# Gravitational force and acceleration in General Relativity.

by yuiop
Tags: acceleration, force, gravitational, relativity
P: 3,967
 Quote by starthaus I showed you why your solution was wrong, you can't "fiddle" or "guess" the correct solutions,
You can if you are lucky and it looks like I was.

Actually it wasn't luck. I just refer to conclusions that are rigorously derived by other people that know what they are doing.
P: 1,568
 Quote by kev If you are refering to this thread http://www.physicsforums.com/showthr...397403&page=11 then it clear that atyy, George Jones and Dalespam showed you were wrong and no one supported your argument. If fact Dalespam gave a lengthy, rigorous and very impressive demonstration that you were wrong starting at #165 of that thread and ending at #169.
I knew you were going to go there. I don't know why you would do that since there were so many calculus errors that you made in that thread. As to the proof produced by Dalespam, I repeatedly pointed out the error in his approach so, in the end, we agreed to disagree. If you want to discuss more on the subject of transformation of centripetal acceleration in SR, we can do it in the appropriate thread.
P: 355
 Quote by kev One thing that always confused me was this statement from http://www.mathpages.com/rr/s6-04/6-04.htm : At the apogee r = R where dr/dt = 0 the quantity in the square brackets is unity and this equation reduces to $$\frac{d^2r}{d\tau^2} = \frac{GM}{r^2} \; \; \; (6)$$ This is a measure of the acceleration of a static test particle at the radial parameter r. I always took that to mean that the proper acceleration of the static test particle is given by (6) but...
Right, this is the radial acceleration of our test particle as measured against the coordinate r. But this isn't the acceleration measured by our test particle. It's what we'd get if we parallel transported the acceleration vector all the way to infinity and then asked the observer at infinity how much acceleration he would measure if he had that acceleration vector.

To get the acceleration measured by our observer at r, you need to measure the acceleration vector in the orthonormal basis used by an observer at r. So the observed acceleration is
$$\frac{d^2 x^\alpha}{d\tau^2} \cdot \hat e^r$$
Where $$|\hat e^r|^2=1$$ and $$\frac{d x^a}{d\tau} \cdot \hat e^r = 0$$. e^r is directed toward larger r value.

It is very important to remember that formula for the dot product involves the metric.
Because our observer is at constant r, just like our observer at infinity, we can choose $$\hat e^r$$ to be in the same direction as the observer @ infinity's radial vector, but
$$\hat e^r$$ has a different length
(remember, a length of a vector is only defined at a point in spacetime. If we put both of these vectors at the same point, then e^r is bigger by a factor of (1-2M/R)^(-1/2))
P: 3,967
 Quote by starthaus I knew you were going to go there. I don't know why you would do that since there were so many calculus errors that you made in that thread. As to the proof produced by Dalespam, I repeatedly pointed out the error in his approach so, in the end, we agreed to disagree. If you want to discuss more on the subject of transformation of centripetal acceleration in SR, we can do it in the appropriate thread.
In that thread it took over 150 posts and a lot of bickering to play the "guess what Starthaus is thinking" game when you could have simply stated what you were thinking in one or two posts and now it seems you want to play that game again in this thread.

So far, in our new "guess what Starthaus is thinking" game, I have guessed in #15 that you are thinking that the relationship between the proper time and the coordinate time of a stationary particle is given by:

$$\frac{dt}{dS} = \frac{k}{1-2m/r}$$

and you have not denied it. Right there is a mistake in your thinking.
Mentor
P: 16,485
 Quote by starthaus As to the proof produced by Dalespam, I repeatedly pointed out the error in his approach so, in the end, we agreed to disagree.
We certainly can go back and do this again if you wish. My derivation and my results are correct, the "error" you pointed out comes straight from the reference (Gron) that you agreed to use, and your error lies in not knowing what proper acceleration is.
P: 355
 Quote by starthaus I checked, the two solutions produce different answers. As to rudeness, look at his tone.
If you got different answers by the 2 methods, then you were doing something wrong. I've checked kev's original answer with a few different sources as well as with my own work, and he made no mistakes at all. I also checked your formulas (the few that you wrote), and they were going in the right direction. You never got very far with them though, and your later comments suggest to me that you probably don't know the correct way to proceed.

In particular, you mention dr/ds as being the proper speed, but this is incorrect. The observer at r has to deal with length contraction in addition to the time dilation that you handled. He measures a dr'/ds, not dr/ds.
Working with 4-vectors instead of the individual coordinates leads to a much cleaner treatment that allows you to transform between observers' measurements easily.

Attacking someone and insisting that they are wrong just because they didn't show their work (which he said he would if you asked) usually doesn't get you a nice response...

Anyway, the question's been cleared up..
P: 1,568
 Quote by kev In that thread it took over 150 posts and a lot of bickering
It is not my problem that you had so much trouble with getting a few basic derivatives correct <shrug>. So many errors in your math makes it so painful teaching you.
P: 1,568
 Quote by DaleSpam We certainly can go back and do this again if you wish. My derivation and my results are correct, the "error" you pointed out comes straight out of the published literature of two different sources,
I can add a very nice derivation from Moller that supports my point and contradicts yours to the one from Rindler.

 and your error lies in not knowing what proper acceleration is.
Nothing productive will come out of this.
P: 1,568
 Quote by LukeD If you got different answers by the 2 methods, then you were doing something wrong. I've checked kev's original answer with a few different sources as well as with my own work, and he made no mistakes at all. I also checked your formulas (the few that you wrote), and they were going in the right direction. You never got very far with them though, and your later comments suggest to me that you probably don't know the correct way to proceed. In particular, you mention dr/ds as being the proper speed, but this is incorrect. The observer at r has to deal with length contraction in addition to the time dilation that you handled. He measures a dr'/ds, not dr/ds.
How do you "manufacture" dr' out of dr? In a rigorous way, not by putting in a scaling factor by hand?

 Working with 4-vectors instead of the individual coordinates leads to a much cleaner treatment that allows you to transform between observers' measurements easily.
Agreed but all you have are the equations of motion that I derived correctly.

 Attacking someone and insisting that they are wrong just because they didn't show their work (which he said he would if you asked) usually doesn't get you a nice response...
There is no work to be shown, the different expressions were put in by hand. You do not know but this is a repeat of another thread on the same exact subject where kev did the same thing, put in the results by hand. It took a painful 150 posts to guide him through a proper derivation.
P: 3,967
 Quote by starthaus It is not my problem that you had so much trouble with getting a few basic derivatives correct . So many errors in your math makes it so painful teaching you.
I admitted in that thread that I am not a mathematician and my calculus is lousy. Maybe you get a kick out of getting me to say that again and again. Your calculus might be good, but you are lousy at understanding the physical interpretation of the equations.

 Quote by LukeD We did this calculation in my GR class last semester. It took me a bit of time to find my book, but as I suspected (the answer is given on p.261 of Hartle with other relevant formulas scattered throughout the chapters), every formula that you wrote is exactly correct.
Thanks Luke
P: 1,568
 Quote by kev I admitted in that thread that I am not a mathematician and my calculus is lousy. Maybe you get a kick out of getting me to say that again and again. Your calculus might be good, but you are lousy at understanding the physical interpretation of the equations.
Physics is about deriving results, not putting them in by hand . For that, you need to know math. As to my physical interpretation, it comes through the ability to derive results rather than cobbling them from websites.
Mentor
P: 16,485
 Quote by starthaus Nothing productive will come out of this.
Well, at least we agree on that.

 Quote by starthaus I can add a very nice derivation from Moller that supports my point and contradicts yours to the one from Rindler.
If you want to back up this claim then you are more than welcome to do so and I will be more than glad to point out once again that you have derived a coordinate acceleration instead of the proper acceleration.
P: 1,568
 Quote by DaleSpam Well, at least we agree on that. If you want to back up this claim then you are more than welcome to do so and I will be more than glad to point out once again that you have derived coordinate acceleration instead of proper acceleration.
Sure, we'll go over it again. Should be fun, once we are done, you an also take up your claim with Moller.
P: 3,967
 Quote by starthaus It took a painful 150 posts to guide him through a proper derivation.
It took 150 posts to figure out that you were talking about some form of coordinate acceleration when everyone else was talking about proper acceleration.

Your derivation concluded that proper centrifugal acceleration is zero and in another place you claimed proper centrifugal acceleration is defined as $a_0 = a\gamma$ and in another place you got $a_0 = a$. You never did arrive at the conclusion that everyone else arrived at, that proper centrifugal acceleration is $a_0 = a\gamma^2$. So can you leave out the condescending tone?
Mentor
P: 16,485
 Quote by starthaus Sure, we'll go over it again. Should be fun, once we are done, you an also take up your claim with Moller.
I just had a kind of flash-back from 2nd grade. "I'll tell my Moller on you".
P: 1,568
 Quote by DaleSpam I just had a kind of flash-back from 2nd grade. "I'll tell my Moller on you".
No, no :lol:. You'll have to tell Moller, not me :lol:
P: 1,568
 Quote by kev It took 150 posts to figure out that you were talking about some form of coordinate acceleration when everyone else was talking about proper acceleration. Your derivation concluded that proper centrifugal acceleration is zero and in another place you claimed proper centrifugal acceleration is defined as $a_0 = a\gamma$ and in another place you got $a_0 = a$. You never did arrive at the conclusion that everyone else arrived at, that proper centrifugal acceleration is $a_0 = a\gamma^2$. So can you leave out the condescending tone?
You need to learn how to stop making false claims.
Mentor
P: 16,485
 Quote by starthaus You need to learn how to stop making false claims.
And the part of The Pot will be played by starthaus.

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