Lorentz believed that length contraction represented a physical contraction of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.BOOK, Miller, Arthur I., Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905-1911), 1998, Springer-Verlag, Mew York, 978-0-387-94870-6, {{rp|62â€“68}} Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotationJOURNAL, Lorentz, H.A., The rotation of the plane of polarization in moving media, Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW), 1902, 4, 669â€“678,weblink 15 November 2018, 1901KNAB....4..669L, and induction of double refraction,JOURNAL, Lorentz, H. A., Electromagnetic phenomena in a system moving with any velocity smaller than that of light, Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW), 1904, 6, 809â€“831,weblink 15 November 2018, 1903KNAB....6..809L, and the induction of torques on charged condensers moving at an angle with respect to the aether. Lorentz was perplexed by experiments such as the Troutonâ€“Noble experiment and the experiments of Rayleigh and Brace which failed to validate his theoretical expectations.}}{{refn|group=note|For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation t'=t-vx/c^2.BOOK, Lorentz, Hendrik, Attempt at a Theory of Electrical and Optical Phenomena in Moving Bodies (Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten KÃ¶rpern), 1895, Investigation of oscillations excited by oscillating ions,weblink (subsection Â§ 31), E. J. Brill, Leiden,weblink Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.BOOK, Bernstein, Jeremy, Secrets of the Old One: Einstein, 1905, 2006, Copernicus Books (imprint of Springer Science + Business Media), 978-0387-26005-1, {{rp|51,80}} Impressed by Lorentz's "most ingenious idea", PoincarÃ© saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, PoincarÃ© did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether. PoincarÃ© made no attempt to redefine the concepts of space and time. To PoincarÃ©, Lorentz transformation described the apparent states of the field for a moving observer. True states remained those defined with respect to the ether.JOURNAL, Darrigol, Olivier, The Genesis of the Theory of Relativity, SÃ©minaire PoincarÃ©, 2005, 1, 1â€“22,weblink 15 November 2018, 2006eins.book....1D, }}In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as Delta s^2:
Delta s^2 ; overset{def}{=} ; c^2 Delta t^2 - (Delta x^2 + Delta y^2 + Delta z^2) This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of Delta s^2 is also an invariant, generated by a variant of the metric signature of spacetime.
The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames. The form of Delta s^2 , being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances.{{refn| group=note|The invariance of Î”s2 under standard Lorentz transformation in analogous to the invariance of squared distances Î”r2 under rotations in Euclidean space. Although space and time have an equal footing in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.}} The invariance of this interval is a property of the general Lorentz transform (also called the PoincarÃ© transformation), making it an isometry of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, i.e. Lorentz boosts, in the x-direction) with all other translations, reflections, and rotations between any Cartesian inertial frame.BOOK, Rindler, Wolfgang, Essential Relativity, 1977, Springer-Verlag, New York, 978-0-387-10090-6, 2nd, {{rp|33â€“34}} In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:
Delta s^2 , = , c^2 Delta t^2 - Delta x^2
Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:
c^2 Delta t^2 - Delta x^2 = c^2 gamma ^2 left(Delta t' + dfrac{v Delta x'}{c^2} right)^2 - gamma ^2 (Delta x' + v Delta t')^2
= gamma ^2 left( c^2 Delta t' ^ {, 2} + 2 v Delta x' Delta t' + dfrac{v^2 Delta x' ^ {, 2}}{c^2} right) - gamma ^2 (Delta x' ^ {, 2} + 2 v Delta x' Delta t' + v^2 Delta t' ^ {, 2})
= gamma ^2 c^2 Delta t' ^ {, 2} - gamma ^2 v^2 Delta t' ^{, 2} - gamma ^2 Delta x' ^ {, 2} + gamma ^2 dfrac{v^2 Delta x' ^ {, 2}}{c^2} = gamma ^2 c^2 Delta t' ^ {, 2} left( 1 - dfrac{v^2}{c^2} right) - gamma ^2 Delta x' ^{, 2} left( 1 - dfrac{v^2}{c^2} right)
= c^2 Delta t' ^{, 2} - Delta x' ^{, 2}
The value of Delta s^2 is hence independent of the frame in which it is measured. In considering the physical significance of Delta s^2, there are three cases to note:{{rp|25â€“39}} - Î”s2 > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that | Delta x / Delta t |
- Î”s2 < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that | Delta x / Delta t | > c , and given the Lorentz transformation Delta t' = gamma (Delta t - v Delta x / c^2) , there exists a v less than c for which Delta t' = 0 (in particular, v = c^2 Delta t / Delta x). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, sqrt { - Delta s^2 }, is called the proper distance, or proper length. For values of v greater than and less than c^2 Delta t / Delta x , the sign of Delta t' changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. The temporal order of timelike-separated events, however, is absolute, since the only way that v could be greater than c^2 Delta t / Delta x would be if v > c .
- Î”s2 = 0: In this case, the two events are said to be lightlike separated. This implies that | Delta x / Delta t | = c , and this relationship is frame independent due to the invariance of s^2 . From this, we observe that the speed of light is c in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.
Relativity of simultaneity
{{See also|Relativity of simultaneity|Ladder paradox}}(File:Relativity of Simultaneity Animation.gif|thumb|Figure 4-1. The three events (A, B, C) are simultaneous in the reference frame of some observer O. In a reference frame moving at v = 0.3c, as measured by O, the events occur in the order C, B, A. In a reference frame moving at v = -0.5c with respect to O, the events occur in the order A, B, C. The white lines, the lines of simultaneity, move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on it. They are the locus of all events occurring at the same time in the respective frame. The gray area is the light cone with respect to the origin of all considered frames.)Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).From {{EquationNote|3|Equation 3}} (the forward Lorentz transformation in terms of coordinate differences)
Delta t' = gamma left(Delta t - frac{v ,Delta x}{c^{2}} right)
It is clear that the two events that are simultaneous in frame S (satisfying {{nowrap|1=Î”t = 0}}), are not necessarily simultaneous in another inertial frame Sâ€² (satisfying {{nowrap|1=Î”tâ€² = 0}}). Only if these events are additionally co-local in frame S (satisfying {{nowrap|1=Î”x = 0}}), will they be simultaneous in another frame Sâ€².The Sagnac effect can be considered a manifestation of the relativity of simultaneity.JOURNAL, Ashby, Neil, Relativity in the Global Positioning System, Living Reviews in Relativity, 6, 1, 1, 10.12942/lrr-2003-1, 28163638, 5253894, 2003, 2003LRR.....6....1A, Since relativity of simultaneity is a first order effect in v, instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber optic gyroscopes, are capable of extreme levels of sensitivity.JOURNAL, Lin, Shih-Chun, Giallorenzi, Thomas G., Sensitivity analysis of the Sagnac-effect optical-fiber ring interferometer, Applied Optics, 1979, 18, 6, 915â€“931, 10.1364/AO.18.000915, 20208844, 1979ApOpt..18..915L, Time dilation
{{See also|Time dilation}}The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that the non-traveling twin sibling has aged much more, the paradox being that at constant velocity we are unable to discern which twin is non-traveling and which twin travels). Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by {{nowrap|1=Î”x = 0}}. To find the relation between the times between these ticks as measured in both systems, {{EquationNote|3|Equation 3}} can be used to find:
Delta t' = gamma, Delta t for events satisfying Delta x = 0 .
This shows that the time (Î”tâ€²) between the two ticks as seen in the frame in which the clock is moving (Sâ€²), is longer than the time (Î”t) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed muons created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.BOOK, Daniel Kleppner, David Kolenkow, yes, An Introduction to Mechanics, 1973, 468â€“70, 978-0-07-035048-9, Length contraction
{{See also|Lorentz contraction}}The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Î”x. To measure the length of this rod in the system Sâ€², in which the rod is moving, the distances xâ€² to the end points of the rod must be measured simultaneously in that system Sâ€². In other words, the measurement is characterized by {{nowrap|1=Î”tâ€² = 0}}, which can be combined with {{EquationNote|3|Equation 3}} to find the relation between the lengths Î”x and Î”xâ€²:
Delta x' = frac{Delta x}{gamma} for events satisfying Delta t' = 0 .
This shows that the length (Î”xâ€²) of the rod as measured in the frame in which it is moving (Sâ€²), is shorter than its length (Î”x) in its own rest frame (S).Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system.Lorentz transformation of velocities
{{See also|Velocity-addition formula}}Consider two frames S and Sâ€² in standard configuration. A particle in S moves in the x direction with velocity vector mathbf{u}. What is its velocity mathbf{u'} in frame Sâ€² ? We can write
{{EquationRef|7|Eq. 7:}} mathbf{|u|} = u = dx / dt .
{{EquationRef|8|Eq. 8:}} mathbf{|u'|} = u' = dx' / dt' .
Substituting expressions for dx' and dt' from {{EquationNote|5|Equation 5}} into {{EquationNote|8|Equation 8,}} followed by straightforward mathematical manipulations and back-substitution from {{EquationNote|7|Equation 7}} yields the Lorentz transformation of the speed u to u':
{{EquationRef|9|Eq. 9:}} u'=frac{dx'}{dt'}=frac{gamma(dx-v dt)}{gamma left (dt-frac{v dx}{c^2} right )}= frac{frac{dx}{dt}-v}{1-left ( frac{v}{c^2} right )left ( frac{dx}{dt} right ) } =frac{u-v}{1- uv / c^2}.
The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing v with -v .
{{EquationRef|10|Eq. 10:}} u=frac{u'+v}{1+ u'v / c^2}.
For mathbf{u} not aligned along the x-axis, we write:{{rp|47-49}}
{{EquationRef|11|Eq. 11:}} mathbf{u} = (u_1, u_2, u_3 ) = ( dx / dt, dy/dt, dz/dt) .
{{EquationRef|12|Eq. 12:}} mathbf{u'} = (u_1', u_2', u_3') = ( dx' / dt', dy'/dt', dz'/dt') .
The forward and inverse transformations for this case are:
{{EquationRef|13|Eq. 13:}} u_1'=frac{u_1 -v}{1-u_1 v / c^2 } , u_2'=frac{u_2}{gamma left( 1-u_1 v / c^2 right) } , u_3'=frac{u_3}{gamma left( 1- u_1 v / c^2 right) } .
{{EquationRef|14|Eq. 14:}} u_1=frac{u_1' +v}{1+ u_1' v / c^2 } , u_2=frac{u_2'}{ gamma left( 1+ u_1' v / c^2 right) } , u_3=frac{u_3'}{gamma left( 1+ u_1' v / c^2 right)} .
{{EquationNote|10|Equation 10}} and {{EquationNote|14|Equation 14}} can be interpreted as giving the resultant mathbf{u} of the two velocities mathbf{v} and mathbf{u'}, and they replace the formula mathbf{u = u' + v} which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and Sâ€² being aligned with each other (although not necessarily in standard configuration).{{rp|47-49}}We note the following points: - If an object (for example, a photon) were moving at the speed of light in one frame {{nowrap|(i.e. u {{=}} Â±c}} {{nowrap|or uâ€² {{=}} Â±c),}} then it would also be moving at the speed of light in any other frame, moving at {{nowrap|{{abs|v}} < c}}.
- The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c.
- If both |u| and |v| (and then also |uâ€²| and |vâ€²|) are small with respect to the speed of light (that is, {{nowrap|e.g., {{abs|{{sfrac|u|c}}}} â‰ª {{math|1}}),}} then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
- Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.
Thomas rotation
{{See also|Thomas rotation}}{{multiple image
| direction = vertical
| width = 220
| footer = Figure 4-2. Thomas-Wigner rotation
| image1 = Thomas-Wigner Rotation 1.svg
| image2 = Thomas-Wigner Rotation 2.svg
}}The composition of two non-collinear Lorentz boosts (i.e. two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. Thomas rotation results from the relativity of simultaneity. In Fig. 4‑2a, a rod of length L in its rest frame (i.e. having a proper length of L) rises vertically along the y‑axis in the ground frame. In Fig. 4‑2b, the same rod is observed from the frame of a rocket moving at speed v to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized in the frame of the rod, relativity of simultaneity causes the observer in the rocket frame to observe (not see) the clock at the right end of the rod as being advanced in time by Lv/c^2 , and the rod is correspondingly observed as tilted.BOOK, Taylor, Edwin F., Wheeler, John Archibald, Spacetime Physics, 1966, W. H. Freeman and Company, San Francisco, 1st, {{rp|98â€“99}}Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the spin of moving particles, where Thomas precession is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope, relating the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.{{rp|169â€“174}}Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".JOURNAL, Shaw, R., Length Contraction Paradox, American Journal of Physics, 1962, 30, 1, 72, 10.1119/1.1941907, 1962AmJPh..30...72S, {{rp|98â€“99}}| width = 220
| footer = Figure 4-2. Thomas-Wigner rotation
| image1 = Thomas-Wigner Rotation 1.svg
| image2 = Thomas-Wigner Rotation 2.svg
Causality and prohibition of motion faster than light
{{See also|Causality (physics)|Tachyonic antitelephone}}(File:Simple light cone diagram.svg|thumb|Figure 4-3. Light cone)In Fig. 4‑3, the time interval between the events A (the "cause") and B (the "effect") is 'time-like'; i.e., there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect).The interval AC in the diagram is 'space-like'; i.e., there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. However, there are no frames accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result.For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).BOOK, Richard C., Tolman, The Theory of the Relativity of Motion, Berkeley, University of California Press, 1917, 54,weblink JOURNAL, G. A. Benford, D. L. Book, W. A. Newcomb, yes, 10.1103/PhysRevD.2.263, The Tachyonic Antitelephone, 1970, Physical Review D, 2, 2, 263, 1970PhRvD...2..263B, A variety of causal paradoxes could then be constructed. {{multiple image
| width = 160
| image_gap = 4
| image1 = Causality violation 1.svg
| image2 = Causality violation 2.svg
| alt2 = Three small white and yellow flowers before green-leaf background
| footer_align = center
| footer = Figure 4-4. Causality violation by the use of fictitious"instantaneous communicators"
}}Consider the spacetime diagrams in Fig. 4‑4. A and B stand alongside a railroad track, when a high speed train passes by, with C riding in the last car of the train and D riding in the leading car. The world lines of A and B are vertical (ct), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (ctâ€²), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground. | image_gap = 4
| image1 = Causality violation 1.svg
| image2 = Causality violation 2.svg
| alt2 = Three small white and yellow flowers before green-leaf background
| footer_align = center
| footer = Figure 4-4. Causality violation by the use of fictitious"instantaneous communicators"
- Fig. 4&8209;4a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the -x' axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives earlier than it was sent.
- Fig. 4&8209;4b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the +x axis, which is a line of simultaneity for the frames of A and B.
Optical effects
Dragging effects
(File:Fizeau experiment schematic.svg|thumb|300px|Figure 5-1. Highly simplified diagram of Fizeau's 1851 experiment.)In 1850, Hippolyte Fizeau and LÃ©on Foucault independently established that light travels more slowly in water than in air, thus validating a prediction of Fresnel's wave theory of light and invalidating the corresponding prediction of Newton's corpuscular theory.JOURNAL, Lauginie, P., Measuring Speed of Light: Why? Speed of what?, Proceedings of the Fifth International Conference for History of Science in Science Education, 2004,weblink 3 July 2015,weblink" title="web.archive.org/web/20150704043700weblink">weblink 4 July 2015, The speed of light was measured in still water. What would be the speed of light in flowing water?In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5‑1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused displacement of the fringes, showing that the motion of the water had affected the speed of the light.According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed through the medium plus the speed of the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If u' = c/n is the speed of light in still water, and v is the speed of the water, and u_{pm} is the water-bourne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then
u_{pm} =frac{c}{n} pm vleft(1-frac{1}{n^2}right) .
Fizeau's results, although consistent with Fresnel's earlier hypothesis of partial aether dragging, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since n depends on wavelength, the aether must be capable of sustaining different motions at the same time.{{refn|group=note|The refractive index dependence of the presumed partial aether-drag was eventually confirmed by Pieter Zeeman in 1914â€“1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to Amsterdam's main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Ã…) through red (6870 Ã…).JOURNAL, Zeeman, Pieter, Fresnel's coefficient for light of different colours. (First part), Proc. Kon. Acad. Van Weten., 17, 1914, 445â€“451,weblink 1914KNAB...17..445Z, JOURNAL, Zeeman, Pieter, Fresnel's coefficient for light of different colours. (Second part), Proc. Kon. Acad. Van Weten., 18, 1915, 398â€“408,weblink 1915KNAB...18..398Z, }} A variety of theoretical explanations were proposed to explain Fresnel's dragging coefficient that were completely at odds with each other. Even before the Michelsonâ€“Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies.BOOK, Stachel, J., The universe of general relativity, 2005, BirkhÃ¤user, Boston, 978-0-8176-4380-5, 1â€“13,weblink Kox, A.J., Eisenstaedt, J, 17 April 2012, Fresnel's (dragging) coefficient as a challenge to 19th century optics of moving bodies, From the point of view of special relativity, Fizeau's result is nothing but an approximation to {{EquationNote|10|Equation 10}}, the relativistic formula for composition of velocities.
u_{pm} = frac{u' pm v}{ 1 pm u'v/c^2 } = frac {c/n pm v}{ 1 pm v/cn } approx c left( frac{1}{n} pm frac{v}{c} right) left( 1 mp frac{v}{cn} right) approx frac{c}{n} pm v left( 1 - frac{1}{n^2} right)
Relativistic aberration of light
(File:Stellar aberration illustration.svg|thumb|Figure 5-2. Illustration of stellar aberration)Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the aberration of light. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light.BOOK, Basic Relativity, 8,weblink 978-0-387-95210-9, 2001, Springer, Richard A. Mould, 2nd, (2) If the source is in motion, the displacement would be the consequence of light-time correction. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver.BOOK, Seidelmann, P. Kenneth, Explanatory Supplement to the Astronomical Almanac, 1992, University Science Books, ill Valley, Calif., 978-0-935702-68-2, 393,weblink The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light,JOURNAL, Ferraro, Rafael, Sforza, Daniel M., European Physical Society logo Arago (1810): the first experimental result against the ether, European Journal of Physics, 26, 195, 10.1088/0143-0807/26/1/020, physics/0412055, 2005, and in 1870, George Airy tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope.WEB, Dolan, Graham, Airy's Water Telescope (1870),weblink The Royal Observatory Greenwich, 20 November 2018, A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,JOURNAL, Hollis, H. P., Airy's water telescope, The Observatory, 1937, 60, 103â€“107,weblinkfull/1937Obs....60..103H/0000105.000.html, 20 November 2018, 1937Obs....60..103H, but was incompatible with the results of the Michelsonâ€“Morley experiment, which apparently demanded complete aether-drag.BOOK, Janssen, Michel, Stachel, John, Stachel, John, Going Critical, 2004, Springer, 978-1-4020-1308-9, The Optics and Electrodynamics of Moving Bodies, https:www.mpiwg-berlin.mpg.de/Preprints/P265.PDF, Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5‑2, these include{{rp|57â€“60}}
cos theta ' = frac{ cos theta + v/c}{ 1 + (v/c)cos theta} OR sin theta ' = frac{sin theta}{gamma [ 1 + (v/c) cos theta ]} OR tan frac{theta '}{2} = left( frac{c - v}{c + v} right)^{1/2} tan frac {theta}{2}
Relativistic Doppler effect
Relativistic longitudinal Doppler effect
The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a time dilation term, and that is the treatment described here.JOURNAL, Sher, D., The Relativistic Doppler Effect, Journal of the Royal Astronomical Society of Canada, 1968, 62, 105â€“111,weblinkfull/1968JRASC..62..105S/0000105.000.html, 11 October 2018, BOOK, Gill, T. P., The Doppler Effect, 1965, Logos Press Limited, London, 6â€“9, https:openlibrary.org/books/OL5947329M/The_Doppler_effect, 12 October 2018, Assume the receiver and the source are moving away from each other with a relative speed v, as measured by an observer on the receiver or the source (The sign convention adopted here is that v, is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then
f_{r} = (1 - v / c_s )f_s
where c_s is the speed of sound.For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be
f_r = gamma(1 - beta) f_s = sqrt{frac{1 - beta}{1 + beta}},f_s.
where
beta = v/c and
gamma = frac{1}{sqrt{1 - beta^2}} is the Lorentz factor.
An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.BOOK, The Feynman Lectures on Physics: Volume 1, Addison-Wesley, Reading, Massachusetts, February 1977, Feynman, Richard P., Richard Feynman, Leighton, Robert B., Robert B. Leighton, Sands, Matthew, Matthew Sands, 2010938208, 9780201021165, 34â€“7 f,weblink Relativistic Effects in Radiation, Transverse Doppler effect
(File:Transverse Doppler effect scenarios 5.svg|thumb|300px|Figure 5-3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.)The transverse Doppler effect is one of the main novel predictions of the special theory of relativity. Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver. Special relativity predicts otherwise. Fig. 5‑3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments. In Fig. 5‑3a, the receiver observes light from the source as being blueshifted by a factor of gamma. In Fig. 5‑3b, the light is redshifted by the same factor.Measurement versus visual appearance
Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of Doppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer. For example, considering time, the elapsed time between two events, measured by identical clocks held by different observers which are moving relative to one another (measured perhaps in seconds), is different.Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. The visual appearance of an object, however, is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.(File:Animated Terrell Rotation - Cube.gif|thumb|330px|Figure 5-4. Comparison of the measured length contraction of a cube versus its visual appearance.)For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be seen as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation.JOURNAL, Terrell, James, Invisibility of the Lorentz Contraction, Physical Review, 15 November 1959, 116, 4, 1041â€“1045, 10.1103/PhysRev.116.1041, 1959PhRv..116.1041T, JOURNAL, Penrose, Roger, The Apparent Shape of a Relativistically Moving Sphere, Proceedings of the Cambridge Philosophical Society, Mathematical Proceedings of the Cambridge Philosophical Society, 24 October 2008, 55, 1, 137, 10.1017/S0305004100033776, 1959PCPS...55..137P, WEB, Cook, Helen, Relativistic Distortion,weblink Mathematics Department, University of British Columbia, 12 April 2017, WEB, Signell, Peter, Appearances at Relativistic Speeds,weblink Project PHYSNET, Michigan State University, East Lansing, MI, 12 April 2017,weblink 12 April 2017, yes, A sphere in motion retains the appearance of a sphere, although images on the surface of the sphere will appear distorted.WEB, Kraus, Ute, The Ball is Round,weblink Space Time Travel: Relativity visualized, Institut fÃ¼r Physik UniversitÃ¤t Hildesheim, 16 April 2017,weblink 16 April 2017, yes, File:M87 jet (1).jpg|thumb|Figure 5-5. Galaxy M87 streams out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.]]Fig. 5‑4 illustrates a cube viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. This illusion has come to be known as Terrell rotation or the Terrellâ€“Penrose effect.Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in Einstein's Cosmos (W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length."Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various radio galaxies, BL Lac objects, quasars, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel.BOOK, Zensus, J. Anton, Pearson, Timothy J., Superluminal Radio Sources, 1987, Cambridge University Press, Cambridge, New York, 9780521345606, 3, 1st, WEB, Chase, Scott I., Apparent Superluminal Velocity of Galaxies,weblink The Original Usenet Physics FAQ, Department of Mathematics, University of California, Riverside, 12 April 2017, WEB, Richmond, Michael, "Superluminal" motions in astronomical sources,weblink Physics 200 Lecture Notes, School of Physics and Astronomy, Rochester Institute of Technology, 20 April 2017,weblink 20 April 2017, yes, In Fig. 5‑5, galaxy M87 streams out a high-speed jet of subatomic particles almost directly towards us, but Penroseâ€“Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5‑4 has been stretched out.WEB, Keel, Bill, Jets, Superluminal Motion, and Gamma-Ray Bursts,weblink Galaxies and the Universe - WWW Course Notes, Department of Physics and Astronomy, University of Alabama, 29 April 2017,weblink 29 April 2017, yes,Dynamics
Section Consequences derived from the Lorentz transformation dealt strictly with kinematics, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself.Equivalence of mass and energy
As an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference.The energy content of an object at rest with mass m equals mc2. Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.In addition to the papers referenced aboveâ€”which give derivations of the Lorentz transformation and describe the foundations of special relativityâ€”Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for {{nowrap|1=E = mc2}}.Massâ€“energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energyâ€“momentum four-vector is {{nowrap|(E/c, 0, 0, 0)}}: it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes {{nowrap|(E/c, Ev/c2, 0, 0)}}. The momentum is equal to the energy multiplied by the velocity divided by c2. As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c2.The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these don't talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations. The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.Does the inertia of a body depend upon its energy content? A. Einstein, Annalen der Physik. 18:639, 1905 (English translation by W. Perrett and G.B. Jeffery) Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong.BOOK, Concepts of Mass in Classical and Modern Physics, Max Jammer, 177â€“178,weblink 978-0-486-29998-3, Courier Dover Publications, 1997, Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.BOOK, Einstein from B to Z, 221, John J. Stachel,weblink 978-0-8176-4143-6, Springer, 2002, Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic massâ€“energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.On the Inertia of Energy Required by the Relativity Principle, A. Einstein, Annalen der Physik 23 (1907): 371â€“384In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig, 1955.How far can one travel from the Earth?
{{See also|Space travel using constant acceleration}}Since one can not travel faster than light, one might conclude that a human can never travel farther from Earth than 40 light years if the traveler is active between the ages of 20 and 60. One would easily think that a traveler would never be able to reach more than the very few solar systems which exist within the limit of 20â€“40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, a hypothetical spaceship can travel thousands of light years during the pilot's 40 active years. If a spaceship could be built that accelerates at a constant 1g, it will, after a little less than a year, be travelling at almost the speed of light as seen from Earth. This is described by:
v(t) = frac{at}{sqrt{1+frac{a^2t^2}{c^2}}}
where v(t) is the velocity at a time t, a is the acceleration of 1g and t is the time as measured by people on Earth.WEB,weblink Acceleration in special relativity: What is the meaning of "uniformly accelerated movement" ?, 26 May 2007, 22 January 2016, Physics Department, ENS Cachan, Baglio, Julien, Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at v = 0.77c relative to Earth. Time dilation will increase the travellers life span as seen from the reference frame of the Earth to 2.7 years, but his lifespan measured by a clock travelling with him will not change. During his journey, people on Earth will experience more time than he does. A 5-year round trip for him will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for him (5 years accelerating, 5 decelerating, twice each) will land him back on Earth having travelled for 335 Earth years and a distance of 331 light years.WEB, Philip Gibbs, Don Koks, yes, The Relativistic Rocket,weblink 30 August 2012, A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1g will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy. This same time dilation is why a muon travelling close to c is observed to travel much farther than c times its half-life (when at rest).The special theory of relativity shows that time and space are affected by motion {{Webarchive|url=https://web.archive.org/web/20121021183616weblink |date=2012-10-21 }}. Library.thinkquest.org. Retrieved on 2013-04-24.Relativity and unifying electromagnetism
Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the LiÃ©nardâ€“Wiechert potential, which is a step towards special relativity.The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant form, i.e. in the language of tensor calculus.BOOK, Formal Structure of Electromagnetics: General Covariance and Electromagnetics, 1962, Dover Publications Inc., 978-0-486-65427-0, E. J. Post,Theories of relativity and quantum mechanics
Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. It is an unsolved problem in physics how general relativity and quantum mechanics can be unified; quantum gravity and a "theory of everything", which require a unification including general relativity too, are active and ongoing areas in theoretical research.The early Bohrâ€“Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics of the time.BOOK, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, 2nd, 114â€“116, R. Resnick, R. Eisberg, John Wiley & Sons, 1985, 978-0-471-87373-0, In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour,JOURNAL, P.A.M. Dirac, P.A.M. Dirac, 1930, A Theory of Electrons and Protons, Proceedings of the Royal Society, A126, 1930RSPSA.126..360D, 95359, 10.1098/rspa.1930.0013, 801, 360â€“365,weblink that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation explained not only the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),JOURNAL, C.D. Anderson, Carl David Anderson, The Positive Electron, Phys. Rev., 43, 491â€“494, 1933, 10.1103/PhysRev.43.491, 6, 1933PhRv...43..491A, and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic quantum mechanics. In non-relativistic quantum mechanics, spin is (Wikt:phenomenological|phenomenological) and cannot be explained.On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles can be created and destroyed throughout space and time.Status
Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c2 in the region of interest.BOOK, Einstein's general theory of relativity: with modern applications in cosmologyauthor2=SigbjÃ¸rn Hervik, yes, Springer, 2007, 978-0-387-69199-2, 195,weblink Extract of page 195 (with units where c=1)
In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10âˆ’20)The number of works is vast, see as example:
JOURNAL, Sidney Coleman, Sheldon L. Glashow, Cosmic Ray and Neutrino Tests of Special Relativity, Physics Letters B, 405, 1997, 249â€“252, hep-ph/9703240, 10.1016/S0370-2693(97)00638-2, 3â€“4, 1997PhLB..405..249C, An overview can be found on this pageand thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) â€“ thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See classical mechanics for a more detailed discussion.Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905,JOURNAL, John D. Norton, 2004, John D., Archive for History of Exact Sciences, Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905, 45â€“105, 59, 1,weblink 10.1007/s00407-004-0085-6, 2004AHES...59...45N, and historians have concluded that Einstein was at least aware of the Michelsonâ€“Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.JOURNAL, Jeroen van Dongen, On the role of the Michelsonâ€“Morley experiment: Einstein in Chicago, 2009, Archive for History of Exact Sciences, 63, 6, 655â€“663, 0908.1545, 2009arXiv0908.1545V, 10.1007/s00407-009-0050-5, In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10âˆ’20)The number of works is vast, see as example:
- The Fizeau experiment (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.
- The famous Michelsonâ€“Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.
- The Troutonâ€“Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
- The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction doesn't lead to birefringence for a co-moving observer, in accordance with the relativity principle.
- Tests of relativistic energy and momentum â€“ testing the limiting speed of particles
- Ivesâ€“Stilwell experiment â€“ testing relativistic Doppler effect and time dilation
- Experimental testing of time dilation â€“ relativistic effects on a fast-moving particle's half-life
- Kennedyâ€“Thorndike experiment â€“ time dilation in accordance with Lorentz transformations
- Hughesâ€“Drever experiment â€“ testing isotropy of space and mass
- Modern searches for Lorentz violation â€“ various modern tests
- Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
- Experiments to test the aether drag hypothesis â€“ no "aether flow obstruction".
Technical discussion of spacetime
Geometry of spacetime
Comparison between flat Euclidean space and Minkowski space
{{see also|line element}}File:Orthogonality and rotation.svg|thumb|350px|Figure 10-1. Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle Ï†, right: in Minkowski spacetime through hyperbolic angle Ï† (red lines labelled c denote the worldlineworldlineSpecial relativity uses a 'flat' 4-dimensional Minkowski space â€“ an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.In 3D space, the differential of distance (line element) ds is defined by
ds^2 = dmathbf{x} cdot dmathbf{x} = dx_1^2 + dx_2^2 + dx_3^2,
where {{nowrap|1=dx = (dx1, dx2, dx3)}} are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X0 derived from time, such that the distance differential fulfills
ds^2 = -dX_0^2 + dX_1^2 + dX_2^2 + dX_3^2,
where {{nowrap|1=dX = (dX0, dX1, dX2, dX3)}} are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10‑1).BOOK, Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 247, 2009, 978-0-470-01460-8, Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. {{Anchor|interval}}Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the PoincarÃ© group) of Minkowski spacetime.The actual form of ds above depends on the metric and on the choices for the X0 coordinate.To make the time coordinate look like the space coordinates, it can be treated as imaginary: {{nowrap|1=X0 = ict}} (this is called a Wick rotation).According to Misner, Thorne and Wheeler (1971, Â§2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take {{nowrap|1=X0 = ct}}, rather than a "disguised" Euclidean metric using ict as the time coordinate.Some authors use {{nowrap|1=X0 = t}}, with factors of c elsewhere to compensate; for instance, spatial coordinates are divided by c or factors of cÂ±2 are included in the metric tensor.BOOK, R. Penrose, The Road to Reality, Vintage books, 2007, 978-0-679-77631-4, The Road to Reality, These numerous conventions can be superseded by using natural units where {{nowrap|1=c = 1}}. Then space and time have equivalent units, and no factors of c appear anywhere.3D spacetime
(File:Special relativity- Three dimensional dual-cone.svg|thumb|Figure 10-2. Three-dimensional dual-cone.)If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space
ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2,
we see that the null geodesics lie along a dual-cone (see Fig. 10‑2) defined by the equation;
ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2
or simply
dx_1^2 + dx_2^2 = c^2 dt^2,
â€Šwhich is the equation of a circle of radius c dt.4D spacetime
If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:
ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2
so
dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2.
(File:Concentric Spheres.svg|thumb|Figure 10-3. Concentric spheres, illustrating in 3-space the null geodesics of a 4-dimensional cone in spacetime.)As illustrated in Fig. 10‑3, the null geodesics can be visualized as a set of continuous concentric spheres with radii = c dt.This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance d = sqrt{x_1^2+x_2^2+x_3^2} away and a time d/c in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the Fig. 10‑2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)The cone in the âˆ’t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought-experiments in special relativity.Note that, in 4d spacetime, the concept of the center of mass becomes more complicated, see center of mass (relativistic).Physics in spacetime
Transformations of physical quantities between reference frames
Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.The Lorentz transformation in standard configuration above, i.e. for a boost in the x direction, can be recast into matrix form as follows:
begin{pmatrix}
ct' x' y' z'end{pmatrix} = begin{pmatrix}gamma & -betagamma & 0 & 0-betagamma & gamma & 0 & 0end{pmatrix}begin{pmatrix}ct x y zend{pmatrix} =begin{pmatrix}gamma ct- gammabeta xgamma x - beta gamma ct y zend{pmatrix}.
X^nu = (X^0, X^1, X^2, X^3)= (ct, x, y, z) = (ct, mathbf{x} ).
where we define {{nowrap|1=X0 = ct}} so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, {{isbn|0-07-032071-3}}Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, pg 51, {{isbn|0-7167-0344-0}}George Sterman, An Introduction to Quantum Field Theory, pg 4 , {{isbn|0-521-31132-2}} Now the transformation of the contravariant components of the position 4-vector can be compactly written as:
X^{mu'}=Lambda^{mu'}{}_nu X^nu
where there is an implied summation on nu from 0 to 3, and Lambda^{mu'}{}_{nu} is a matrix.More generally, all contravariant components of a four-vector T^nu transform from one frame to another frame by a Lorentz transformation:
T^{mu'} = Lambda^{mu'}{}_{nu} T^nu
Examples of other 4-vectors include the four-velocity U^mu, defined as the derivative of the position 4-vector with respect to proper time:
U^mu = frac{dX^mu}{dtau} = gamma(v)( c , v_x , v_y, v_z ) = gamma(v) (c, mathbf{v} ).
where the Lorentz factor is:
gamma(v)= frac{1}{sqrt{1- left( frac{v}{c} right )^2}} qquad v^2 = v_x^2 + v_y^2 + v_z^2.
The relativistic energy E = gamma(v)mc^2 and relativistic momentum mathbf{p} = gamma(v)m mathbf{v} of an object are respectively the timelike and spacelike components of a contravariant four momentum vector:
P^mu = m U^mu = mgamma(v)(c,v_x,v_y,v_z)= left (frac{E}{c},p_x,p_y,p_z right ) = left (frac{E}{c}, mathbf{p} right ).
where m is the invariant mass.The four-acceleration is the proper time derivative of 4-velocity:
A^mu = frac{d U^mu}{dtau}.
The transformation rules for three-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix.The four-gradient of a scalar field Ï† transforms covariantly rather than contravariantly:
begin{pmatrix} frac{1}{c}frac{partial phi}{partial t'} & frac{partial phi}{partial x'} & frac{partial phi}{partial y'} & frac{partial phi}{partial z'}end{pmatrix} = begin{pmatrix} frac{1}{c}frac{partial phi}{partial t} & frac{partial phi}{partial x} & frac{partial phi}{partial y} & frac{partial phi}{partial z}end{pmatrix}begin{pmatrix}
gamma & +betagamma & 0 & 0+betagamma & gamma & 0 & 0end{pmatrix}.which is the transpose of:
(partial_{mu'} phi) = Lambda_{mu'}{}^{nu} (partial_nu phi)qquad partial_{mu} equiv frac{partial}{partial x^{mu}}.
only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation:
T_{mu'} = Lambda_{mu'}{}^{nu} T_nu,
where Lambda_{mu'}{}^{nu} is the reciprocal matrix of Lambda^{mu'}{}_{nu}.The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation lawSPACETIME AND GEOMETRY: AN INTRODUCTION TO GENERAL RELATIVITY >AUTHOR=SEAN M. CARROLL DATE=2004 PAGE=22,weblink
T^{alpha' beta' cdots zeta'}_{theta' iota' cdots kappa'} = Lambda^{alpha'}{}_{mu} Lambda^{beta'}{}_{nu} cdots Lambda^{zeta'}{}_{rho} Lambda_{theta'}{}^{sigma} Lambda_{iota'}{}^{upsilon} cdots Lambda_{kappa'}{}^{phi} T^{mu nu cdots rho}_{sigma upsilon cdots phi}
where Lambda_{chi'}{}^{psi} is the reciprocal matrix of Lambda^{chi'}{}_{psi}. All tensors transform by this rule.An example of a four dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order antisymmetric tensor.The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stressâ€“energy tensor for the electromagnetic field, namely the electromagnetic stressâ€“energy tensor.Metric
The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric Î· has components (valid in any inertial reference frame) which can be arranged in a {{nowrap|4 Ã— 4}} matrix:
eta_{alphabeta} = begin{pmatrix}
-1 & 0 & 0 & 0end{pmatrix}which is equal to its reciprocal, eta^{alphabeta}, in those frames. Throughout we use the signs as above, different authors use different conventions â€“ see Minkowski metric alternative signs.The PoincarÃ© group is the most general group of transformations which preserves the Minkowski metric:
eta_{alphabeta} = eta_{mu'nu'} Lambda^{mu'}{}_alpha Lambda^{nu'}{}_beta
and this is the physical symmetry underlying special relativity.The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is:
T^{alpha}S_{alpha}=T^{alpha}eta_{alphabeta}S^{beta} = T_{alpha}eta^{alphabeta}S_{beta} = text{invariant scalar}
Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Î› appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself:
|mathbf{T}| = sqrt{T^{alpha}T_{alpha}}
One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:
T^{alpha}{}_{alpha},T^{alpha}{}_{beta}T^{beta}{}_{alpha},T^{alpha}{}_{beta}T^{beta}{}_{gamma}T^{gamma}{}_{alpha} = text{invariant scalars},
similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants.Relativistic kinematics and invariance
The coordinate differentials transform also contravariantly:
dX^{mu'}=Lambda^{mu'}{}_nu dX^nu
so the squared length of the differential of the position four-vector dXÎ¼ constructed using
dmathbf{X}^2 = dX^mu ,dX_mu = eta_{munu},dX^mu ,dX^nu = -(c dt)^2+(dx)^2+(dy)^2+(dz)^2,
is an invariant. Notice that when the line element dX2 is negative that {{sqrt|âˆ’dX2}} is the differential of proper time, while when dX2 is positive, {{sqrt|dX2}} is differential of the proper distance.The 4-velocity UÎ¼ has an invariant form:
{mathbf U}^2 = eta_{numu} U^nu U^mu = -c^2 ,,
which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by Ï„ produces:
2eta_{munu}A^mu U^nu = 0.
So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.Relativistic dynamics and invariance
The invariant magnitude of the momentum 4-vector generates the energyâ€“momentum relation:
mathbf{P}^2 = eta^{munu}P_mu P_nu = -left (frac{E}{c} right )^2 + p^2 .
We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
mathbf{P}^2 = - left (frac{E_mathrm{rest}}{c} right )^2 = - (m c)^2 .
We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.The rest energy is related to the mass according to the celebrated equation discussed above:
E_mathrm{rest} = m c^2.
Note that the mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:
F_nu = frac{d P_{nu}}{d tau} = m A_nu
In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, i.e. dp/dt while the four force is defined by the rate of change of momentum with respect to proper time, i.e. dp/dÏ„.In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is âˆ’1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.See also
People: Hendrik Lorentz | Henri PoincarÃ© | Albert Einstein | Max Planck | Hermann Minkowski | Max von Laue | Arnold Sommerfeld | Max Born | Gustav Herglotz | Richard C. Tolman
Relativity: Theory of relativity | History of special relativity | Principle of relativity | Doubly special relativity | General relativity | Frame of reference | Inertial frame of reference | Lorentz transformations | Bondi k-calculus | Einstein synchronisation | Rietdijkâ€“Putnam argument | Special relativity (alternative formulations) | Criticism of relativity theory | Relativity priority dispute
Physics: Einstein's thought experiments | Newtonian Mechanics | spacetime | speed of light | simultaneity | center of mass (relativistic) | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Lorentz ether theory | Moving magnet and conductor problem | Shape waves | Relativistic heat conduction | Relativistic disk | Thomas precession | Born rigidity | Born coordinates
Mathematics: Derivations of the Lorentz transformations | Minkowski space | four-vector | world line | light cone | Lorentz group | PoincarÃ© group | geometry | tensors | split-complex number | Relativity in the APS formalism
Philosophy: actualism | conventionalism | formalism
Paradoxes: Twin paradox | Ehrenfest paradox | Ladder paradox | Bell's spaceship paradox | Velocity composition paradox | Lighthouse paradox
Primary sources
{{Reflist|group=p|35em}}References
{{Reflist|35em}}Notes
{{reflist|group=note|35em}}Textbooks
- Einstein, Albert (1920). s:Relativ.
- Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. {{isbn|1-56731-136-9}}
- Logunov, Anatoly A. (2005) Henri PoincarÃ© and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soleviev, edited by V. A. Petrov) Nauka, Moscow.
- Charles Misner, Kip Thorne, and John Archibald Wheeler (1971) Gravitation. W. H. Freeman & Co. {{isbn|0-7167-0334-3}}
- Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications.
- Wolfgang Rindler (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. {{isbn|978-0-19-853952-0}}; {{isbn|0-19-853952-5}}
- Harvey R. Brown (2005). Physical relativity: spaceâ€“time structure from a dynamical perspective, Oxford University Press, {{isbn|0-19-927583-1}}; {{isbn|978-0-19-927583-0}}
- BOOK, Qadir, Asghar, Asghar Qadir, Relativity: An Introduction to the Special Theory, World Scientific, World Scientific Publications, 1989, Singapore, 128,weblink 978-9971-5-0612-4, 1989rist.book.....Q,
- BOOK, French, A. P., Special Relativity (M.I.T. Introductory Physics), 1st, W. W. Norton & Company, 1968, 978-0393097931,
- Silberstein, Ludwik (1914) The Theory of Relativity.
- BOOK, Space, Time and Spacetime, Lawrence Sklar,weblink 978-0-520-03174-6, University of California Press, 1977,
- BOOK, Philosophy of Physics, Lawrence Sklar,weblink 978-0-8133-0625-4, Westview Press, 1992,
- Taylor, Edwin, and John Archibald Wheeler (1992) Spacetime Physics (2nd ed.). W.H. Freeman & Co. {{isbn|0-7167-2327-1}}
- Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. {{isbn|0-7167-4345-0}}
Journal articles
- JOURNAL, Alvager, T., Farley, F. J. M., 1964, Kjellman, J., Wallin, L., Test of the Second Postulate of Special Relativity in the GeV region, Physics Letters, 12, 3, 260, 10.1016/0031-9163(64)91095-9, 1964PhL....12..260A, etal,
- JOURNAL, 10.1086/430652, Darrigol, Olivier, 2004, The Mystery of the PoincarÃ©â€“Einstein Connection, Isis, 95, 4, 614â€“26, 16011297,
- JOURNAL, 10.1103/PhysRevA.56.4405, Wolf, Peter, Petit, Gerard, 1997, Satellite test of Special Relativity using the Global Positioning System, Physical Review A, 56, 6, 4405â€“09, 1997PhRvA..56.4405W,
- Special Relativity Scholarpedia
- Special relativity: Kinematics Wolfgang Rindler, Scholarpedia, 6(2):8520.
External links
{{Wikisourcepar|Relativity: The Special and General Theory}}{{Wikisource portal|Relativity}}{{Wikiversity|Special Relativity}}{{Wiktionary|special relativity}}Original works
- Zur Elektrodynamik bewegter KÃ¶rper Einstein's original work in German, Annalen der Physik, Bern 1905
- On the Electrodynamics of Moving Bodies English Translation as published in the 1923 book The Principle of Relativity.
Special relativity for a general audience (no mathematical knowledge required)
- Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
- Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
- Audio: Cain/Gay (2006) â€“ Astronomy Cast. Einstein's Theory of Special Relativity
Special relativity explained (using simple or more advanced mathematics)
- Greg Egan's Foundations.
- The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus.
- Relativity Calculator: Special Relativity â€“ An algebraic and integral calculus derivation for {{nowrap|1=E = mc2}}.
- MathPages â€“ Reflections on Relativity A complete online book on relativity with an extensive bibliography.
- Special Relativity An introduction to special relativity at the undergraduate level.
- {{gutenberg|no=5001|name=Relativity: the Special and General Theory}}, by Albert Einstein
- Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.
- Understanding Special Relativity The theory of special relativity in an easily understandable way.
- An Introduction to the Special Theory of Relativity (1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format).
- Lecture Notes on Special Relativity by J D Cresser Department of Physics Macquarie University.
- SpecialRelativity.net - An overview with visualizations and minimal mathematics.
Visualization
- Raytracing Special Relativity Software visualizing several scenarios under the influence of special relativity.
- Real Time Relativity The Australian National University. Relativistic visual effects experienced through an interactive program.
- Spacetime travel A variety of visualizations of relativistic effects, from relativistic motion to black holes.
- Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images.
- Warp Special Relativity Simulator A computer program to show the effects of traveling close to the speed of light.
- {{YouTube|C2VMO7pcWhg|Animation clip}} visualizing the Lorentz transformation.
- Original interactive FLASH Animations from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.
- lightspeed An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.
- Animation showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.
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