- #1
Mike_Fontenot
- 249
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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
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