Rindler coordinates

Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a hyperbolically accelerated reference frame

Regarding 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),

|\right)\\

x & =\sqrt{\left(X+\frac{1}{\alpha}\right)^{2}-T^{2-\frac{1}{\alpha}\\

y & =Y\\

z & =Z

\end{align}

\end{array}</math>|

----

!Rindler coordinates

The Fermat metric

The fact that in the Rindler chart, the projections of null geodesics into any spatial hyperslice for the Rindler observers are simply semicircular arcs can be verified directly from the general solution just given, but there is a very simple way to see this. A static spacetime is one in which a vorticity-free timelike Killing vector field can be found. In this case, we have a uniquely defined family of (identical) spatial hyperslices orthogonal to the corresponding static observers (who need not be inertial observers). This allows us to define a new metric on any of these hyperslices which is conformally related to the original metric inherited from the spacetime, but with the property that geodesics in the new metric (note this is a Riemannian metric on a Riemannian three-manifold) are precisely the projections of the null geodesics of spacetime. This new metric is called the Fermat metric, and in a static spacetime endowed with a coordinate chart in which the line element has the form

:<math> ds^2 = g_{00} \, dt^2 + g_{jk} \, dx^j \, dx^k,\;\; j,\; k \in \{1, 2, 3\} </math>

the Fermat metric on <math> t = 0</math> is simply

:<math> d\rho^2 = \frac{1}{-g_{00\left(g_{jk} \, dx^j \, dx^k\right)</math>

(where the metric coeffients are understood to be evaluated at <math> t = 0</math>).

In the Rindler chart, the timelike translation <math> \partial_t</math> is such a Killing vector field, so this is a static spacetime (not surprisingly, since Minkowski spacetime is of course trivially a static vacuum solution of the Einstein field equation). Therefore, we may immediately write down the Fermat metric for the Rindler observers:

:<math> d\rho^2 = \frac{1}{x^2}\left(dx^2 + dy^2 + dz^2\right),\;\; \forall x > 0,\;\; \forall y, z</math>

But this is the well-known line element of hyperbolic three-space H3 in the upper half space chart. This is closely analogous to the well known upper half plane chart for the hyperbolic plane H2, which is familiar to generations of complex analysis students in connection with conformal mapping problems (and much more), and many mathematically minded readers already know that the geodesics of H2 in the upper half plane model are simply semicircles (orthogonal to the circle at infinity represented by the real axis).

Symmetries

Since the Rindler chart is a coordinate chart for Minkowski spacetime, we expect to find ten linearly independent Killing vector fields. Indeed, in the Cartesian chart we can readily find ten linearly independent Killing vector fields, generating respectively one parameter subgroups of time translation, three spatials, three rotations and three boosts. Together these generate the (proper isochronous) Poincaré group, the symmetry group of Minkowski spacetime.

However, it is instructive to write down and solve the Killing vector equations directly. We obtain four familiar looking Killing vector fields

:<math> \partial_t, \; \; \partial_y, \; \; \partial_z, \; \; -z \, \partial_y + y \, \partial_z </math>

(time translation, spatial translations orthogonal to the direction of acceleration, and spatial rotation orthogonal to the direction of acceleration) plus six more:

:<math>\begin{align}

&\exp(\pm t) \, \left( \frac{y}{x} \, \partial_t \pm \left[ y \, \partial_x - x \, \partial_y \right] \right)\\

&\exp(\pm t) \, \left( \frac{z}{x} \, \partial_t \pm \left[ z \, \partial_x - x \, \partial_z \right] \right)\\

&\exp(\pm t) \, \left( \frac{1}{x} \, \partial_t \pm \partial_x \right)

\end{align}</math>

(where the signs are chosen consistently + or −). We leave it as an exercise to figure out how these are related to the standard generators; here we wish to point out that we must be able to obtain generators equivalent to <math> \partial_T</math> in the Cartesian chart, yet the Rindler wedge is obviously not invariant under this translation. How can this be? The answer is that like anything defined by a system of partial differential equations on a smooth manifold, the Killing equation will in general have locally defined solutions, but these might not exist globally. That is, with suitable restrictions on the group parameter, a Killing flow can always be defined in a suitable local neighborhood, but the flow might not be well-defined globally. This has nothing to do with Lorentzian manifolds per se, since the same issue arises in the study of general smooth manifolds.

Notions of distance

One of the many valuable lessons to be learned from a study of the Rindler chart is that there are in fact several distinct (but reasonable) notions of distance which can be used by the Rindler observers.

frame|left|Operational meaning of the radar distance between two Rindler observers (navy blue vertical lines). The Rindler horizon is shown at left (red vertical line). The world line of the radar pulse is also depicted, together with the (properly scaled) light cones at events A, B, C.

The first is the one we have tacitly employed above: the induced Riemannian metric on the spatial hyperslices <math> t = t_0</math>. We will call this the ruler distance since it corresponds to this induced Riemannian metric, but its operational meaning might not be immediately apparent.

From the standpoint of physical measurement, a more natural notion of distance between two world lines is the radar distance. This is computed by sending a null geodesic from the world line of our observer (event A) to the world line of some small object, whereupon it is reflected (event B) and returns to the observer (event C). The radar distance is then obtained by dividing the round trip travel time, as measured by an ideal clock carried by our observer.

(In Minkowski spacetime, fortunately, we can ignore the possibility of multiple null geodesic paths between two world lines, but in cosmological models and other applications things are not so simple. We should also caution against assuming that this notion of distance between two observers gives a notion which is symmetric under interchanging the observers.)

In particular, consider a pair of Rindler observers with coordinates <math> x = x_0, \; y = 0,\; z = 0</math> and <math> x = x_0 + h, \; y = 0,\; z = 0</math> respectively. (Note that the first of these, the trailing observer, is accelerating a bit harder, in order to keep up with the leading observer). Setting <math> dy = dz = 0</math> in the Rindler line element, we readily obtain the equation of null geodesics moving in the direction of acceleration:

:<math> t - t_0 = \log\left(\frac{x}{x_0}\right) </math>

Therefore, the radar distance between these two observers is given by

:<math> x_0 \, \log \left(1 + \frac{h}{x_0} \right) = h - \frac{h^2}{2 \, x_0} + O \left( h^3 \right) </math>

This is a bit smaller than the ruler distance, but for nearby observers the discrepancy is negligible.

A third possible notion of distance is this: our observer measures the angle subtended by a unit disk placed on some object (not a point object), as it appears from his location. We call this the optical diameter distance. Because of the simple character of null geodesics in Minkowski spacetime, we can readily determine the optical distance between our pair of Rindler observers (aligned with the direction of acceleration). From a sketch it should be plausible that the optical diameter distance scales like <math display="inline"> h + \frac{1}{x_0} + O \left( h^3 \right) </math>. Therefore, in the case of a trailing observer estimating distance to a leading observer (the case <math> h > 0</math>), 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 essentially 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.

History

Overview

;Kottler–Møller and Rindler coordinates

Albert Einstein (1907) who formulated the corresponding orthonormal tetrad, transformation formulas and metric (, ). Also Karl Bollert (1922) Sommerfeld (1910) defined that the coordinates allowed for the transformation between inertial and hyperbolic coordinates must satisfy <math>T<X</math>. Kottler (1914) defined this region as <math>X^{2}-T^{2}>0</math>, and pointed out the existence of a "border plane" () <math>c^2/\alpha+x</math>, beyond which no signal can reach the observer in hyperbolic motion. This was called the "horizon of the observer" () by Bollert (1922). Rindler (1966) Metric () was rediscovered by Harry Lass (1963),

|<math>{\scriptstyle \begin{matrix}\sigma=\tau\left(1+\frac{\gamma\xi}{c^{2\right)\\

\sigma=\tau e^{\gamma\xi/c^{2\\

c\left(1+\frac{\gamma\xi}{c^{2\right)

\end{matrix</math>

!Born (1909)

|<math>{\scriptstyle \begin{matrix}x=-q\xi,\ y=\eta,\ z=\zeta,\ t=\frac{p}{c^{2\xi\\

\left(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</math>

!Herglotz (1909)

|<math>{\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</math>

!Sommerfeld (1910)

|<math>\scriptstyle\begin{align}

x & =r\cos\varphi\\

y & =y'\\

z & =z'\\

l & =r\sin\varphi\\

\varphi & =i\psi,\ l =ict

\end{align}</math>

!von Laue (1911)

|<math>\scriptstyle\begin{align}

X & =R\cos\varphi\\

L & =R\sin\varphi\\

R^{2} & =X^{2}+L^{2}\\

\tan\varphi & =\frac{L}{X}

\end{align}</math>

!Einstein (1912)

|<math>{\scriptstyle \begin{matrix}d\xi^{2}-d\tau^{2}=dx^{2}-c^{2}dt^{2}\\

\boldsymbol{\downarrow}\\

c=c_{0}+ax\\

\boldsymbol{\downarrow}\\

\begin{align}\xi & =x+\frac{ac}{2}t^{2}\\

\eta & =y\\

\zeta & =z\\

\tau & =ct

\end{align}

\end{matrix</math>

!Kottler (1912)

|<math>{\scriptstyle \begin{align}x^{(1)} & =x_{0}^{(1)}\\

x^{(2)} & =x_{0}^{(2)}\\

x^{(3)} & =b\cos i\varphi\\

x^{(4)} & =b\sin i\varphi

\end{align</math>

!Lorentz (1913)

|<math>{\scriptstyle \begin{matrix}dc=\frac{g}{c}dz\\

\hline \begin{align}z & =a\left(z'-z_{0}^{\prime}\right)\\

ct & =b\left(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'=k\left(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}

\end{align}

\end{matrix</math>

|valign="top"|

{| class="wikitable"

!Kottler (1914a)

|<math>{\scriptstyle \begin{matrix}\begin{align}x^{(1)} & =x_{0}^{(1)}\\

x^{(2)} & =x_{0}^{(2)}\\

x^{(3)} & =b\cos iu\\

x^{(4)} & =b\sin iu

\end{align}

\boldsymbol{\downarrow}\\

ds^{2}=-c^{2}d\tau^{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}\cdot\frac{1}{c}

\end{matrix</math>

!Kottler (1914b)

|<math>\scriptstyle\begin{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)\cosh\mathfrak{u}\\

cT & =\left(b+\mathfrak{Z}'\right)\sinh\mathfrak{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}T\\

\mathfrak{Y}' & =Y_{0}-y_{0}+q_{y}T\\

b+\mathfrak{Z}' & =\sqrt{\left(Z_{0}+q_{x}T\right)^{2}-c^{2}T^{2\\

c\mathfrak{T}' & =b\operatorname{artanh}\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}T\right)\\

\boldsymbol{\downarrow}\\

dS^{2}=(d\mathfrak{X}')^{2}+(d\mathfrak{Y}')^{2}+(d\mathfrak{Z}')^{2}-c^{2}\left(\frac{b+\mathfrak{Z}'}{b^{2\right)^{2}(d\mathfrak{T}')^{2}

\end{matrix}</math>

!Kottler (1916, 1918)

|<math>\scriptstyle\begin{matrix}\begin{align}

x & =x'\\

y & =y'\\

\frac{c^{2{\gamma}+z & =\left(\frac{c^{2{\gamma}+z'\right)\cosh\frac{\gamma t'}{c}\\

ct & =\left(\frac{c^{2{\gamma}+z'\right)\sinh\frac{\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}</math>

|valign="top"|

{| class="wikitable"

!Pauli (1921)

|<math>\scriptstyle\begin{matrix}\begin{align}

x^{1} & =\varrho\cos\varphi\\

x^{4} & =\varrho\sin\varphi

\end{align}\\

\boldsymbol{\downarrow}\\

ds^{2}=\left(d\xi^{1}\right)^{2}+\left(d\xi^{2}\right)^{2}+\left(d\xi^{3}\right)^{2}+\left(\xi^{1}\right)^{2}\left(d\xi^{4}\right)^{2}\\

\left(\xi^{(1)}=\varrho,\ \xi^{(2)}=x^{(2)},\ \xi^{(3)}=x^{(3)},\ \xi^{(4)}=\varphi\right)

\end{matrix}</math>

!Bollert (1922)

|<math>{\scriptstyle \begin{matrix}ds^{2}=c^{2}\left(1+\frac{\gamma_{0}x}{c^{2\right)d\tau^{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_{11V'=0\\

\left(g_{22}=-1,\ g_{11}=-1,\ V=0,\ V=ax+b\right)\\

\boldsymbol{\downarrow}\\

ds^{2}=dx_{4}^{2}(ax+b)^{2}-dx^{2}-dy^{2}-dz^{2}

\end{matrix</math>

!Mohorovičić (1922, 1923); Bollert (1922b)

|<math>{\scriptstyle \begin{matrix}\text{Mohorovičić (1922):}\\

g_{11}=g_{44}=V^{2},\ VV-V'^{2}=0,\ V\left(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</math>

!Lemaître (1924)

|<math>{\scriptstyle \begin{matrix}\begin{align}1+g\xi= & (1+gx)\cosh gt\\

g\tau= & (1+gx)\sinh gt

\end{align}

\boldsymbol{\downarrow}\\

ds^{2}=-dx^{2}-dy^{2}-dz^{2}+(1+gx)^{2}dt^{2}

\end{matrix</math>

!Einstein & Rosen (1935)

|<math>\scriptstyle\begin{matrix}\begin{align}

\xi_{1} & =x_{1}\cosh\alpha x_{4}\\

\xi_{2} & =x_{2}\\

\xi_{3} & =x_{3}\\

\xi_{4} & =x_{1}\sinh\alpha 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}</math>

!Møller (1952)

|<math>\scriptstyle\begin{matrix}\alpha_{ik}=\left(\begin{matrix}U_{4}/ic & 0 & 0 & iU_{1}/c\\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0\\

U_{1}/ic & 0 & 0 & U_{4}/ic

\end{matrix}\right)\\

U_{i}=\left(c\sinh\frac{g\tau}{c},\ 0,0,\ ig\cosh\frac{g\tau}{c}\right)\\

\boldsymbol{\downarrow}\\

X_{i}=\mathbf{f}_{i}(t)+x^{\prime\kappa}\alpha_{\kappa i}(\tau)\\

\boldsymbol{\downarrow}\\

\begin{align}

X & =\frac{c^{2{g}\left(\cosh\frac{gt}{c}-1\right)+x\cosh\frac{gt}{c}\\

Y & =y\\

Z & =z\\

T & =\frac{c}{g}\sinh\frac{gt}{c}+x\frac{\sinh\frac{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}</math>

References

</references>

Historical sources

</references>

Rindler coordinates

The Fermat metric

Symmetries

Notions of distance

Generalization to curved spacetimes

History

Overview

See also

References

Historical sources

Further reading