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 c = 1, the inertial coordinates are (X,Y,Z,T),
and the hyperbolic coordinates are (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 T = 0 at position X = 1/α (with 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 T = 0 at position
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.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), 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 ==
The worldline of a body in hyperbolic motion
having constant proper acceleration
α
{\displaystyle \alpha }
in the
X
{\displaystyle X}
-direction as a function of proper time
τ
{\displaystyle \tau }
and rapidity
α
τ
{\displaystyle \alpha \tau }
can be given by
T
=
x
sinh
⁡
(
α
τ
)
,
X
=
x
cosh
⁡
(
α
τ
)
{\displaystyle T=x\sinh(\alpha \tau ),\quad
X=x\cosh(\alpha \tau )}
where
x
=
1
/
α
{\displaystyle x=1/\alpha }
is constant and
α
τ
{\displaystyle \alpha \tau }
is variable, with the worldline resembling
the hyperbola
X
2
−
T
2
=
x
2
{\displaystyle X^{2}-T^{2}=x^{2}}
. Sommerfeld showed that the equations can
be reinterpreted by defining
x
{\displaystyle x}
as variable and
α
τ
{\displaystyle \alpha \tau }
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
τ
=
t
{\displaystyle \tau =t}
, the transformation formulas between the
inertial coordinates and the hyperbolic coordinates
are consequently:
with the inverse
t
=
1
α
artanh
⁡
(
T
X
)
,
x
=
X
2
−
T
2
,
y
=
Y
,
z
=
Z
{\displaystyle t={\frac {1}{\alpha }}\operatorname
{artanh} \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
d
s
2
=
−
d
T
2
+
d
X
2
+
d
Y
2
+
d
Z
2
{\displaystyle ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2}}
, the metric in the hyperbolically accelerated
frame follows
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
0
<
X
<
∞
,
−
X
<
T
<
X
{\displaystyle {\scriptstyle 0\,<\,X\,<\,\infty
,\;-X\,<\,T\,<\,X}}
, which is often called the Rindler wedge,
if
α
{\displaystyle \alpha }
represents the proper acceleration (along
the hyperbola
x
=
1
{\displaystyle x=1}
) of the Rindler observer whose proper time
is defined to be equal to Rindler coordinate
time. To maintain this world line, the observer
must accelerate with a constant proper acceleration,
with Rindler observers closer to
x
=
0
{\displaystyle x=0}
(the Rindler horizon) having greater proper
acceleration. All the Rindler observers are
instantaneously at rest at time
T
=
0
{\displaystyle T=0}
in the inertial frame, and at this time a
Rindler observer with proper acceleration
α
i
{\displaystyle \alpha _{i}}
will be at position
X
=
1
/
α
i
{\displaystyle X=1/\alpha _{i}}
(really
X
=
c
2
/
α
i
{\displaystyle X=c^{2}/\alpha _{i}}
, but we assume units where
c
=
1
{\displaystyle c=1}
), which is also that observer's constant
distance from the Rindler horizon in Rindler
coordinates. If all Rindler observers set
their clocks to zero at
T
=
0
{\displaystyle T=0}
, then when defining a Rindler coordinate
system we have a choice of which Rindler observer's
proper time will be equal to the coordinate
time
t
{\displaystyle t}
in Rindler coordinates, and this observer's
proper acceleration defines the value of
α
{\displaystyle \alpha }
above (for other Rindler observers at different
distances from the Rindler horizon, the coordinate
time will equal some constant multiple of
their own proper time). It is a common convention
to define the Rindler coordinate system so
that the Rindler observer whose proper time
matches coordinate time is the one who has
proper acceleration
α
=
1
{\displaystyle \alpha =1}
, so that
α
{\displaystyle \alpha }
can be eliminated from the equations.
The above equation has been simplified for
c
=
1
{\displaystyle c=1}
. The unsimplified equation is more convenient
for finding the Rindler Horizon distance,
given an acceleration
α
{\displaystyle \alpha }
.
t
=
c
α
artanh
⁡
(
c
T
X
)
≈
X
≫
c
T
c
2
T
α
X
X
≈
c
2
T
α
t
≈
T
≈
t
c
2
α
{\displaystyle {\begin{aligned}t&={\frac {c}{\alpha
}}\operatorname {artanh} \left({\frac {cT}{X}}\right)\;{\overset
{X\,\gg \,cT}{\approx }}\;{\frac {c^{2}T}{\alpha
X}}\\X&\approx {\frac {c^{2}T}{\alpha t}}\;{\overset
{T\,\approx \,t}{\approx }}\;{\frac {c^{2}}{\alpha
}}\end{aligned}}}
The remainder of the article will follow the
convention of setting both
α
=
1
{\displaystyle \alpha =1}
and
c
=
1
{\displaystyle c=1}
, so units for
X
{\displaystyle X}
and
x
{\displaystyle x}
will be 1 unit
=
c
2
/
α
=
1
{\displaystyle =c^{2}/\alpha =1}
. Be mindful that setting
α
=
1
{\displaystyle \alpha =1}
light-second/second2 is very different from
setting
α
=
1
{\displaystyle \alpha =1}
light-year/year2. Even if we pick units where
c
=
1
{\displaystyle c=1}
, the magnitude of the proper acceleration
α
{\displaystyle \alpha }
will depend on our choice of units: for example,
if we use units of light-years for distance,
(
X
{\displaystyle X}
or
x
{\displaystyle x}
) and years for time, (
T
{\displaystyle T}
or
t
{\displaystyle t}
), this would mean
α
=
1
{\displaystyle \alpha =1}
light year/year2, equal to about 9.5 meters/second2,
while if we use units of light-seconds for
distance, (
X
{\displaystyle X}
or
x
{\displaystyle x}
), and seconds for time, (
T
{\displaystyle T}
or
t
{\displaystyle t}
), this would mean
α
=
1
{\displaystyle \alpha =1}
light-second/second2, or 299 792 458 meters/second2).
== Variants of transformation formulas ==
A more general derivation of the transformation
formulas is given, when the corresponding
Fermi-Walker tetrad is formulated from which
the Fermi coordinates or Proper coordinates
can be derived. Depending on the choice of
origin of these coordinates, one can derive
the metric, the time dilation between the
time at the origin
d
t
0
{\displaystyle dt_{0}}
and
d
t
{\displaystyle dt}
at point
x
{\displaystyle x}
, and the coordinate light speed
|
d
x
|
/
|
d
t
|
{\displaystyle |dx|/|dt|}
(this variable speed of light does not contradict
special relativity, because it is only an
artifact of the accelerated coordinates employed,
while in inertial coordinates it remains constant).
Instead of Fermi coordinates, also Radar coordinates
can be used, which are obtained by determining
the distance using light signals (see section
Notions of distance), by which metric, time
dilation and speed of light do not depend
on the coordinates anymore – in particular,
the coordinate speed of light remains identical
with the speed of light
(
c
=
1
)
{\displaystyle (c=1)}
in inertial frames:
== The Rindler observers ==
In the new chart (1a) with
c
=
1
{\displaystyle c=1}
and
α
=
1
{\displaystyle \alpha =1}
, it is natural to take the coframe field
d
σ
0
=
−
x
d
t
,
d
σ
1
=
d
x
,
d
σ
2
=
d
y
,
d
σ
3
=
d
z
{\displaystyle d\sigma ^{0}=-x\,dt,\;\;d\sigma
^{1}=dx,\;\;d\sigma ^{2}=dy,\;\;d\sigma ^{3}=dz}
which has the dual frame field
e
→
0
=
1
x
∂
t
,
e
→
1
=
∂
x
,
e
→
2
=
∂
y
,
e
→
3
=
∂
z
{\displaystyle {\vec {e}}_{0}={\frac {1}{x}}\partial
_{t},\;\;{\vec {e}}_{1}=\partial _{x},\;\;{\vec
{e}}_{2}=\partial _{y},\;\;{\vec {e}}_{3}=\partial
_{z}}
This defines a local Lorentz frame in the
tangent space at each event (in the region
covered by our Rindler chart, namely the Rindler
wedge). The integral curves of the timelike
unit vector field
e
→
0
{\displaystyle \scriptstyle {\vec {e}}_{0}}
give a timelike congruence, consisting of
the world lines of a family of observers called
the Rindler observers. In the Rindler chart,
these world lines appear as the vertical coordinate
lines
x
=
x
0
,
y
=
y
0
,
z
=
z
0
{\displaystyle \scriptstyle x\;=\;x_{0},\;y\;=\;y_{0},\;z\;=\;z_{0}}
. Using the coordinate transformation above,
we find that these correspond to hyperbolic
arcs in the original Cartesian chart.
As with any timelike congruence in any Lorentzian
manifold, this congruence has a kinematic
decomposition (see Raychaudhuri equation).
In this case, the expansion and vorticity
of the congruence of Rindler observers vanish.
The vanishing of the expansion tensor implies
that each of our observers maintains constant
distance to his neighbors. The vanishing of
the vorticity tensor implies that the world
lines of our observers are not twisting about
each other; this is a kind of local absence
of "swirling".
The acceleration vector of each observer is
given by the covariant derivative
∇
e
→
0
e
→
0
=
1
x
e
→
1
{\displaystyle \nabla _{{\vec {e}}_{0}}{\vec
{e}}_{0}={\frac {1}{x}}{\vec {e}}_{1}}
That is, each 
Rindler observer is accelerating in the
∂
x
{\displaystyle \scriptstyle \partial _{x}}
direction. Individually speaking, each observer
is in fact accelerating with constant magnitude
in this direction, so their world lines are
the Lorentzian analogs of circles, which are
the curves of constant path curvature in the
Euclidean geometry.
Because the Rindler observers are vorticity-free,
they are also hypersurface orthogonal. The
orthogonal spatial hyperslices are
t
=
t
0
{\displaystyle \scriptstyle t\;=\;t_{0}}
; these appear as horizontal half-planes in
the Rindler chart and as half-planes through
T
=
X
=
0
{\displaystyle \scriptstyle T\;=\;X\;=\;0}
in the Cartesian chart (see the figure above).
Setting
d
t
=
0
{\displaystyle \scriptstyle dt\;=\;0}
in the line element, we see that these have
the ordinary Euclidean geometry,
d
σ
2
=
d
x
2
+
d
y
2
+
d
z
2
,
∀
x
>
0
,
∀
y
,
z
{\displaystyle \scriptstyle d\sigma ^{2}\;=\;dx^{2}\,+\,dy^{2}\,+\,dz^{2},\;\forall
x\,>\,0,\;\forall y,\,z}
. Thus, the spatial coordinates in the Rindler
chart have a very simple interpretation consistent
with the claim that the Rindler observers
are mutually stationary. We will return to
this rigidity property of the Rindler observers
a bit later in this article.
== A "paradoxical" property ==
Note that Rindler observers with smaller constant
x coordinate are accelerating harder to keep
up. This may seem surprising because in Newtonian
physics, observers who maintain constant relative
distance must share the same acceleration.
But in relativistic physics, we see that the
trailing endpoint of a rod which is accelerated
by some external force (parallel to its symmetry
axis) must accelerate a bit harder than the
leading endpoint, or else it must ultimately
break. This is a manifestation of Lorentz
contraction. As the rod accelerates its velocity
increases and its length decreases. Since
it is getting shorter, the back end must accelerate
harder than the front. Another way to look
at it is: the back end must achieve the same
change in velocity in a shorter period of
time. This leads to a differential equation
showing, that at some distance, the acceleration
of the trailing end diverges, resulting in
the Rindler horizon.
This phenomenon is the basis of a well known
"paradox", Bell's spaceship paradox. However,
it is a simple consequence of relativistic
kinematics. One way to see this is to observe
that the magnitude of the acceleration vector
is just the path curvature of the corresponding
world line. But the world lines of our Rindler
observers are the analogs of a family of concentric
circles in the Euclidean plane, so we are
simply dealing with the Lorentzian analog
of a fact familiar to speed skaters: in a
family of concentric circles, inner circles
must bend faster (per unit arc length) than
the outer ones.
== Minkowski observers ==
It is worthwhile to also introduce an alternative
frame, given in the Minkowski chart by the
natural choice
f
→
0
=
∂
T
,
f
→
1
=
∂
X
,
f
→
2
=
∂
Y
,
f
→
3
=
∂
Z
{\displaystyle {\vec {f}}_{0}=\partial _{T},\;{\vec
{f}}_{1}=\partial _{X},\;{\vec {f}}_{2}=\partial
_{Y},\;{\vec {f}}_{3}=\partial _{Z}}
Transforming these vector fields using the
coordinate transformation given above, we
find that in the Rindler chart (in the Rinder
wedge) this frame becomes
f
→
0
=
1
x
cosh
⁡
(
t
)
∂
t
−
sinh
⁡
(
t
)
∂
x
f
→
1
=
−
1
x
sinh
⁡
(
t
)
∂
t
+
cosh
⁡
(
t
)
∂
x
f
→
2
=
∂
y
,
f
→
3
=
∂
z
{\displaystyle {\begin{aligned}{\vec {f}}_{0}&={\frac
{1}{x}}\cosh(t)\,\partial _{t}-\sinh(t)\,\partial
_{x}\\{\vec {f}}_{1}&=-{\frac {1}{x}}\sinh(t)\,\partial
_{t}+\cosh(t)\,\partial _{x}\\{\vec {f}}_{2}&=\partial
_{y},\;{\vec {f}}_{3}=\partial _{z}\end{aligned}}}
Computing the kinematic decomposition of the
timelike congruence defined by the timelike
unit vector field
f
→
0
{\displaystyle \scriptstyle {\vec {f}}_{0}}
, we find that the expansion and vorticity
again vanishes, and in addition the acceleration
vector vanishes,
∇
f
→
0
f
→
0
=
0
{\displaystyle \scriptstyle \nabla _{{\vec
{f}}_{0}}{\vec {f}}_{0}\;=\;0}
. In other words, this is a geodesic congruence;
the corresponding observers are in a state
of inertial motion. In the original Cartesian
chart, these observers, whom we will call
Minkowski observers, are at rest.
In the Rindler chart, the world lines of the
Minkowski observers appear as hyperbolic secant
curves asymptotic to the coordinate plane
x
=
0
{\displaystyle \scriptstyle x\;=\;0}
. Specifically, in Rindler coordinates, the
world line of the Minkowski observer passing
through the event
t
=
t
0
,
x
=
x
0
,
y
=
y
0
,
z
=
z
0
{\displaystyle \scriptstyle t\;=\;t_{0},\;x\;=\;x_{0},\;y\;=\;y_{0},\;z\;=\;z_{0}}
is
t
=
artanh
⁡
(
s
x
0
)
,
−
x
0
<
s
<
x
0
x
=
x
0
2
−
s
2
,
−
x
0
<
s
<
x
0
y
=
y
0
z
=
z
0
{\displaystyle {\begin{aligned}t&=\operatorname
{artanh} \left({\frac {s}{x_{0}}}\right),\;-x_{0}<s<x_{0}\\x&={\sqrt
{x_{0}^{2}-s^{2}}},\;-x_{0}<s<x_{0}\\y&=y_{0}\\z&=z_{0}\end{aligned}}}
where
s
{\displaystyle \scriptstyle s}
is the proper time of this Minkowski observer.
Note that only a small portion of his history
is covered by the Rindler chart. This shows
explicitly why the Rindler chart is not geodesically
complete; timelike geodesics run outside the
region covered by the chart in finite proper
time. Of course, we already knew that the
Rindler chart cannot be geodesically complete,
because it covers only a portion of the original
Cartesian chart, which is a geodesically complete
chart.
In the case depicted in the figure,
x
0
=
1
{\displaystyle \scriptstyle x_{0}\;=\;1}
and we have drawn (correctly scaled and boosted)
the light cones at
s
∈
{
−
1
2
,
0
,
1
2
}
{\displaystyle \scriptstyle s\,\in \,\left\{-{\frac
{1}{2}},\;0,\;{\frac {1}{2}}\right\}}
.
== The Rindler horizon ==
The Rindler coordinate chart has a coordinate
singularity at x = 0, where the metric tensor
(expressed in the Rindler coordinates) has
vanishing determinant. This happens because
as x → 0 the acceleration of the Rindler
observers diverges. As we can see from the
figure illustrating the Rindler wedge, the
locus x = 0 in the Rindler chart corresponds
to the locus T2 = X2, X > 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 ≥ 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 ≥ X if they stopped their
acceleration and crossed this line themselves)
nor could they have ever sent signals to events
with T ≤ −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.
== Geodesics ==
The geodesic equations in the Rindler chart
are easily obtained from the geodesic Lagrangian;
they are
t
¨
+
2
x
x
˙
t
˙
=
0
,
x
¨
+
x
t
˙
2
=
0
,
y
¨
=
0
,
z
¨
=
0
{\displaystyle {\ddot {t}}+{\frac {2}{x}}\,{\dot
{x}}\,{\dot {t}}=0,\;{\ddot {x}}+x\,{\dot
{t}}^{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.
From the first, third, and fourth we immediately
obtain the first integrals
t
˙
=
E
x
2
,
y
˙
=
P
,
z
˙
=
Q
{\displaystyle {\dot {t}}={\frac {E}{x^{2}}},\;\;{\dot
{y}}=P,\;\;{\dot {z}}=Q}
But from the line element we have
ϵ
=
−
x
2
t
˙
2
+
x
˙
2
+
y
˙
2
+
z
˙
2
{\displaystyle \scriptstyle \epsilon \;=\;-x^{2}\,{\dot
{t}}^{2}\,+\,{\dot {x}}^{2}\,+\,{\dot {y}}^{2}\,+\,{\dot
{z}}^{2}}
where
ϵ
∈
{
−
1
,
0
,
1
}
{\displaystyle \scriptstyle \epsilon \;\in
\;\left\{-1,\,0,\,1\right\}}
for timelike, null, and spacelike geodesics,
respectively. This gives the fourth first
integral, namely
x
˙
2
=
(
ϵ
+
E
2
x
2
)
−
P
2
−
Q
2
{\displaystyle {\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
E
2
x
2
−
P
2
−
Q
2
{\displaystyle \scriptstyle {\frac {E^{2}}{x^{2}}}\,-\,P^{2}\,-\,Q^{2}}
with nonzero
E
{\displaystyle \scriptstyle E}
, we see that the x coordinate ranges over
the interval
0
<
x
<
E
P
2
+
Q
2
{\displaystyle \scriptstyle 0\,<\,x\,<\,{\frac
{E}{\sqrt {P^{2}\,+\,Q^{2}}}}}
.
The complete seven parameter family giving
any null geodesic through any event in the
Rindler wedge, is
t
−
t
0
=
artanh
⁡
(
1
E
[
s
(
P
2
+
Q
2
)
−
E
2
−
(
P
2
+
Q
2
)
x
0
2
]
)
+
artanh
⁡
(
1
E
E
2
−
(
P
2
+
Q
2
)
x
0
2
)
x
=
x
0
2
+
2
s
E
2
−
(
P
2
+
Q
2
)
x
0
2
−
s
2
(
P
2
+
Q
2
)
y
−
y
0
=
P
s
;
z
−
z
0
=
Q
s
{\displaystyle {\begin{aligned}t-t_{0}&=\operatorname
{artanh} \left({\frac {1}{E}}\left[s\left(P^{2}+Q^{2}\right)-{\sqrt
{E^{2}-\left(P^{2}+Q^{2}\right)x_{0}^{2}}}\right]\right)+\\&\quad
\quad \operatorname {artanh} \left({\frac
{1}{E}}{\sqrt {E^{2}-(P^{2}+Q^{2})x_{0}^{2}}}\right)\\x&={\sqrt
{x_{0}^{2}+2s{\sqrt {E^{2}-(P^{2}+Q^{2})x_{0}^{2}}}-s^{2}(P^{2}+Q^{2})}}\\y-y_{0}&=Ps;\;\;z-z_{0}=Qs\end{aligned}}}
Plotting the tracks of some representative
null geodesics through a given event (that
is, projecting to the hyperslice
t
=
0
{\displaystyle \scriptstyle t\,=\,0}
), we obtain a picture which looks suspiciously
like the family of all semicircles through
a point and orthogonal to the Rindler horizon.
(See the figure.)
== 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
d
s
2
=
g
00
d
t
2
+
g
j
k
d
x
j
d
x
k
,
j
,
k
∈
{
1
,
2
,
3
}
{\displaystyle ds^{2}=g_{00}\,dt^{2}+g_{jk}\,dx^{j}\,dx^{k},\;\;j,\;k\in
\{1,2,3\}}
the Fermat metric on
t
=
0
{\displaystyle \scriptstyle t\;=\;0}
is simply
d
ρ
2
=
1
−
g
00
(
g
j
k
d
x
j
d
x
k
)
{\displaystyle d\rho ^{2}={\frac {1}{-g_{00}}}\left(g_{jk}\,dx^{j}\,dx^{k}\right)}
(where the metric coeffients are understood
to be evaluated at
t
=
0
{\displaystyle \scriptstyle t\;=\;0}
).
In the Rindler chart, the timelike translation
∂
t
{\displaystyle \scriptstyle \partial _{t}}
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:
d
ρ
2
=
1
x
2
(
d
x
2
+
d
y
2
+
d
z
2
)
,
∀
x
>
0
,
∀
y
,
z
{\displaystyle d\rho ^{2}={\frac {1}{x^{2}}}\left(dx^{2}+dy^{2}+dz^{2}\right),\;\;\forall
x>0,\;\;\forall y,z}
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
∂
t
,
∂
y
,
∂
z
,
−
z
∂
y
+
y
∂
z
{\displaystyle \partial _{t},\;\;\partial
_{y},\;\;\partial _{z},\;\;-z\,\partial _{y}+y\,\partial
_{z}}
(time translation, spatial translations orthogonal
to the direction of acceleration, and spatial
rotation orthogonal to the direction of acceleration)
plus six more:
exp
⁡
(
±
t
)
(
y
x
∂
t
±
[
y
∂
x
−
x
∂
y
]
)
exp
⁡
(
±
t
)
(
z
x
∂
t
±
[
z
∂
x
−
x
∂
z
]
)
exp
⁡
(
±
t
)
(
1
x
∂
t
±
∂
x
)
{\displaystyle {\begin{aligned}&\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{aligned}}}
(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
∂
T
{\displaystyle \scriptstyle \partial _{T}}
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.
The first is the one we have tacitly employed
above: the induced Riemannian metric on the
spatial hyperslices
t
=
t
0
{\displaystyle \scriptstyle t\;=\;t_{0}}
. 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, let us consider a pair of Rindler
observers with coordinates
x
=
x
0
,
y
=
0
,
z
=
0
{\displaystyle \scriptstyle x\;=\;x_{0},\;y\;=\;0,\;z\;=\;0}
and
x
=
x
0
+
h
,
y
=
0
,
z
=
0
{\displaystyle \scriptstyle x\;=\;x_{0}\,+\,h,\;y\;=\;0,\;\;z\;=\;0}
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
d
y
=
d
z
=
0
{\displaystyle \scriptstyle dy\;=\;dz\;=\;0}
in the Rindler line element, we readily obtain
the equation of null geodesics moving in the
direction of acceleration:
t
−
t
0
=
log
⁡
(
x
x
0
)
{\displaystyle t-t_{0}=\log \left({\frac {x}{x_{0}}}\right)}
Therefore, the radar distance between these
two observers is given by
x
0
log
⁡
(
1
+
h
x
0
)
=
h
−
h
2
2
x
0
+
O
(
h
3
)
{\displaystyle x_{0}\,\log \left(1+{\frac
{h}{x_{0}}}\right)=h-{\frac {h^{2}}{2\,x_{0}}}+O\left(h^{3}\right)}
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
h
+
1
x
0
+
O
(
h
3
)
{\displaystyle \scriptstyle h\,+\,{\frac {1}{x_{0}}}\,+\,O\left(h^{3}\right)}
. Therefore, in the case of a trailing observer
estimating distance to a leading observer
(the case
h
>
0
{\displaystyle \scriptstyle h\,>\,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.
== History ==
=== Overview ===
Kottler-Møller and Rindler coordinatesAlbert
Einstein (1907) studied the effects within
a uniformly accelerated frame, obtaining equations
for coordinate dependent time dilation and
speed of light equivalent to (2c), and in
order to make the formulas independent of
the observer's origin, he obtained time dilation
(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" (German: hyperbolisch beschleunigtes
Bezugsystem) equivalent to (2d). Born's work
was further elaborated by Arnold Sommerfeld
(1910) and Max von Laue (1911) who both obtained
(2d) using imaginary numbers, which was summarized
by Wolfgang Pauli (1921) who besides coordinates
(2d) also obtained metric (2e) using imaginary
numbers. Einstein (1912) studied a static
gravitational field and obtained the Kottler-Møller
metric (2b) as well as approximations to formulas
(2a) using a coordinate dependent speed of
light. Hendrik Lorentz (1913) obtained coordinates
similar to (2d, 2e, 2f) while studying Einstein's
equivalence principle and the uniform gravitational
field.
A detailed description was given by Friedrich
Kottler (1914), who formulated the corresponding
orthonormal tetrad, transformation formulas
and metric (2a, 2b). Also Karl Bollert (1922)
obtained the metric (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 (2a, 2b). Albert Einstein and Nathan
Rosen (1935) described (2d, 2e) as the "well
known" expressions for a homogeneous gravitational
field. After Christian Møller (1943) obtained
(2a, 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 (2d, 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.
HorizonBorn (1909) showed that the inner points
of a Born rigid body in hyperbolic motion
can only be in the region
X
/
(
X
2
−
T
2
)
>
0
{\displaystyle X/\left(X^{2}-T^{2}\right)>0}
. Sommerfeld (1910) defined that the coordinates
allowed for the transformation between inertial
and hyperbolic coordinates must satisfy
T
<
X
{\displaystyle T<X}
. Kottler (1914) defined this region as
X
2
−
T
2
>
0
{\displaystyle X^{2}-T^{2}>0}
, and pointed out the existence of a "border
plane" (German: Grenzebene)
c
2
/
α
+
x
{\displaystyle c^{2}/\alpha +x}
, beyond which no signal can reach the observer
in hyperbolic motion. This was called the
"horizon of the observer" (German: Horizont
des Beobachters) by Bollert (1922). Rindler
(1966) demonstrated the relation between such
a horizon and the horizon in Kruskal coordinates.
Radar coordinatesUsing Bollert's formalism,
Stjepan Mohorovičić (1922) made a different
choice for some parameter and obtained metric
(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
(2b, now called Kottler-Møller metric) is
incorrect, which was rebutted by Bollert (1922).
Metric (2h) was rediscovered by Harry Lass
(1963), who also gave the corresponding coordinates
(2g) which are sometimes called "Lass coordinates".
Metric (2h), as well as (2a, 2b), was also
derived by Fritz Rohrlich (1963). Eventually,
the Lass coordinates (2g, 2h) were identified
with Radar coordinates by Desloge & Philpott
(1987).
=== Table with historical formulas ===
== See also ==
Bell's spaceship paradox, for a sometimes
controversial subject often studied using
Rindler coordinates.
Born coordinates, for another important coordinate
system adapted to the motion of certain accelerated
observers in Minkowski spacetime.
Congruence (general relativity)
Ehrenfest paradox, for a sometimes controversial
subject often studied using Born coordinates.
Frame fields in general relativity
General relativity resources
Milne model
Raychaudhuri equation
Unruh effect
