Speed of Light in an Accelerated System
Tom Roberts
Date: June 23, 2000 [repost from December 13, 1998]
Author: Tom Roberts, tjro...@lucent.com
INTRODUCTION
------------
This article will present an exact, no-compromises calculation of two physically-
realizable measurements of the speed of light in an accelerated system. The
specific accelerated system used is called uniform acceleration, or hyperbolic
motion. This type of motion occurs when the origin of the system has a constant
proper acceleration. This is called hyperbolic motion as the origin of the
accelerated system moves along a hyperbola in inertial coordinates.
Several interesting points will be shown:
1. The speed of light measured with standard clocks is not isotropic, except in
the limit as the distance goes to zero and the measurement is made simultaneously
with synchronizing the clocks.
2. For a light source ahead of the detector, an infinite speed will be measured if
one waits long enough after synchronizing the clocks.
3. For a light source far enough behind the detector, the light pulse never reaches
the detector -- the detector actually outruns it; this is analogous to the event
horizon of a black hole, in that one region of spacetime is not causally connected
to another region.
4. Standard clocks which are accelerated will not remain synchronized and will tick
at different rates when they are at different "heights".
5. Coordinate clocks will remain synchronized, and can be E-synched. If they are used
to measure the speed of light, it is locally isotropic with a value other than c,
but over a finite distance it is neither isotropic nor c.
A few months ago there was a long and protracted argument in this newsgroup about the
claim that E-synched clocks always yield an isotropic speed of light. Item 5 is a
specific counterexample to this claim. Remarkably, accelerated clocks slaved to each
other with light signals will measure anisotropy in the speed of light.
Items 2 and 3 are consequences of the fact that in differential geometry a given
coordinate system does not necessarily cover all of the manifold. The accelerated
coordinates presented below are incapable of covering all of spacetime. This also
has been a recent subject of debate in this newsgroup.
Item 4 is applicable to the current discussion on clock rates and rotating observers,
which has all the stigmata of becoming a protracted debate.
[Author's note: the above discussions were current in December 1998.]
An accelerated observer is discussed in most intermediate and advanced relativity
texts, in particular:
R.A.Mould, _Basic_Relativity_, Chapter 8.
W.Rindler, _Essential_Relativity_.
Misner, Thorne, and Wheeler [MTW], _Gravitation_, Chapter 6.
I will use results and exercises from the latter, but because of the limitations of
ASCII will use different notation (my x' is their Greek xi^1, etc.). I will also use
units in which c=1. This entire article considers only a flat spacetime, and is a
calculation using SR; occasionally the relationship to GR concepts will be mentioned.
All measurements consider light propagating in vacuum.
PHYSICAL DESCRIPTION AND COORDINATE DEFINITIONS
-----------------------------------------------
I assume an inertial frame using Cartesian coordinates (t,x,y,z) is valid throughout
the region of interest. The accelerated system has Cartesian coordinates (t',x',y',z')
with x' parallel to x, etc. The origin of this system accelerates with constant proper
acceleration g in the +x direction. At time t=0=t' the two coordinate systems coincide,
and all primed coordinate clocks are synchronized with their instantaneously-comoving
and collocated unprimed inertial clocks (which are of course synchronized in the
unprimed inertial frame).
The accelerated (primed) coordinates are determined by the conditions that the origin
have a constant proper acceleration g, that the time coordinate at the origin be
determined by a standard clock located there, that the spatial coordinates be parallel
to their inertial counterparts, and that the lengths of objects at rest in the
accelerated system remain the same as their proper length in an inertial frame (i.e.
the same as measured in the unprimed system at time t=0). This latter condition
requires that the primed coordinates have an Euclidean 3-metric with spatial
coordinates orthogonal to the time coordinate. That is sufficient to determine the
coordinate transformations between the primed and unprimed coordinates (see MTW 6.6;
Mould Apndx E). Only the primed to unprimed transforms are needed in this article:
t = (1/g + x') sinh(gt')
x = (1/g + x') cosh(gt') - 1/g (eq 1)
y = y'
z = z'
The presence of the hyperbolic sinh and cosh indicate why this is called hyperbolic
motion. In the (x,t) plane the point x'=0 traces out a hyperbola asymptotic to the
past and future light cones of the point x=-1/g. This point is called the "focus" of
the acceleration, and has some remarkable properties (see below). If g is one earth-
surface gravity, this point is about one light-year from the origin.
Note that t' is the proper time of a standard clock at the origin of the primed
coordinates (but as a coordinate it is valid everywhere in the coordinate system).
In these coordinates the metric is:
ds^2 = -(1 + gx')^2 dt'^2 + dx'^2 + dy'^2 + dz'^2 (eq 2)
>From this it is clear that a standard clock at rest in these coordinates will tick
at a rate (1+gx') faster than does its collocated coordinate clock.
Nothing unusual has happened to rulers at rest in these coordinates, as the
coordinates are orthogonal and the spatial metric is Euclidean. But clocks at
rest in the primed system do behave differently from those in an inertial frame.
A standard clock at rest with fixed x'=H accumulates proper time at the rate
(1+gH) compared to the collocated coordinate clock, which is the same as the
standard clock at x'=0. If H>0 the clock at x'=H "ticks faster" than the clock at
the origin and we say it is "higher"; if H<0 it "ticks slower" than the clock at
the origin and we say it is "lower". The inertial observer would call these two
clocks "ahead" and "behind", respectively.
"ticks faster" is a rather nebulous concept, as it is not an observer-
independent (coordinate-independent) relationship. I use it in the sense
that one measures the elapsed proper time between two events on the common
worldline and then takes the limit of the ratio (std_clock/coord_clock) as
the events approach each other (to be collocated implies that both are at
rest in these coordinates). This is equivalent to setting up the integral
for the proper-time of the standard clock using the coordinates, and then
examining the integrand -- it is sqrt(g_00) of the metric tensor (eq 2 for
these accelerated coordinates).
>From the metric it is easy to compute the coordinate speed of light. Coordinate
clocks remain in synch, and are paced by the standard clock at the origin. The
simplest way to do this is to send a periodic light signal (in vacuum) from the
origin clock and have the other clocks be paced by this signal. One can then E-synch
them in the usual way and they will remain in synch (see Mould 8.6).
The usual way is to have clock A send a light ray to clock B where it is
immediately reflected back to A. B is then set so it would have shown the
midpoint of the two times on A when the ray was reflected.
Note also the usual relationship between coordinate clocks and standard clocks:
coordinate time is integrable (i.e. has a definite value at each point in the manifold;
it is a field on the manifold), and coordinate clocks remain mutually synchronized;
standard clocks do not remain synchronized, but do indicate an invariant.
Using coordinate clocks and distances, the _local_ speed of light is easily obtained
from the metric (ds^2=0 for a light ray):
c_local_coordinates = sqrt(dx'^2+dy'^2+dz'^2)/dt' = (1 + gx') (eq 3)
This is isotropic locally, but is not c (= 1) unless g=0 or the measurement is made at
the origin. Clearly this is merely a consequence of the difference between standard
clocks and coordinate clocks (coordinate rulers and standard rulers are identical).
Looking at the metric again, we can use standard clocks and distances instead of
coordinate ones. Because of the form of the metric in these coordinates, the result
of a _local_ measurement is:
c_local_standard = c_local_coordinates (coord_time/proper_time) = 1 (eq 4)
This is isotropic and independent of position, and is the standard value. This is
directly related to the fact that in GR any infinitesimal region of spacetime has
local Minkowski geometry with the speed of light = 1 (measured with standard clocks
and rulers).
But these are only _local_ values. It is not really feasible to actually perform
such an infinitesimal measurement using real tools (e.g. accuracy is completely
lost as path-length approaches 0); clock synchronization is also an issue. It is
interesting and instructive to calculate a realistic measurement of the speed of
light using standard clocks separated by a finite distance. In general one needs
to integrate the metric over the path to be used for the measurement, but the
presence of the inertial frame permits a shortcut: we can use the above transform
equations to relate the experimental measurement in the accelerated system to one
in the inertial frame where we know the speed of light (= 1).
Note the difference between standard clocks and coordinate clocks:
A standard clock is of the "standard" construction, and it ticks off the proper
time of its worldline. All standard clocks are identical.
A coordinate clock is _not_ of "standard" construction, but has been modified
so it ticks off the coordinate time of whatever coordinate system it represents,
at its current location in that coordinate system. Coordinate clocks are in
general not identical with each other, or with standard clocks.
In inertial frames this distinction is not needed, as in an inertial frame any
standard clock at rest in the frame is also a coordinate clock (this is basically
the statement that in an inertial frame the metric tensor has only diagonal
components with value +/-1). But in differential geometry this is not so, and the
distance between two points need not be the same as the coordinate difference
between the same points (the metric tensor components can have values other than
+/-1, and can be non-diagonal).
Note, for instance, that the GPS makes this distinction. Its satellite clocks are
physically different from earthbound clocks in that their tick rate has been
modified for their different gravitational potential (plus other minor corrections).
In The GPS, all clocks act as coordinate clocks, in the ECI coordinate system (in
which coordinate clocks do not respond to the gravitation of the earth). This can
only be done over a finite region of spacetime, and then only over distances small
compared to the curvature. The reason they _must_ use coordinate clocks and not
standard clocks is that the latter cannot remain in synch -- the accelerated system
of this article shares this property (see below).
A PHYSICALLY-REALIZABLE MEASUREMENT USING STANDARD CLOCKS
---------------------------------------------------------
Consider two standard clocks separated by a distance L and both at rest in the
accelerated system with one located at the origin, and the other along a line
with angle theta wrt the acceleration. Let the origin clock be equipped with a
light detector and the other with a light source. Let all equipment be in the
z'=0 plane, and synchronize both clocks with their instantaneously-comoving and
collocated inertial clocks at t=0=t'.
The speed of light measured with this equipment is clearly L / (Td - Te), where
L = proper distance between the two clocks.
Td = time indicated on the detector clock when the ray is detected.
Te = time indicated on the emitter clock when the ray is emitted.
[note upper-case T here]
This is just a standard 1-way measurement of the speed of light in the accelerated
system.
>From the above transform equations (eq 1) and the description, we have:
Inertial coordinates of the detection of the light ray (x'=0,y'=0):
td = (1/g) sinh(gt')
xd = (1/g) cosh(gt') - 1/g t' = time of detection (eq 5)
yd = 0
Inertial coordinates of the emission of the light ray:
te = (1/g + L cos(theta)) sinh(gt')
xe = (1/g + L cos(theta)) cosh(gt') - 1/g t' = time of emission (eq 6)
ye = L sin(theta)
[note lower-case t here]
But we want to relate the t' values to the proper time indicated on the two clocks.
>From MTW exercise 6.6, we have:
d\tau/dt' = 1 + gx' \tau = proper time of a standard clock (eq 7)
As our clocks have fixed x', this is easily integrated; remembering the synch:
Td = t' of detection (eq 8)
Te = (t' of emission) (1 + gx') x' = point of emission = L cos(theta)
These are trivially substituted in the above equations for the two t' values.
Now that we know the inertial coordinates of the emission and detection events given
the corresponding clock readings, it is a simple procedure to substitute them into the
usual formula for a light ray in the unprimed coordinates:
(xd - xe)^2 + (yd - ye)^2 = (td - te)^2 (eq 9)
This is now merely an exercise in algebra. What we really want is the measured time
difference; a half-page of algebra yields:
Td - Te = (1/g) arccosh(1 + (1/2)(gL)^2/f) - gL cos(theta) Te / f (eq 10)
where f == (1 + gL cos(theta))
[always take the positive value of the arccosh]
Remember that the measured speed of light is:
c_measured = L / (Td - Te) (eq 11)
IMPLICATIONS OF THE STANDARD-CLOCK MEASUREMENT
----------------------------------------------
>From eq 10 and 11 we have
lim[g->0] c_measured = 1 (eq 12)
So in the inertial case c_measured is the usual value (= 1).
For cos(theta) > 0, there is a value of Te for which Td-Te = 0. This implies that
c_measured is infinite in this case. What has happened is that we waited so long
after synchronizing the two clocks that they have drifted apart (due to their
different "heights") so much that the measurement is basically meaningless. For
cos(theta)=1 and gL << 1, this time is about Te ~ 1/g; for the earth's surface
acceleration due to gravity this is about 1 year. Note that this time is
independent of L; for larger L the clocks diverge faster, but have further to go.
But note:
lim[L->0] c_measured = 1 / (1 - gTe cos(theta)) (eq 13)
which is not isotropic and is not the usual value unless Te=0 or cos(theta)=0.
And for a large enough Te (~ 1/g) it still blows up -- even for an infinitesimal
separation these accelerated standard clocks cannot maintain consistency for times
longer than ~1/g.
If (gL cos(theta)) < -1, there is no solution. We also have:
lim[L->-1/g] c_measured = 0 (eq 14)
What is happening is that as the emitter approaches the focus of the accelerated
system (from above), the detector at the origin has enough time to approach
lightspeed (wrt the inertial frame) before the light ray hits it, and it takes
extraordinarily long to reach the detector. For an emitter to the left of the
focus (x=-1/g), the detector will actually outrun the light ray and the measurement
never completes. This is an horizon, and is directly analogous to the event horizon
of a black hole. The entire region x < -1/g (== x' < -1/g) is causally unrelated to
the region x' >= 0 (that is, events in the two regions cannot communicate in either
direction via light rays). Note that this is dependent on the unrealistic assumption
that the constant acceleration goes on forever; cease accelerating and the horizon
disappears. Note that the transform (eq 1) is singular at x' = -1/g; this is a
coordinate singularity similar to that in Schwarzschild coordinates at a black
hole's horizon; no physical singularity exists there.
For cos(theta) > 0, c_measured increases with increasing Te. For cos(theta) < 0,
c_measured decreases with increasing Te. So the consequences of waiting after the
clocks are synchronized vary with position. This is directly related to the way
standard clocks tick relative to coordinate clocks in the accelerated system. For
cos(theta) = 0 the two clocks are of equal "height", and c_measured is independent
of Te.
A PHYSICALLY-REALIZABLE MEASUREMENT USING COORDINATE CLOCKS
-----------------------------------------------------------
The above measurement is physically realizable, but may not be the simplest
experiment to perform. It may be easier or more accurate to use a single clock
and to pace the other clock from the first. That is essentially how the
coordinate clocks work, so I will consider a physical situation similar to the
previous measurement, but will use coordinate clocks rather than standard clocks.
Consider two coordinate clocks separated by a distance L and both at rest in the
accelerated system with one located at the origin, and the other along a line with
angle theta wrt the acceleration. Let the origin clock be equipped with a light
detector and the other with a light source. Let all equipment be in the z'=0 plane,
and synchronize both clocks with their instantaneously-comoving and collocated
inertial clocks at t=0=t'.
The speed of light measured with this equipment is clearly L / (Td - Te), where
L = proper distance between the two clocks.
Td = coordinate time when the ray is detected.
Te = coordinate time when the ray is emitted.
[note upper-case T here]
This is just a standard 1-way measurement of the speed of light in the accelerated
system.
>From the above transform equations (eq 1) and the description, we have:
Inertial coordinates of the detection of the light ray (x'=0,y'=0):
td = (1/g) sinh(gTd)
xd = (1/g) cosh(gTd) - 1/g (eq 15)
yd = 0
Inertial coordinates of the emission of the light ray:
te = (1/g + L cos(theta)) sinh(gTe)
xe = (1/g + L cos(theta)) cosh(gTe) - 1/g (eq 16)
ye = L sin(theta)
Now that we know the inertial coordinates of the emission and detection events given
the corresponding clock readings, it is a simple procedure to substitute them into
the usual formula for a light ray in the unprimed coordinates:
(xd - xe)^2 + (yd - ye)^2 = (td - te)^2 (eq 17)
This is now merely an exercise in algebra. What we really want is the measured time
difference; a half-page of algebra yields:
Td - Te = (1/g) arccosh(1 + (1/2)(gL)^2/(1 + gL cos(theta))) (eq 18)
Remember that the measured speed of light is:
c_measured = L / (Td - Te) (eq 19)
Note that (eq 17) is just (eq 10) without the term in Te -- c_measured using
coordinate clocks is independent of when the measurement occurs.
Compared to the standard-clock measurement above, this has slightly different
implications: c_measured is still anisotropic, and the limit g->0 of c_measured
is still 1. But now the limit L->0 of c_measured is also 1, isotropically and
independent of when the measurement is made, and for cos(theta)>0 there is never
an infinite result -- unlike standard clocks these coordinate clocks remain
consistent forever. The horizon at x'=-1/g remains, reflecting the fact that
these accelerated coordinates cannot cover the entire spacetime.
CONCLUSIONS
-----------
The speed of light measured in a physically-realizable way is not isotropic in
an accelerated system. This is true whether one uses standard clocks or
coordinate clocks (i.e. clocks slaved to a single standard clock at the origin).
Accelerated standard clocks, however, cannot remain usefully synchronized
forever, and the speed of light measured with them is generally not constant
in time. Coordinate clocks do remain synchronized, and yield a measurement
which is constant in time but also not isotropic. Neither kind of measurement
can be used for large distances behind the origin; an event horizon intervenes
and the detector will outrun the light beam.
Tom Roberts tjro...@lucent.com