What is the surprising result for c_hat according to an accelerating observer?

[Mike_Fontenot]

I haven't previously ever derived an expression for c_hat, the velocity of any given light-pulse, relative to an accelerating observer, according to that accelerating observer. I hadn't done that derivation before, because I never felt the need for that result, and I didn't regard the issue as particularly important.

But a few days ago, I finally decided to do the derivation.

I chose to do the derivation in a very "first-principles" sort of way, starting with a Minkowski diagram of the accelerating traveler's world-line, plotted in some (arbitrary) inertial frame, much as I did when I originally derived the CADO equation. I had expected it to be fairly quick and easy, but it wasn't.

I had expected to get an expression for c_hat that contained the observer's acceleration "a" (as measured on his accelerometer), as a parameter. That's because of the fact that, for the idealized case of an instantaneous velocity change (for which the acceleration is a Dirac delta function), the distance (according to the accelerating observer) to any object (including a light-pulse), instantaneously changes ... seemingly implying an infinite value of c_hat.

The result of the derivation, for the case where the acceleration "a" is finite, but arbitrarily large, was very surprising to me ... so much so that at first, I didn't believe the result I was getting. (In hindsight, though, I can now see that it SHOULDN'T have been a surprise).

The result is that

c_hat = c.

I.e., according to an observer accelerating at "a" (in ly/y/y, say), the velocity of any light-pulse, relative to that observer, is always equal to c, just as it is for an inertial observer.

Mike Fontenot

 

[ghwellsjr]

...the velocity of any light-pulse, relative to that observer, is always equal to c, just as it is for an inertial observer.

But according to Einstein, we cannot observe, measure or know the velocity of a light-pulse, even for an inertial observer. Arbitrarily defining the speed of a light-pulse to be equal to c is the basis for the Theory of Special Relativity.

See A critique of Mike Fontenot's CADO scheme.

 

[Mike_Fontenot]

[...]

I still wonder why the above result for c_hat is so different from the result v_hat (the relative velocity between accelerating traveler and home-twin, according to the traveler), that I derived a long time ago, and gave here:

https://www.physicsforums.com/showpost.php?p=3231234&postcount=329

(I denoted v_hat by "V" in that posting).

The v_hat result is

v_hat = v * (1 - L*a/gamma) .

I had suspected that c_hat MIGHT turn out to be something like

c_hat = c*(1-L*a/gamma).

I still find it troubling that my results for c_hat and v_hat are SO different ... that just doesn't seem intuitively reasonable. I can't quite manage to put this matter to rest.

Mike Fontenot

 

[DrGreg]

I still wonder why the above result for c_hat is so different from the result v_hat (the relative velocity between accelerating traveler and home-twin, according to the traveler), that I derived a long time ago, and gave here:

https://www.physicsforums.com/showpost.php?p=3231234&postcount=329

(I denoted v_hat by "V" in that posting).

The v_hat result is

v_hat = v * (1 - L*a/gamma) .

I had suspected that c_hat MIGHT turn out to be something like

c_hat = c*(1-L*a/gamma).

I still find it troubling that my results for c_hat and v_hat are SO different ... that just doesn't seem intuitively reasonable. I can't quite manage to put this matter to rest.

Mike Fontenot

I haven't checked any of the maths or theory behind this, but simply going on the contents of this post alone, surely they are the same? When v = c, what is the value of γ?

 

[Mike_Fontenot]

I THINK I've spotted a mistake in my derivation of c_hat. But it'll take some work (hopefully tomorrow morning) to correct it. Stay tuned.

Mike Fontenot

 

[Mike_Fontenot]

[...]
[...] surely they are the same? When v = c, what is the value of γ?

When I wrote this:

"(In hindsight, though, I can now see that it SHOULDN'T have been a surprise)" ,

in my original posting, there were two things that, in hindsight, made that result seem not so surprising. One of those things has now evaporated because of the mistake I THINK I've spotted in my derivation. The other thing had to do with thinking about the v_hat vs v equation, and what it says (for any given (but fixed) value of "a") about what happens to v_hat when v gets very near c ... same thing that you've noticed. That one may indeed be a good indication that my original result IS correct. But in order to have a completely rigorous result, I need to see if that suspected mistake I think I spotted in my derivation really IS a mistake, and if so, figure out how to correct the derivation. But it may well turn out that a corrected rigorous derivation will still give the same original result. We'll see.

Mike Fontenot

 

[Mike_Fontenot]

[...]

When I wrote this:

"I had suspected that c_hat MIGHT turn out to be something like

c_hat = c*(1-L*a/gamma)" ,

in my original posting, that WAS silly, because gamma isn't defined for v = c, as you pointed out. But I still wouldn't want to rely on the result of simply plugging v = c into the v_hat equation, since that equation was derived specifically for the purpose of determining the relative velocity between the accelerating traveler and an INERTIAL observer, which assumes v < c. But I'm fairly confident I can come up with a completely rigorous derivation of c_hat, hopefully soon.

Thanks for your comment.

Mike Fontenot

 

[Mike_Fontenot]

I finally was able to derive an expression for the velocity of a light pulse, according to an accelerating observer. I've given the result here:

https://www.physicsforums.com/showpost.php?p=3300663&postcount=1 .

It took me MUCH longer than I had expected, mostly because I was plagued by MANY more careless mistakes and missteps than usual.

So far, it has passed all the sanity-checks that I've been able to come up with, including some numerical results that I can compare against.

So, I guess the title of this thread didn't turn out to be true ... the equation for c_hat actually ISN'T surprising ... it's pretty much what I would have originally hoped it would be. And it is consistent with what you can infer from the limiting idealizations of instantaneous velocity changes.

Mike Fontenot