Rindler coordinates

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Rindler coordinates
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In relativistic physics, the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this hyperbolically accelerated frame can be compared to effects arising in a homogeneous gravitational field. For general overview of accelerations in flat spacetime, see Acceleration (special relativity) and Proper reference frame (flat spacetime).In this article, the speed of light is defined by {{math|c {{=}} 1}}, the inertial coordinates are {{math|(X,Y,Z,T)}}, and the hyperbolic coordinates are {{math|(x,y,z,t)}}. These hyperbolic coordinates can be separated into two main variants depending on the accelerated observer's position: If the observer is located at time {{math|T {{=}} 0}} at position {{math|X {{=}} 1/α}} (with {{math|α}} as the constant proper acceleration measured by a comoving accelerometer), then the hyperbolic coordinates are often called Rindler coordinates with the corresponding Rindler metric. If the observer is located at time {{math|T {{=}} 0}} at position {{math|X {{=}} 0}}, then the hyperbolic coordinates are sometimes called Møller coordinates or Kottler-Møller coordinates with the corresponding Kottler-Møller metric. An alternative chart often related to observers in hyperbolic motion is obtained using Radar coordinates which are sometimes called Lass coordinates. Both the Kottler-Møller coordinates as well as Lass coordinates are denoted as Rindler coordinates as well.For instance, Birrill & Davies (1982), pp. 110-111 or Padmanabhan (2010), p. 126 denote equations ({{equationNote|2g}}, {{equationNote|2h}}) as Rindler coordinates or Rindler frame; Tilbrook (1997) pp. 864-864 or Jones & Wanex (2006) denote equations ({{equationNote|2a}}, {{equationNote|2b}}) as Rindler coordinatesRegarding the history, such coordinates were introduced soon after the advent of special relativity, when they were studied (fully or partially) alongside the concept of hyperbolic motion: In relation to flat Minkowski spacetime by Albert Einstein (1907, 1912), Max Born (1909), Arnold Sommerfeld (1910), Max von Laue (1911), Hendrik Lorentz (1913), Friedrich Kottler (1914), Wolfgang Pauli (1921), Karl Bollert (1922), Stjepan Mohorovičić (1922), Georges Lemaître (1924), Einstein & Nathan Rosen (1935), Christian Møller (1943, 1952), Fritz Rohrlich (1963), Harry Lass (1963), and in relation to both flat and curved spacetime of general relativity by Wolfgang Rindler (1960, 1966). For details and sources, see section on history.

Characteristics of the Rindler frame

(File:Rindler chart.svg|thumb|Rindler chart, for alpha=0.5 in equation ({{equationNote|1a}}), plotted on a Minkowski diagram. The dashed lines are the Rindler horizons)The worldline of a body in hyperbolic motion having constant proper acceleration alpha in the X-direction as a function of proper time tau and rapidity alphatau can be given by
T=xsinh(alphatau),quad X=xcosh(alphatau)
where x=1/alpha is constant and alphatau is variable, with the worldline resembling the hyperbola X^{2}-T^{2}=x^2. Sommerfeld showed that the equations can be reinterpreted by defining x as variable and alphatau as constant, so that it represents the simultaneous "rest shape" of a body in hyperbolic motion measured by a comoving observer. By using the proper time of the observer as the time of the entire hyperbolically accelerated frame by setting tau=t, the transformation formulas between the inertial coordinates and the hyperbolic coordinates are consequently:{{NumBlk|:|T=xsinh(alpha t),quad X=xcosh(alpha t),quad Y=y,quad Z=z|{{equationRef|1a}}}}with the inverse
t=frac{1}{alpha}operatorname{arctanh}left(frac{T}{X}right),quad x=sqrt{X^{2}-T^{2}},quad y=Y,quad z=Z
Differentiated and inserted into the Minkowski metric ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2}, the metric in the hyperbolically accelerated frame follows{{NumBlk|:|ds^{2}=-(alpha x)^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}|{{equationRef|1b}}}}These transformations define the Rindler observer as an observer that is "at rest" in Rindler coordinates, i.e., maintaining constant x, y, z and only varying t as time passes. The coordinates are valid in the region {scriptstyle 0, 0 in the Cartesian chart, which consists of two null half-planes, each ruled by a null geodesic congruence.For the moment, we simply consider the Rindler horizon as the boundary of the Rindler coordinates. If we consider the set of accelerating observers who have a constant position in Rindler coordinates, none of them can ever receive light signals from events with T â‰¥ X (on the diagram, these would be events on or to the left of the line T = X which the upper red horizon lies along; these observers could however receive signals from events with T â‰¥ X if they stopped their acceleration and crossed this line themselves) nor could they have ever sent signals to events with T â‰¤ −X (events on or to the left of the line T = −X which the lower red horizon lies along; those events lie outside all future light cones of their past world line). Also, if we consider members of this set of accelerating observers closer and closer to the horizon, in the limit as the distance to the horizon approaches zero, the constant proper acceleration experienced by an observer at this distance (which would also be the G-force experienced by such an observer) would approach infinity. Both of these facts would also be true if we were considering a set of observers hovering outside the event horizon of a black hole, each observer hovering at a constant radius in Schwarzschild coordinates. In fact, in the close neighborhood of a black hole, the geometry close to the event horizon can be described in Rindler coordinates. Hawking radiation in the case of an accelerating frame is referred to as Unruh radiation. The connection is the equivalence of acceleration with gravitation.Dieter Brill, "Black Hole Horizons and How They Begin", Astronomical Review (2012); Online Article, cited Sept.2012.


The geodesic equations in the Rindler chart are easily obtained from the geodesic Lagrangian; they are
ddot + frac{2}{x} , dot{x} , dot = 0, ; ddot{x} + x , dot^2 = 0, ; ddot{y} = 0, ; ddot{z} = 0
Of course, in the original Cartesian chart, the geodesics appear as straight lines, so we could easily obtain them in the Rindler chart using our coordinate transformation. However, it is instructive to obtain and study them independently of the original chart, and we shall do so in this section.(File:UHS geodesics.png|frame|left|Some representative null geodesics (black hyperbolic semicircular arcs) projected into the spatial hyperslice t=0 of the Rindler observers. The Rindler horizon is shown as a magenta plane.)From the first, third, and fourth we immediately obtain the first integrals
dot = frac{E}{x^2}, ; ; dot{y} = P, ; ; dot{z} = Q
But from the line element we have scriptstyle epsilon ;=; -x^2 , dot^2 ,+, dot{x}^2 ,+, dot{y}^2 ,+, dot{z}^2 where scriptstyleepsilon ;in; left{-1,, 0,, 1right} for timelike, null, and spacelike geodesics, respectively. This gives the fourth first integral, namely
dot{x}^2 = left(epsilon + frac{E^2}{x^2} right) - P^2 - Q^2.
This suffices to give the complete solution of the geodesic equations.In the case of null geodesics, from scriptstylefrac{E^2}{x^2} ,-, P^2 ,-, Q^2 with nonzero scriptstyle E, we see that the x coordinate ranges over the interval scriptstyle 0 ,, 0), the optical distance is a bit larger than the ruler distance, which is a bit larger than the radar distance. The reader should now take a moment to consider the case of a leading observer estimating distance to a trailing observer.There are other notions of distance, but the main point is clear: while the values of these various notions will in general disagree for a given pair of Rindler observers, they all agree that every pair of Rindler observers maintains constant distance. The fact that very nearby Rindler observers are mutually stationary follows from the fact, noted above, that the expansion tensor of the Rindler congruence vanishes identically. However, we have shown here that in various senses, this rigidity property holds at larger scales. This is truly a remarkable rigidity property, given the well-known fact that in relativistic physics, no rod can be accelerated rigidly (and no disk can be spun up rigidly) — at least, not without sustaining inhomogeneous stresses. The easiest way to see this is to observe that in Newtonian physics, if we "kick" a rigid body, all elements of matter in the body will immediately change their state of motion. This is of course incompatible with the relativistic principle that no information having any physical effect can be transmitted faster than the speed of light.It follows that if a rod is accelerated by some external force applied anywhere along its length, the elements of matter in various different places in the rod cannot all feel the same magnitude of acceleration if the rod is not to extend without bound and ultimately break. In other words, an accelerated rod which does not break must sustain stresses which vary along its length. Furthermore, in any thought experiment with time varying forces, whether we "kick" an object or try to accelerate it gradually, we cannot avoid the problem of avoiding mechanical models which are inconsistent with relativistic kinematics (because distant parts of the body respond too quickly to an applied force).Returning to the question of the operational significance of the ruler distance, we see that this should be the distance which our observers will obtain should they very slowly pass from hand to hand a small ruler which is repeatedly set end to end. But justifying this interpretation in detail would require some kind of material model.

Generalization to curved spacetimes

Rindler coordinates as described above can be generalized to curved spacetime, as Fermi normal coordinates. The generalization essential involves constructing an appropriate orthonormal tetrad and then transporting it along the given trajectory using the Fermi–Walker transport rule. For details, see the paper by Ni and Zimmermann in the references below. Such a generalization actually enables one to study inertial and gravitational effects in an Earth-based laboratory, as well as the more interesting coupled inertial-gravitational effects.



Kottler-Møller and Rindler coordinates
Albert Einstein (1907) studied the effects within a uniformly accelerated frame, obtaining equations for coordinate dependent time dilation and speed of light equivalent to ({{equationNote|2c}}), and in order to make the formulas independent of the observer's origin, he obtained time dilation ({{equationNote|2i}}) in formal agreement with Radar coordinates. While introducing the concept of Born rigidity, Max Born (1909) noted that the formulas for hyperbolic motion can be used as transformations into a "hyperbolically accelerated reference system" () equivalent to ({{equationNote|2d}}). Born's work was further elaborated by Arnold Sommerfeld (1910) and Max von Laue (1911) who both obtained ({{equationNote|2d}}) using imaginary numbers, which was summarized by Wolfgang Pauli (1921) who besides coordinates ({{equationNote|2d}}) also obtained metric ({{equationNote|2e}}) using imaginary numbers. Einstein (1912) studied a static gravitational field and obtained the Kottler-Møller metric ({{equationNote|2b}}) as well as approximations to formulas ({{equationNote|2a}}) using a coordinate dependent speed of light. Hendrik Lorentz (1913) obtained coordinates similar to ({{equationNote|2d}}, {{equationNote|2e}}, {{equationNote|2f}}) while studying Einstein's equivalence principle and the uniform gravitational field.A detailed description was given by Friedrich Kottler (1914),Kottler (1912), pp. 1715; Kottler (1914a), Table I; pp. 747-748; Kottler (1914b), pp. 488-489, 503; Kottler (1916), pp. 958-959; (1918), pp. 453-454; who formulated the corresponding orthonormal tetrad, transformation formulas and metric ({{equationNote|2a}}, {{equationNote|2b}}). Also Karl Bollert (1922) obtained the metric ({{equationNote|2b}}) in his study of uniform acceleration and uniform gravitational fields. In a paper concerned with Born rigidity, Georges Lemaître (1924) obtained coordinates and metric ({{equationNote|2a}}, {{equationNote|2b}}). Albert Einstein and Nathan Rosen (1935) described ({{equationNote|2d}}, {{equationNote|2e}}) as the "well known" expressions for a homogeneous gravitational field. After Christian Møller (1943) obtained ({{equationNote|2a}}, {{equationNote|2b}}) in as study related to homogeneous gravitational fields, he (1952) as well as Misner & Thorne & Wheeler (1973) used Fermi-Walker transport to obtain the same equations.While these investigations were concerned with flat spacetime, Wolfgang Rindler (1960) analyzed hyperbolic motion in curved spacetime, and showed (1966) the analogy between the hyperbolic coordinates ({{equationNote|2d}}, {{equationNote|2e}}) in flat spacetime with Kruskal coordinates in Schwarzschild space. This influenced subsequent writers in their formulation of Unruh radiation measured by an observer in hyperbolic motion, which is similar to the description of Hawking radiation of black holes.
Born (1909) showed that the inner points of a Born rigid body in hyperbolic motion can only be in the region X/left(X^{2}-T^{2}right)>0.Born (1909), p. 35 Sommerfeld (1910) defined that the coordinates allowed for the transformation between inertial and hyperbolic coordinates must satisfy T0, and pointed out the existence of a "border plane" () c^2/alpha+x, beyond which no signal can reach the observer in hyperbolic motion. This was called the "horizon of the observer" () by Bollert (1922).Bollert (192{{equationNote|2b}}), pp. 194-196 Rindler (1966) demonstrated the relation between such a horizon and the horizon in Kruskal coordinates.
Radar coordinates
Using Bollert's formalism, Stjepan Mohorovičić (1922) made a different choice for some parameter and obtained metric ({{equationNote|2h}}) with a printing error, which was corrected by Bollert (1922b) with another printing error, until a version without printing error was given by Mohorovičić (1923). In addition, Mohorovičić erroneously argued that metric ({{equationNote|2b}}, now called Kottler-Møller metric) is incorrect, which was rebutted by Bollert (1922).Bollert (1922b), p. 189 Metric ({{equationNote|2h}}) was rediscovered by Harry Lass (1963), who also gave the corresponding coordinates ({{equationNote|2g}}) which are sometimes called "Lass coordinates". Metric ({{equationNote|2h}}), as well as ({{equationNote|2a}}, {{equationNote|2b}}), was also derived by Fritz Rohrlich (1963). Eventually, the Lass coordinates ({{equationNote|2g}}, {{equationNote|2h}}) were identified with Radar coordinates by Desloge & Philpott (1987).

Table with historical formulas

{| style="text-align: center;"|{scriptstyle begin{matrix}sigma=tauleft(1+frac{gammaxi}{c^{2}}right)sigma=tau e^{gammaxi/c^{2}}cleft(1+frac{gammaxi}{c^{2}}right)end{matrix}}!Born (1909)Born (1909), p. 25|{scriptstyle begin{matrix}x=-qxi, y=eta, z=zeta, t=frac{p}{c^{2}}xileft(p=x_{tau}, q=-t_{tau}=sqrt{1+p^{2}/c^{2}}right)boldsymbol{downarrow}x^{2}-c^{2}t^{2}=xi^{2}end{matrix}}!Herglotz (1909)Herglotz (1909), pp. 408, 414|{scriptstyle begin{matrix}begin{align}x & =x'y & =y't-z & =(t'-z')e^{vartheta}t+z & =(t'+z')e^{-vartheta}end{align}boldsymbol{downarrow}x=x_{0},quad y=y_{0},quad z=sqrt{z_{0}^{2}+t^{2}}end{matrix}}!Sommerfeld (1910)Sommerfeld (1910), pp. 670-671|scriptstylebegin{align}x & =rcosvarphiy & =y'z & =z'l & =rsinvarphivarphi & =ipsi, l =ictend{align}!von Laue (1911)von Laue (1911), p. 109|scriptstylebegin{align}X & =RcosvarphiL & =RsinvarphiR^{2} & =X^{2}+L^{2}tanvarphi & =frac{L}{X}end{align}!Einstein (1912)Einstein (1912), pp. 358-359|{scriptstyle begin{matrix}dxi^{2}-dtau^{2}=dx^{2}-c^{2}dt^{2}boldsymbol{downarrow}c=c_{0}+axboldsymbol{downarrow}begin{align}xi & =x+frac{ac}{2}t^{2}eta & =yzeta & =ztau & =ctend{align}end{matrix}}!Kottler (1912)Kottler (1912), pp. 1715|{scriptstyle begin{align}x^{(1)} & =x_{0}^{(1)}x^{(2)} & =x_{0}^{(2)}x^{(3)} & =bcos ivarphix^{(4)} & =bsin ivarphiend{align}}!Lorentz (1913)Lorentz (1913), pp. 34-38; 50-52|{scriptstyle begin{matrix}dc=frac{g}{c}dzhline begin{align}z & =aleft(z'-z_{0}^{prime}right)ct & =bleft(z'-z_{0}^{prime}right)a & =frac{1}{2}left(e^{kt'}+e^{-kt}right)b & =frac{1}{2}left(e^{kt'}-e^{-kt}right)end{align}boldsymbol{downarrow}c'=kleft(z'-z_{0}^{prime}right), z'-z_{0}^{prime}=frac{c^{2}}{g}boldsymbol{downarrow}begin{align} & dx^{2}+dy^{2}+dz^{2}-c^{2}dt
& =dx^{prime2}+dy^{prime2}+dz^{prime2}-c^{prime2}dt^{prime2}
{| class="wikitable" !Einstein (1907)Einstein (1907), §§ 18-21
|{scriptstyle begin{matrix}begin{align}x^{(1)} & =x_{0}^{(1)}x^{(2)} & =x_{0}^{(2)}x^{(3)} & =bcos iux^{(4)} & =bsin iuend{align}boldsymbol{downarrow}ds^{2}=-c^{2}dtau^{2}=b^{2}(du)^{2}boldsymbol{downarrow}begin{matrix}c_{1}^{(1)}=0, & & c_{1}^{(2)}=0, & & c_{1}^{(3)}=-sin iu, & & c_{1}^{(4)}=cos iu,c_{2}^{(1)}=0, & & c_{2}^{(2)}=0, & & c_{2}^{(3)}=-cos iu, & & c_{2}^{(4)}=-sin iu,end{matrix}boldsymbol{downarrow}dS^{2}=(dX')^{2}+(dY')^{2}+(dZ')^{2}-left(c+frac{Z'c}{b}right)^{2}dT'boldsymbol{downarrow}c'=c+frac{Z'c^{2}}{b}cdotfrac{1}{c}end{matrix}}!Kottler (1914b)Kottler (1914b), pp. 488-489, 503|scriptstylebegin{matrix}begin{matrix}c_{1}^{(1)}=0, & & c_{1}^{(2)}=0, & & c_{1}^{(3)}=frac{1}{i}sinh u, & & c_{1}^{(4)}=cosh u,c_{2}^{(1)}=0, & & c_{2}^{(2)}=0, & & c_{2}^{(3)}=frac{1}{i}cosh u, & & c_{2}^{(4)}=-sinh u,c_{3}^{(1)}=1, & & c_{3}^{(2)}=0, & & c_{3}^{(3)}=0, & & c_{3}^{(4)}=0,c_{4}^{(1)}=0, & & c_{4}^{(2)}=1, & & c_{4}^{(3)}=0, & & c_{4}^{(4)}=0,end{matrix}boldsymbol{downarrow}X=x+Delta^{(2)}c_{2}+Delta^{(3)}c_{3}+Delta^{(4)}c_{4}boldsymbol{downarrow}begin{align}X & =x_{0}+mathfrak{X}'Y & =y_{0}+mathfrak{Y}'Z & =left(b+mathfrak{Z}'right)coshmathfrak{u}cT & =left(b+mathfrak{Z}'right)sinhmathfrak{u}end{align}left(Delta^{(2)}=mathfrak{X}', Delta^{(3)}=mathfrak{Y}', Delta^{(4)}=mathfrak{Z}'right)boldsymbol{downarrow}begin{align}mathfrak{X}' & =X_{0}-x_{0}+q_{x}Tmathfrak{Y}' & =Y_{0}-y_{0}+q_{y}Tb+mathfrak{Z}' & =sqrt{left(Z_{0}+q_{x}Tright)^{2}-c^{2}T^{2}}cmathfrak{T}' & =boperatorname{arctanh}frac{cT}{Z_{0}+q_{x}T}end{align}left(X=X_{0}+q_{x}T, Y=Y_{0}+q_{y}T, Z=Z_{0}+q_{x}Tright)boldsymbol{downarrow}dS^{2}=(dmathfrak{X}')^{2}+(dmathfrak{Y}')^{2}+(dmathfrak{Z}')^{2}-c^{2}left(frac{b+mathfrak{Z}'}{b^{2}}right)^{2}(dmathfrak{T}')^{2}end{matrix}!Kottler (1916, 1918)Kottler (1916), pp. 958-959; (1918), pp. 453-454|scriptstylebegin{matrix}begin{align}x & =x'y & =y'frac{c^{2}}{gamma}+z & =left(frac{c^{2}}{gamma}+z'right)coshfrac{gamma t'}{c}ct & =left(frac{c^{2}}{gamma}+z'right)sinhfrac{gamma t'}{c}end{align}boldsymbol{downarrow}ds^{2}=dx^{prime2}+dy^{prime2}+dz^{prime2}-left(c+frac{gamma}{c}z'right){}^{2}dt^{prime2}end{matrix}
{| class="wikitable"!Kottler (1914a)Kottler (1914a), Table I; pp. 747-748
|scriptstylebegin{matrix}begin{align}x^{1} & =varrhocosvarphix^{4} & =varrhosinvarphiend{align}boldsymbol{downarrow}ds^{2}=left(dxi^{1}right)^{2}+left(dxi^{2}right)^{2}+left(dxi^{3}right)^{2}+left(xi^{1}right)^{2}left(dxi^{4}right)^{2}left(xi^{(1)}=varrho, xi^{(2)}=x^{(2)}, xi^{(3)}=x^{(3)}, xi^{(4)}=varphiright)end{matrix}!Bollert (1922)Bollert (1922a), p. 261, 266|{scriptstyle begin{matrix}ds^{2}=c^{2}left(1+frac{gamma_{0}x}{c^{2}}right)dtau^{2}-dx^{2}-dy^{2}-dz^{2}hline ds^{2}=g_{44}dx_{4}^{2}+g_{11}dx_{1}^{2}+g_{22}left(dx_{2}^{2}+dx_{3}^{2}right)boldsymbol{downarrow}V''-frac{g_{11}}{2g_{11}}V'=0left(g_{22}=-1, g_{11}=-1, V''=0, V=ax+bright)boldsymbol{downarrow}ds^{2}=dx_{4}^{2}(ax+b)^{2}-dx^{2}-dy^{2}-dz^{2}end{matrix}}!Mohorovičić (1922, 1923); Bollert (1922b)Mohorovičić (1922), p. 92, without x_1 in the exponent due to a printing error, which was corrected by Bollert (1922b), p.189, as well as Mohorovičić (1923), p. 54|{scriptstyle begin{matrix}text{Mohorovičić (1922):}g_{11}=g_{44}=V^{2}, VV''-V'^{2}=0, Vleft(x_{1}right)=e^{ax_{1}}boldsymbol{downarrow}ds^{2}=e^{2a}left(-dx_{4}^{2}+dx_{1}^{2}right)+dx_{2}^{2}+dx_{3}^{2}\text{corrected by Bollert (1922b):}ds^{2}=e^{2ax}left(-dx_{4}^{2}+dx_{1}^{2}right)+dx_{2}^{2}+dx_{3}^{2}\text{final correction by Mohorovičić (1923):}ds^{2}=e^{2ax_{1}}left(-dx_{4}^{2}+dx_{1}^{2}right)+dx_{2}^{2}+dx_{3}^{2}end{matrix}}!Lemaître (1924)Lemaitre (1921), pp. 166, 168|{scriptstyle begin{matrix}begin{align}1+gxi= & (1+gx)cosh gtgtau= & (1+gx)sinh gtend{align}boldsymbol{downarrow}ds^{2}=-dx^{2}-dy^{2}-dz^{2}+(1+gx)^{2}dt^{2}end{matrix}}!Einstein & Rosen (1935)Einstein & Rosen (1935, p. 74|scriptstylebegin{matrix}begin{align}xi_{1} & =x_{1}coshalpha x_{4}xi_{2} & =x_{2}xi_{3} & =x_{3}xi_{4} & =x_{1}sinhalpha x_{4}end{align}boldsymbol{downarrow}ds^{2}=-dx_{1}^{2}-dx_{2}^{2}-dx_{3}^{2}+alpha{}^{2}x_{1}^{2}dx_{4}^{2}end{matrix}!Møller (1952)Møller (1952), pp. 121-123; 255-258|scriptstylebegin{matrix}alpha_{ik}=left(begin{matrix}U_{4}/ic & 0 & 0 & iU_{1}/cU_{1}/ic & 0 & 0 & U_{4}/icend{matrix}right)U_{i}=left(csinhfrac{gtau}{c}, 0,0, igcoshfrac{gtau}{c}right)boldsymbol{downarrow}X_{i}=mathbf{f}_{i}(t)+x^{primekappa}alpha_{kappa i}(tau)boldsymbol{downarrow}begin{align}X & =frac{c^{2}}{g}left(coshfrac{gt}{c}-1right)+xcoshfrac{gt}{c}Y & =yZ & =zT & =frac{c}{g}sinhfrac{gt}{c}+xfrac{sinhfrac{gt}{c}}{c}end{align}boldsymbol{downarrow}ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}left(1+gx/c^{2}right)^{2}\end{matrix}
{| class="wikitable"!Pauli (1921)Pauli (1921), pp. 647-648

See also


BOOK, von Laue, M., 1921, Die Relativitätstheorie, Band 1, fourth edition of "Das Relativitätsprinzip”, Vieweg,weblink ; First edition 1911, second expanded edition 1913, third expanded edition 1919.{{Citationauthorlink=Wolfgang Pauli|year=1921title= Die Relativitätstheorievolume=5|issue=2|url=}} In English: BOOK, Pauli, W., Theory of Relativity, Fundamental Theories of Physics, 165, Dover Publications, 1981, 1921, 0-486-64152-X, BOOK, Møller, C., The theory of relativity, 1955, 1952, Oxford Clarendon Press,weblink JOURNAL, Harry Lass, 1963, Accelerating Frames of Reference and the Clock Paradox, American Journal of Physics, 31, 4, 274–276, 10.1119/1.1969430, 1963AmJPh..31..274L, JOURNAL, Rohrlich, Fritz, 1963, The principle of equivalence, Annals of Physics, 22, 2, 169–191, 10.1016/0003-4916(63)90051-4, 1963AnPhy..22..169R, JOURNAL, Rindler, W., 1960, Hyperbolic Motion in Curved Space Time, Physical Review, 119, 6, 2082–2089, 10.1103/PhysRev.119.2082, 1960PhRv..119.2082R, JOURNAL, Rindler, W., 1966, Kruskal Space and the Uniformly Accelerated Frame, American Journal of Physics, 34, 12, 1174–1178, 10.1119/1.1972547, 1966AmJPh..34.1174R, BOOK, Misner, C. W., Thorne, K. S., Wheeler, J. A., Gravitation, 1973, Freeman, 0716703440, JOURNAL, Desloge, Edward A., Philpott, R. J., 1987, Uniformly accelerated reference frames in special relativity, American Journal of Physics, 55, 3, 252–261, 10.1119/1.15197, 1987AmJPh..55..252D, JOURNAL, David Tilbrook, 1997, General Coordinatisations of the Flat Space-Time of Constant Proper-acceleration, Australian Journal of Physics, 50, 5, 851–868, 10.1071/P96111, JOURNAL, Dolby, Carl E., Gull, Stephen F., 2001, On radar time and the twin "paradox", American Journal of Physics, 69, 12, 1257–1261, gr-qc/0104077, 10.1119/1.1407254, 2001AmJPh..69.1257D, JOURNAL, Massimo Pauri, Michele Vallisneri, 2000, Märzke-Wheeler coordinates for accelerated observers in special relativity, Foundations of Physics Letters, 13, 5, 401–425, gr-qc/0006095, 10.1023/A:1007861914639, JOURNAL, Minguzzi, E., 2005, The Minkowski metric in non-inertial observer radar coordinates, American Journal of Physics, 73, 12, 1117–1121, 10.1119/1.2060716, physics/0412024, 2005AmJPh..73.1117M, BOOK, Don Koks, Explorations in Mathematical Physics, 2006, Springer, 0387309438, 235–269, BOOK, Leonard Susskind, James Lindesay, An Introduction to Black Holes, Information and the String Theory Revolution: The Holographic Universe, 2005, World Scientific, 9812561315, 8–10, BOOK, Øyvind Grøn, Lecture Notes on the General Theory of Relativity, 772, Lecture Notes in Physics, 2010, Springer, 978-0387881348, 86–91, JOURNAL, Muñoz, Gerardo, Jones, Preston, 2010, The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field, American Journal of Physics, 78, 4, 377–383, 1003.3022, 10.1119/1.3272719, 2010AmJPh..78..377M, BOOK, Kopeikin,S., Efroimsky, M., Kaplan, G., Relativistic Celestial Mechanics of the Solar System, 2011, John Wiley & Sons, 978-3527408566, BOOK, Padmanabhan, T., Gravitation: Foundations and Frontiers, 2010, Cambridge University Press, 978-1139485395, JOURNAL, Jones, Preston, Wanex, Lucas F., 2006, The Clock Paradox in a Static Homogeneous Gravitational Field, Foundations of Physics Letters, 19, 1, 75–85, physics/0604025, 10.1007/s10702-006-1850-3, 2006FoPhL..19...75J, BOOK, N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, 1982, Cambridge University Press, 1107392810, JOURNAL, Blum, A. S., Renn, J., Salisbury, D. C., Schemmel, M., & Sundermeyer, K., 2012, 1912: A turning point on Einstein's way to general relativity, Annalen der Physik, 524, 1, A12–A13, 10.1002/andp.201100705, 2012AnP...524A..11B,

Historical sources

{{Citation|author=Einstein, Albert|year=1908|orig-year=1907|title=Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen|journal=Jahrbuch der Radioaktivität und Elektronik|volume=4|pages=411–462|bibcode = 1908JRE.....4..411E|url=}}; English translation On the relativity principle and the conclusions drawn from it at Einstein paper project.{{Citation|author=Einstein, Albert|year=1912|title=Lichtgeschwindigkeit und Statik des Gravitationsfeldes|journal=Annalen der Physikissue=7doi=10.1002/andp.19123430704|bibcode=1912AnP...343..355E}}, English translation The Speed of Light and the Statics of the Gravitational Field at Einstein paper project.JOURNAL, Born, Max, 1909, Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips, Wikisource translation: s:Translat, Annalen der Physik, 335, 11, 1–56, 10.1002/andp.19093351102, 1909AnP...335....1B, JOURNAL, Sommerfeld, Arnold, 1910, Zur Relativitätstheorie II: Vierdimensionale Vektoranalysis, Wikisource translation: s:Translat, Annalen der Physik, 338, 14, 649–689, 10.1002/andp.19103381402, 1910AnP...338..649S,weblink BOOK, Laue, Max von, 1911, Das Relativitätsprinzip, Vieweg, Braunschweig,weblink {{Citation|author=Herglotz, Gustav|year=1910|orig-year=1909|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper|trans-title=Wikisource translation: s:Translat|journal=Annalen der Physik|volume=336|issue=2 |pages=393–415|doi=10.1002/andp.19103360208|bibcode = 1910AnP...336..393H }}BOOK, Lorentz, Hendrik Antoon, 1914, 1913, Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem (1913), B.G. Teubner, Leipzig/Berlin, s:de:Das Relativitätsprinzip (Lorentz), JOURNAL, Kottler, Friedrich, 1912, Über die Raumzeitlinien der Minkowski'schen Welt, Wikisource translation: s:Translat, Wiener Sitzungsberichte 2a, 121, 1659–1759, 2027/mdp.39015051107277, JOURNAL, Kottler, Friedrich, 1914a, Relativitätsprinzip und beschleunigte Bewegung, Annalen der Physik, 349, 13, 701–748,weblink 10.1002/andp.19143491303, 1914AnP...349..701K, JOURNAL, Kottler, Friedrich, 1914b, Fallende Bezugssysteme vom Standpunkte des Relativitätsprinzips, Annalen der Physik, 350, 20, 481–516,weblink 10.1002/andp.19143502003, 1914AnP...350..481K, JOURNAL, Kottler, Friedrich, 1916, Über Einsteins Äquivalenzhypothese und die Gravitation, Annalen der Physik, 355, 16, 955–972,weblink 1916AnP...355..955K, 10.1002/andp.19163551605, JOURNAL, Kottler, Friedrich, 1918, Über die physikalischen Grundlagen der Einsteinschen Relativitätstheorie, Annalen der Physik, 361, 14, 401–461,weblink 1918AnP...361..401K, 10.1002/andp.19183611402, BOOK, von Laue, M., 1921, Die Relativitätstheorie, Band 1, fourth edition of "Das Relativitätsprinzip”, Vieweg,weblink ; First edition 1911, second expanded edition 1913, third expanded edition 1919.BOOK, Pauli, W., 1921, Encyclopädie der mathematischen Wissenschaften, Die Relativitätstheorie, 539–776, 5.2,weblink New edition 2013: Editor: Domenico Giulini, Springer, 2013 {{ISBN|3642583555}}.JOURNAL, Karl Bollert, 1922a, Das homogene Gravitationsfeld und die Lorentztransformationen, Zeitschrift für Physik, 10, 1, 256–266, 10.1007/BF01332567, 1922ZPhy...10..256B,weblink JOURNAL, Karl Bollert, 1922b, Die Entstehung der Lorentzverkürzung und die strenge Behandlung des Uhrenparadoxons, Zeitschrift für Physik, 12, 1, 189–206, 10.1007/BF01328090, 1923ZPhy...12..189B, JOURNAL, Mohorovičić, S., 1922, Das "homogene" Gravitationsfeld und die Lorentztransformation, Zeitschrift für Physik, 11, 1, 88–92, 10.1007/BF01328404, 1922ZPhy...11...88M,weblink JOURNAL, Mohorovičić, S., 1923, Äther, Materie, Gravitation und Relativitätstheorie, Zeitschrift für Physik, 18, 1, 34–63, 10.1007/BF01327684, 1923ZPhy...18...34M, {{Citation|author=Lemaître, G.|year=1924|title=The motion of a rigid solid according to the relativity principle|journal=Philosophical Magazine |series=Series 6|volume=48|issue=283|pages=164–176|doi=10.1080/14786442408634478}}JOURNAL, Einstein, Albert, Rosen, Nathan, A Particle Problem in the General Theory of Relativity, Physical Review, 48, 1, 73–77, 1935, 10.1103/PhysRev.48.73, 1935PhRv...48...73E, JOURNAL, Møller, C., 1943, On homogeneous gravitational fields in the general theory of relativity and the clock paradox, Dan. Mat. Fys. Medd., 8, 3–25, BOOK, Møller, C., The theory of relativity, 1955, 1952, Oxford Clarendon Press,weblink

Further reading

Useful background:
  • BOOK, Boothby, William M., An Introduction to Differentiable Manifolds and Riemannian Geometry, New York: Academic Press, 1986, 0-12-116052-1, See Chapter 4 for background concerning vector fields on smooth manifolds.
  • BOOK, Frankel, Theodore, Gravitational Curvature: an Introduction to Einstein's Theory, San Francisco : W. H. Freeman, 1979, 0-7167-1062-5, See Chapter 8 for a derivation of the Fermat metric.
Rindler coordinates:
  • BOOK, Rindler, Wolfgang, Essential Relativity, New York, Van Nostrand Reinhold Co, 1969,
isbn=978-0-387-90201-2, 10.1007/978-1-4757-1135-6,
  • BOOK, Misner, Charles, Thorne, Kip S., Wheeler, John Archibald, yes, Gravitation, San Francisco: W. H. Freeman, 1973, 0-7167-0344-0, See Section 6.6.
  • BOOK, Rindler, Wolfgang, Relativity: Special, General and Cosmological, Oxford: Oxford University Press, 2001, 0-19-850836-0,
  • JOURNAL, Ni, Wei-Tou, Zimmermann, Mark, Inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer, Physical Review D, 1978, 17, 6, 1473–1476, 10.1103/PhysRevD.17.1473, 1978PhRvD..17.1473N,
Rindler horizon:
  • JOURNAL, Jacobson, Ted, Parenti, Renaud, yes, Horizon Entropy, Found. Phys., 2003, 33, 2, 323–348, 10.1023/A:1023785123428, eprint version
  • JOURNAL, Barceló, Carlos, Liberati, Stefano, Visser, Matt, yes, Analogue Gravity, Living Reviews in Relativity, gr-qc/0505065, 10.12942/lrr-2005-12, 2005LRR.....8...12B, 8, 1, 12, 2005,

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