Velocity of Light, for an Accelerating Observer

[Mike_Fontenot]

I have finally been able to derive the velocity of a light pulse, according to an accelerating observer. The reference frame for the accelerating observer is taken to be the CADO reference frame, which is the only choice that I consider to be acceptable. The CADO reference frame is described in

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7 ,

and in my paper,

"Accelerated Observers in Special Relativity",
PHYSICS ESSAYS, December 1999, p629.

The result is

c_hat = c + a * R_hat / c ,

where c is the velocity of light, according to any inertial observer, "a" is the acceleration in ly/yr (as measured on the accelerating observer's accelerometer), and R_hat is the distance to the light pulse, according to the accelerating observer. R_hat, "a", c, and c_hat are positive when in the accelerating observer's positive spatial direction.

When you are using units for which |c| = 1, and when you simplify things (like I usually do) by ignoring all the c's in the equations (i.e., when you use "dimensionless units"), some care in simplifying the above equation is necessary. For a positive-going light pulse, c = 1, and the above equation simplifies to

c_hat = 1 + a * R_hat , for a positive-going light pulse.

But for a negative-going light pulse, c = -1, and the equation simplifies to

c_hat = -1 - a * R_hat , for a negative-going light pulse.

(If you want to specify the acceleration in g's, you can use the fact that 1g is approximately 1.031 ly/y, and 1 ly/y is approximately 0.970g.)

Note that the equation says that, for a light pulse passing by the observer, c_hat = c, regardless of the value of the acceleration a. But when the light pulse is at some non-zero distance from the accelerating observer, c_hat and c will differ, and the difference will be proportional to the distance R_hat.

For some given R_hat, the difference between C_hat and c is proportional to the acceleration a. Note that, since "a" can be an arbitrarily large positive or negative number, c_hat can be in the opposite direction from c, and c_hat can be arbitrarily large.

Mike Fontenot

 

[zhermes]

Why would light have a different velocity from an accelerated reference frame?
Why wouldn't this be observable from Michelson interferometers?

 

[GrayGhost]

Why would light have a different velocity from an accelerated reference frame? Why wouldn't this be observable from Michelson interferometers?

Because the photons are not far enough away (and/or the acceleration not pronounced enough). In Mike's model, the accelerating observer must measure any photon at speed c when measured AT his location, as relativity requires it. It's distant photons that must wildly fluctuate in speed because of rapid accelerations. This because in Mike's theory, the accelerating POV is taken as the collective composite of contiguous-momentary-inertial-POVs transitioned. So for the same reason twin A jumps wildly per B at B's rapid (or instant) turrnabout, the distant photon's speed can abrupty change. The change in POV due to frame transitioning produces all these effects, per he who undergoes proper acceleration. In the MMX, the range of the moving photons is very very very small. so the effect should be notable IMO.

We'll see what Mike says.

GrayGhost

 

[GrayGhost]

I have finally been able to derive the velocity of a light pulse, according to an accelerating observer. The result is

c_hat = c + a * R_hat / c

Sounds reasonable Mike, given the process and convention assumed. I notice there is nothing inherent that identifies any time interval. Does this equation work only for instant jumps in acceleration?

GrayGhost

 

[Mike_Fontenot]

[...]
I notice there is nothing inherent that identifies any time interval. Does this equation work only for instant jumps in acceleration?

The equation is true for ANY given acceleration profile a(t), where t is the age of the accelerating observer. The limiting case of an instantaneous velocity change at t = t0 just corresponds to the case where a(t) is a Dirac delta function, A * delta(t- t0), where A is some constant.

The quantity c_hat is likewise a function of t, with a definite value at each instant t in the accelerating observer's life. The value of c_hat at any given instant t = t1 doesn't depend on anything that happened during any past times t < t1, or that will happen during any future times t > t1.

It's basically the same situation that occurs with the accelerating observer's conclusions, at any instant t = t1 in his life, about the current age of a distant person (and about the current distance to that distant person): the accelerating observer's conclusion is (by definition) the same as the conclusion of the inertial reference frame with which he is momentarily stationary at that instant t1 ... his "MSIRF" at that particular instant, denoted as MSIRF(t1).

Mike Fontenot

 

[Mike_Fontenot]

[...]
We'll see what Mike says.

Yeah, you got it right.

R_hat and "a" are essentially zero for Michelson-Morley.

 

[Mike_Fontenot]

[...]
Does this equation work only for instant jumps in acceleration?

Sorry ... I misread your above comment. I thought you were talking about instantaneous VELOCITY changes, rather than instantaneous ACCELERATION changes.

The function a(t), where t is an instant in the accelerating observer's life, IS completely arbitrary, and the observer is free to choose whatever function a(t) that he desires. There are three different possibilities:

(1) The acceleration is zero, except at a finite number of instants in the observer's life. At those isolated instants, the acceleration is a delta function ... i.e., the acceleration is infinite there, but has only an infinitesimal duration, so that the area under the delta function is equal to one. This results in the idealized (limiting) case of instantaneous velocity changes, with constant velocities everywhere in between those instantaneous changes. At those instants of instantaneous velocity change, c_hat and v_hat are both infinite (either positively or negatively infinite). At all other instants, c_hat = c, and v_hat = v (where "v" is the velocity of the distant inertial person (the "home twin"), relative to the traveler, according to the distant inertial person).

(2) The acceleration is "piece-wise constant", which means that the acceleration instantaneously changes at a finite number of instants in the observer's life, but is constant between those instants of instantaneous acceleration change. At those instants t_sub_i where the acceleration changes, the acceleration exactly at the instants t_sub_i is ambiguous: you have to specify whether you mean the limit of a(t), as t approaches t_sub_i from below, or as t approaches t_sub_i from above. c_hat and v_hat will likewise have different limits, depending on whether the limit is taken from below or from above.

(3) The acceleration function a(t) is continuous, but otherwise completely arbitrary. Here, both c_hat and v_hat are also continuous, and there is no ambiguity in a(t), c_hat(t), or v_hat(t) anywhere.

Case (1) is the easiest, and can usually easily be analyzed by hand, or with a calculator. (This is the usual twin "paradox" scenario).

Case (2) can be handled analytically, in closed form, and is easy to do with a computer program. It can also be done with a good calculator, but that way is too laborious to do routinely.

Case (3) can't (in general) be done analytically, in closed form. It CAN be done, but it requires numerical integration, on a computer. Fortunately, it is rare that case (3) is really needed.

Mike Fontenot

 

[GrayGhost]

Because the photons are not far enough away (and/or the acceleration not pronounced enough). In Mike's model, the accelerating observer must measure any photon at speed c when measured AT his location, as relativity requires it. It's distant photons that must wildly fluctuate in speed because of rapid accelerations. This because in Mike's theory, the accelerating POV is taken as the collective composite of contiguous-momentary-inertial-POVs transitioned. So for the same reason twin A jumps wildly per B at B's rapid (or instant) turrnabout, the distant photon's speed can abrupty change. The change in POV due to frame transitioning produces all these effects, per he who undergoes proper acceleration. In the MMX, the range of the moving photons is very very very small, so the effect should be notable IMO. We'll see what Mike says.

zhermes,

Ooops. Major Type-O there. The highlight above should have read ... so the effect should NOT be notable IMO.

It was too late to EDIT the original post, so sorry about that.

GrayGhost

 

[Mike_Fontenot]

I have finally been able to derive the velocity of a light pulse, according to an accelerating observer.
[...]
The result is

c_hat = c + a * R_hat / c ,

where c is the velocity of light, according to any inertial observer, "a" is the acceleration in ly/y/y (as measured on the accelerating observer's accelerometer), and R_hat is the position of the light pulse, relative to the accelerating observer, according to the accelerating observer. R_hat, "a", c, and c_hat are positive when in the accelerating observer's positive spatial direction.
[...]
(If you want to specify the acceleration in g's, you can use the fact that 1g is approximately 1.031 ly/y/y, and 1 ly/y/y is approximately 0.970g.)

In order to use the above result, you have to know how to determine R_hat. Deriving an expression for R_hat is a bit contorted, so I thought I should probably provide that expression. The result is

R_hat = gamma * (1 + v/c) * { - L + c * (T - T_lo) } ,

where

L = L(t) is the position of the accelerating traveler, at any given instant t in the accelerating observer's life, relative to some (arbitrary) inertial observer, according to that inertial observer,

T = T(t) is the age of that inertial observer, when the accelerating observer's age is t, according to that inertial observer,

T_lo is a constant (once the given light pulse has been specified), equal to the age of the inertial observer, when she is passed by the given light pulse,

v = v(t) is the velocity of the accelerating observer, at the given instant t of his life, relative to the inertial observer, according to that inertial observer,

gamma = gamma(t) is the gamma factor corresponding to the above relative velocity v(t), and

c is the velocity of the light pulse, relative to the inertial observer, according to that observer.

Note that c is the VELOCITY of the light pulse, NOT the MAGNITUDE of the velocity ... i.e., for the assumed one-dimensional motion, c can be positive OR negative (plus or minus one, when the units are ly and years).

v(t) can also be positive or negative, as can all the other variables in the equation, except for gamma.

The above equation has passed all my sanity checks, so far. If anyone thinks they find an unreasonable result when using the equation, I'd appreciate hearing about it.

Mike Fontenot

 

[Mike_Fontenot]

In order to use the above result, you have to know how to determine R_hat. Deriving an expression for R_hat is a bit contorted, so I thought I should probably provide that expression. The result is

R_hat = gamma * (1 + v/c) * { - L + c * (T - T_lo) } ,

where
[...]

I should probably provide some indication of how the quantities v(t), gamma(t), T(t), and L(t) can be determined (where t denotes any given instant in the accelerating observer's life), since those quantities are needed in the equation for the position of a given light pulse, according to the accelerating observer (denoted as R_hat(t)). (The quantities c and T_lo are also needed, in the computation of R_hat(t), but they are constants which follow directly from the specification of the particular given light pulse.)

The quantities v(t), gamma(t), T(t) and L(t) are the SAME quantities that are needed in the basic CADO equation (although in the CADO equation, I denote the quantity T(t) as CADO_H(t)). The CADO equation is described here:

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7

and here:

https://www.physicsforums.com/showpost.php?p=2923277&postcount=1

The details of how to compute the quantities v(t), gamma(t), T(t), and L(t), for any arbitrary acceleration profile a(t), are given in my paper:

"Accelerated Observers in Special Relativity", PHYSICS ESSAYS, December 1999, p629.

(Since the quantities v, gamma, T, and L are the conclusions of an inertial observer, it is fairly widely known how to determine them. Taylor & Wheeler use the same basic techniques in their "Spacetime Physics" book (in particular, in their worked example of how far a traveler can go, by constantly accelerating at 1g in a straight line)).

I'll just give an indication here of how it's done:

Given a(t), the "rapidity" (which I denote as eta(t)) is the integral of a(t) / |c|. (Rapidity is needed, rather than velocity, because velocity doesn't add linearly across inertial reference frames. Rapidity does.)

Once you have eta(t), you can get v(t) from v(t) = tanh{eta(t)}, and then you can get gamma(t) from v(t), in the usual way.

The quantity T(t) is the integral of gamma(t), and the quantity L(t) is the integral of |c| * v(t) * gamma(t).

For completely general acceleration profiles a(t), the above integrals must be determined via numerical integration.

When a(t) is piecewise-constant, the above integrals can be done analytically, with the result being that T(t) is expressed in terms of sinh{eta(t)}, and that L(t) is expressed in terms of cosh{eta(t)}.

For the simple idealized limiting case of a finite number of instantaneous velocity changes, the above integrals aren't even needed: all of the required quantities are almost trivial to determine by hand in that case.

Once you have v(t), T(t) and L(t), you can use the CADO equation to determine the current age of an inertial observer (the "home twin"), according to the accelerating observer (which is denoted as CADO_T(t) in the CADO equation). And knowing gamma(t), you can determine the distance to the inertial (home) twin, at the instant t, according to the accelerating observer (which is denoted as L_hat(t) = L(t) / gamma(t)). And if you differentiate L_hat(t) with respect to t, you can get the expression for v_hat(t), the velocity of the accelerating observer, relative to the inertial home twin, according to the accelerating observer, at any instant t of the accelerating observer's life.

If you then identify a particular light pulse (by specifying c and T_lo), you can compute the position of the light pulse, according to the accelerating observer (denoted as R_hat(t)). The expression I gave for the velocity of the given light pulse (denoted as c_hat(t)), was obtained by differentiating R_hat(t) with respect to t.

Mike Fontenot