Gravitational force and acceleration in General Relativity.

[Altabeh]

See post #9 in this thread! That's essentially the same method that some others have mentioned in this thread, using the facts that

1. proper acceleration is the magnitude of 4-acceleration
2. 4-acceleration is the covariant derivative of 4-velocity with respect to proper time
3. 4-velocity is the derivative of worldline-coordinates with respect to proper time

Very well-worded and clear definitions that I agree with, as well. Maybe it's now too late to get involved in this catchy brawl but since the thread goes as fast as it can, I've lost almost the subject of battle but definitely I don't think starthaus has been telling beyond these, has he?

AB

 

[Al68]

You know where you can put your condescending tone. I asked you to drop it but you continue with this.

So, you're complaining about someone else's condescending tone? Seriously? After this:

It's not my fault that you don't understand basic calculus.
Is that you can't read math or are you just trolling?
Do you have some comprehension problem?
I think that you are are clearly confused on the subject.
Hey, thanks for playing anyway, better luck next time.
What in the definition : "proper acceleration is the derivative of proper speed wrt coordinate tiime" did you not understand?

I'm way too lazy to quote a significant fraction of your rude and condescending remarks, so that's just a small sampling.

It's obvious who the most condescending person in this thread is, by far.

 

[starthaus]

Very well-worded and clear definitions that I agree with, as well. Maybe it's now too late to get involved in this catchy brawl but since the thread goes as fast as it can, I've lost almost the subject of battle but definitely I don't think starthaus has been telling beyond these, has he?

AB

You are right, I haven't. I simply showed a few methods of deriving the expressions, rather than putting them in by hand, mindlessly. The most interesting approach is the one explained in my blog, using variational mechanics. The approach seems even more powerful (and certainly has a richer physics content) than even the approach based on covariant derivatives.

 

[Altabeh]

Rindler (11.15) p.230 clearly disagrees with what you are saying. See his derivation for proper acceleration.

Where in that page does he claim that the equation (11.15) belongs to proper acceleration of a particle following a time-like geodesic or whatever you might think of other than saying that it's just the field strength?

AB

 

[Altabeh]

In the second post of this thread, starthaus brings up the first of theories "kev's wrong because I'm right" by notifying us (!) of the following error (!) that he probably finds uninteresting just because kev is the poster:

This is very wrong. You need to start with the metric:


ds^2=\alpha dt^2-\frac{1}{\alpha}dr^2-r^2d\phi^2

\alpha=1-\frac{2m}{r}

From this you construct the Lagrangian:

L=\alpha (\frac{dt}{ds})^2-\frac{1}{\alpha}(\frac{dr}{ds})^2-r^2(\frac{d\phi}{ds})^2

From the above Lagrangian, you get immediately the equations of motion:

\alpha \frac{dt}{ds}=k

r^2 \frac{d\phi}{ds}=h

whre h,k are constants.

This is in response to the very correct formula,

<br /> a= \frac{GM}{r^2}\left(1-\frac{2GM}{rc^2}\right)<br />

which shows the initial coordinate acceleration of a rest particle released at the distance r according to an observer at infinity. Well if I'm not wrong about realizing what you seem to be pointing at in kev's post to be incorrect, I can very beautifully mathematically prove that the above formula is correct. I know that you're a knowledgeable guy but sometimes I feel you really get gutless to confess to what I call "the ordinary mistakes" that every physicist can make multiple times in his long-lived career. At least I'm trying to hint at the moot point everyone else here has been leaving out: If you've been trying to make kev learn from you, why don't I teach you something that you as an expert in calculus and everything probably don't know? If you're ready, I'm going to start debating about your numerous mistakes here!

I'm not going to be spineless when I find myself getting corrected as in a thread I really found the whole view of mine on the issue of "proper vs coordinate" wrong due to a debate with JesseM. You see that we confess to the wrongness or correctness of our mistakes as every normal physicist of Einsteinian level does.

AB

 

[Altabeh]

There is only one non-zero component of:
\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}=\Gamma^{r}_{tt}U^{t}U^{t}=\frac{c^2 (r-R) R}{2 r^3}

So, substituting back in we obtain the four-acceleration in the Schwarzschild coordinates:
\mathbf A = \left(0,\frac{c^2 R}{2 r^2},0,0\right) = \left(0,\frac{G M}{r^2},0,0\right)

Would someone please make it clear for me why the second term in

\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}

disappears in the end?

AB

 

[starthaus]

Would someone please make it clear for me why the second term in

\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}

disappears in the end?

AB

It doesn't "disappear", \mathbf X=(t,r_0,0,0) with r_0 fixed, so all you are left with is the term in U^tU^t

 

[Altabeh]

In fact, I checked starthaus's derivation and it is flawless and mine is a sort of similar approach but definitely doesn't result in an expression different from his for the coordinate acceleration. All I nag about is the mistakes he makes when facing with the definition of proper acceleration in GR as he clearly does this in his article "General Euler-Lagrange Solution for Calculating Coordinate Acceleration" where he says the proper acceleration is \frac{d^2r}{ds^2} while it's obviously not. As is clear, the proper acceleration here is to be defined by

\frac{Dr}{Ds}=\frac{d^2r}{ds^2}+\Gamma^{r}_{\mu\nu}U^{\mu}U^{\nu}.

(This was the idea that I came to when getting involved in a debate with JesseM.)

If we can settle these issues, I'll go on.

AB

 

[Altabeh]

It doesn't "disappear", \mathbf X=(t,r_0,0,0) with r_0 fixed, so all you are left with is the term in U^tU^t

No, I was asking this question that if I want to get from

\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}=\Gamma ^{r}_{tt}U^{t}U^{t}=\frac{c^2 (r-R) R}{2 r^3}

to

\mathbf A = \left(0,\frac{c^2 R}{2 r^2},0,0\right) = \left(0,\frac{G M}{r^2},0,0\right)

it is required to have the term \frac{-c^2R^2}{2 r^3} disappear. How does it occur?

AB

 

[starthaus]

No, I was asking this question that if I want to get from

\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}=\Gamma ^{r}_{tt}U^{t}U^{t}=\frac{c^2 (r-R) R}{2 r^3}

to

\mathbf A = \left(0,\frac{c^2 R}{2 r^2},0,0\right) = \left(0,\frac{G M}{r^2},0,0\right)

it is required to have the term \frac{-c^2R^2}{2 r^3} disappear. How does it occur?

AB

It doesn't either. R=\frac{2GM}{rc^2}

 

[starthaus]

As is clear, the proper velocity here is to be defined by

\frac{Dr}{Ds}=\frac{dr}{ds}+\Gamma^{r}_{\mu\nu}U^{\mu}U^{\nu}.

You sure about that? :-)

 

[Altabeh]

It doesn't either. R=\frac{2GM}{rc^2}

Look, I know all of this. The problem that I'm fed up with is how the second expression is derived from the first.

AB

 

[Altabeh]

You sure about that? :-)

I am, aren't you?]

Edit: All typos and stuff were edited. You see, I intentionally can make an error and you come and point it out to me somehow and I then stand corrected. What happened? Do you know more than rest of us, then? LOL

Anyways, I hope I'll know today whether the question in the post 305 is right!

AB

 

[starthaus]

In fact, I checked starthaus's derivation and it is flawless

1. So, why are you wasting time?

and mine is a sort of similar approach but definitely doesn't result in an expression different from his for the coordinate acceleration.

2. I must have missed your derivation, where is it?

All I nag about is the mistakes he makes when facing with the definition of proper acceleration in GR as he clearly does this in his article "General Euler-Lagrange Solution for Calculating Coordinate Acceleration" where he says the proper acceleration is \frac{d^2r}{ds^2} while it's obviously not.

Well, it wasn't that obvious to me when I started developing the alternate approach since I was trying to avoid covariant derivatives. Otherwise the approach would not be original, would it? Besides, I was interested in getting an alternate definition for proper acceleration. To date, none of the definitions (see my exchange with Rolfe2) has been satisfactory. This may mean that the variational mechanics method is good for deriving coordinate acceleration (something that the covariant derivative method can't do) and the covariant derivative method is good at deriving the proper acceleration for the hovering particle only. So, we need both methods.

As is clear, the proper acceleration here is to be defined by

\frac{Dr}{Ds}=\frac{d^2r}{ds^2}+\Gamma^{r}_{\mu\nu}U^{\mu}U^{\nu}.

(This was the idea that I came to when getting involved in a debate with JesseM.)

If we can settle these issues, I'll go on.

AB

3. You editted your post from inadvertently trying to correct my definition of proper "speed" to trying to correct my definition of proper "acceleration". Do you think anyone is missing your moving the goalposts between posts?
4. BTW , your new definition (of proper acceleration) is still wrong, though you copied from post 38.

 

[yuiop]

... a particle hovering at r=r_0 experiences a proper acceleration:

a_0=-\frac{m}{r_0^2}/\sqrt{1-2m/r_0}

The disagreement is about what happens when the particle starts falling in the non-uniform gravitational field.

Good to see you have finally come to the conclusion that the equation I gave in #1 for proper acceleration (based mainly on information from the mathpages website) is correct in the context that it was given in. The claim you started in #2 that the equations I gave in #1 are wrong, has been shown by the vast majority of posters in this thread (i.e. everyone but you), to be invalid.

You have expressed an interest to extend the equations beyond the context of #1, to the more general situation. In order to do that, we should first establish a "baseline" of what we do agreee on. A good place to start, would be for you to openly concede that the equations in #1 are correct in their context.

 

[starthaus]

Good to see you have finally come to the conclusion that the equation I gave in #1 for proper acceleration (based mainly on information from the mathpages website) is correct in the context that it was given in. The claim you started in #2 that the equations I gave in #1 are wrong, has been shown by the vast majority of posters in this thread (i.e. everyone but you), to be invalid.

Nope. You got the proper acceleration by using the hack a_0=a\gamma^3

You have expressed an interest to extend the equations beyond the context of #1, to the more general situation.

This is precisely what the method based on variational mechanics does. Derivation from scratch, not selective "cut and paste" from the mathpages+hacks.

 

[starthaus]

I am, aren't you?]

Edit: All typos and stuff were edited.

Nope, you still have quite a few. Some are typos but the others are serious conceptual errors.

You see, I intentionally can make an error

Intentionally? You still have some more "intentional" errors. You may want to continue copying from post 38, it is all correct there.

 

[yuiop]

Your equation:

a_0=-\frac{m}{r^2}\frac{\sqrt{1-2m/r_0}}{1-2m/r}

does not work, because the proper acceleration of a free falling particle is zero. (Attach an accelerometer to a free falling particle and see what it reads.)

...
The math used for deriving the proper and coordinate acceleration in the radial field does not support your claim above. If you think otherwise try deriving the formula for the proper acceleration of a particle dropped from r=r_0 in a variable field as a function of the radial coordinate r. Show that your obtain a_0=0 for any r

Others have already correctly stated that the proper acceleration of a particle in freefall is zero by definition, but I accept your challenge for the more general case. This is not a formal derivation or proof. Just my best shot based on my physical understanding of the situation and my interpretation of the equations given by references like mathpages. (i.e. using selective "cut and paste" from the mathpages+hacks.)

For easy visualisation, consider the case of a small rocket descending into a gravitational well, with its thrust (Fo) directed in the opposite direction to the local gravitational acceleration (a&#039;_g) so that the descent acceleration of the rocket at r, as measured by a stationary observer at r, is d^2r&#039;/dt&#039;^2 using his local clocks and rulers. (All primed variables indicate measurements made by a stationary local observer at r in the Schwarzschild coordinates). I can now say:

F_o = \left( \frac{d^2r&#039;}{dt&#039; ^2} - a&#039;_g \right) m_o

where Fo is the proper force directed upwards and m_o is the rest mass of the rocket .

Dividing both sides by m_o gives the proper acceleration a_o of the rocket as:

a_o = \frac{d^2r&#039;}{dt&#039; ^2} - a&#039;_g

where positive acceleration is defined here as upwards and negative acceleration means downwards.

A combination of http://www.mathpages.com/rr/s6-07/6-07.htm" gives the velocity dependent Schwarzschild coordinate gravitational acceleration a_g as:

a_g = - \frac{M}{r^2}(1-2M/r)\left(1-\frac{3(dr/dt)^2}{(1-2M/r)^2}\right)

Assuming the following relationships between the local vertical measurements (dr’, dt’, a’) made by a stationary observer at r and the coordinate measurements (dr, dt, a) made by a Schwarzschild observer at infinity using the gravitational boost factor \gamma_g = 1/\sqrt{(1-2M/r)}:

dr&#039; = dr \; \gamma_g

dt&#039; = dt \; \gamma_g^{-1}

dr&#039;/dt&#039; = dr/dt \; \gamma_g^{2}

a&#039; = a \; \gamma_g^{3}

it is easy to obtain:

a&#039;_g = - \frac{M}{r^2}\left(\frac{1-3(dr&#039;/dt&#039;)^2}{\sqrt{1-2M/r}}\right)

where dr’/dt’ is the descent velocity of the rocket as it passes radial coordinate r as measured by a stationary observer located at r.

The more general proper acceleration can now be expressed in the form:

a_o = \frac{d^2r&#039;}{dt&#039; ^2} - a&#039;_g = \frac{d^2r&#039;}{dt&#039; ^2} + \frac{M}{r^2}\left(\frac{1-3(dr&#039;/dt&#039;)^2}{\sqrt{1-2M/r}}\right)

When the rocket is holding position at r by using its thrust to hover at r, dr’/dt’ = 0 and d^2r&#039;/dt&#039; ^2 =0 and its proper acceleration is:

a_o = \frac{d^2r&#039;}{dt&#039; ^2} - a&#039;_g = 0 + \frac{M}{r^2}\left(\frac{1-3(dr&#039;/dt&#039;)^2}{\sqrt{1-2M/r}}\right) = \frac{M}{r^2}\frac{1}{\sqrt{1-2M/r}}

which is what most people here expect.

In the specific case of freefall only, dr’/dt’ can be calculated using:

\frac{dr&#039;}{dt&#039;} = {\frac{\sqrt{2M (1/r-1/R)}}{\sqrt{1-2M/r}}

where R is a constant parameter of the trajectory path representing the height of the apogee and the following relationship is true:

\frac{d^2r&#039;}{dt&#039; ^2} = - \frac{M}{r^2}\left(\frac{1-3(dr&#039;/dt&#039;)^2}{\sqrt{1-2M/r}}\right)

When the rocket turns off its engines and falls freely in a vacuum, the proper acceleration of the rocket is:

a_o = \frac{d^2{r}&#039;}{d{t}&#039; ^2} - a&#039;_g = \frac{d^2{r}&#039;}{d{t}&#039; ^2} + \frac{M}{r^2}\left(\frac{1-3(dr&#039;/dt&#039;)^2}{\sqrt{1-2M/r}}\right) = 0

for all R>2m and r>2m.

 

[yuiop]

Nope. You got the proper acceleration by using the hack a_0=a\gamma^3

You are missing the point. I am asking you to establish a "baseline of agreement" by conceding that the equations in #1 and the equations given in mathpages are correct, even if you do not agree with my method of using mathpages equations plus my hacks to obtain them.

 

[starthaus]

Assuming the following relationships between the local vertical measurements (dr’, dt’, a’) made by a stationary observer at r and the coordinate measurements (dr, dt, a) made by a Schwarzschild observer at infinity using the gravitational boost factor \gamma_g = 1/\sqrt{(1-2M/r)}:

dr&#039; = dr \; \gamma_g

dt&#039; = dt \; \gamma_g^{-1}

dr&#039;/dt&#039; = dr/dt \; \gamma_g^{2}

a&#039; = a \; \gamma_g^{3}

a&#039; = a \; \gamma_g^{3}? Again?
Try deriving this and you'll see that it isn't correct.

 

[Dale]

No, I was asking this question that if I want to get from

\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}=\Gamma ^{r}_{tt}U^{t}U^{t}=\frac{c^2 (r-R) R}{2 r^3}

to

\mathbf A = \left(0,\frac{c^2 R}{2 r^2},0,0\right) = \left(0,\frac{G M}{r^2},0,0\right)

it is required to have the term \frac{-c^2R^2}{2 r^3} disappear. How does it occur?

AB

Sorry, this is my fault, I wrote the expression down incompletely. This should have been:

\Gamma^{\mu}_{\nu\lambda}U^{\nu}U^{\lambda}=\Gamma ^{r}_{tt}U^{t}U^{t}=\frac{c^2 (r-R) R}{2 r^3} \left(1-\frac{R}{r}\right)^{-1/2} \left(1-\frac{R}{r}\right)^{-1/2}

The previous expression was only the Christoffel symbol, and did not include the contributions from U.

 

[yuiop]

a&#039; = a \; \gamma_g^{3}? Again?
Try deriving this and you'll see that it isn't correct.

... and yet unlike you, I obtain the correct results for proper acceleration by using that assumption.

Defend your assertion that it wrong, by showing how you would show:

a&#039; \neq a \; \gamma_g^{3}

(eg give a counter example)

 

[starthaus]

... and yet unlike you, I obtain the correct results for proper acceleration by using that assumption.

Defend your assertion that it wrong, by showing how would derive:

a&#039; \neq a \; \gamma_g^{3}

It doesn't work this way, kev. You used the formula, show how you derived it. You showed all the intermediate steps except the last one. Show the last step and I'll show you were your error is.

 

[yuiop]

... and yet unlike you, I obtain the correct results for proper acceleration by using that assumption.

Defend your assertion that it wrong, by showing how (you) would derive:

a&#039; \neq a \; \gamma_g^{3}

for the colinear vertical (radial) case.

I have already declared that I no mathematician and that my methods are informal. It would take a better mathematician/ relativist like Dalespam, George, rofle2, DrGreg, JesseM or Altabeh to do a more formal proof and determine if I am on the right track or not.

 

[Altabeh]

1. So, why are you wasting time?

Because if no one prevents you from making nonsense, I have to do it, don't I?

2. I must have missed your derivation, where is it?

Takes time to write. It'll come tonight! You’re not the only one who is familiar with the variational methods in Physicsforums, are you!? :-)

3. You editted your post from inadvertently trying to correct my definition of proper "speed" to trying to correct my definition of proper "acceleration". Why are you moving the goalposts? Do you think I missed that?

Nupe, I just meant to write "acceleration" in place of "velocity" in the very first edition of my post but intentionally changed it to a wrong thing to only know if you were going to point it out ironically to me which you unfortunately did! It has nothing to do with the proper velocity in case you see I wrote the proper 4-acceleration’s components. So I’m not going to show any respect to you here and in fact I’ll throw at your face whatever mistake you make like the one you’ve been making us play with inadvertently for weeks now. Yet this whole test that I put you to shows whose tone is condescending. All you’re trying now to do doesn't change your false claim about what proper acceleration is.

4. Your new definition (of proper acceleration) is still wrong, though you copied from post 38.[/QUOTE]

I’m not into copy-paste stuff like you. The definition of proper acceleration can be found in http://en.wikipedia.org/wiki/Proper_acceleration" page.

 

[starthaus]

4. Your new definition (of proper acceleration) is still wrong, though you copied from post 38.

I’m not into copy-paste stuff like you. The definition of proper acceleration can be found in http://en.wikipedia.org/wiki/Proper_acceleration" page.

You still have it wrong. Not one mistake, several.

 

[Altabeh]

your moving the goalposts between posts?
4. BTW , your new definition (of proper acceleration) is still wrong, though you copied from post 38.

Don't play with words. That is not the general definition. Here we are just dealing with the radial velocity of a particle near a gravitating body and that definition is to be assigned to this.

All you do is to make people busy on some monkey job you take along yourself to these forums and just give us some garbage and then say that is a mathematically backed physics while still playing down other's arguments over your nonsense. As an example, you repeatedly referred to me copying the definition of proper acceleration from the post 38:

Intentionally? You still have some more "intentional" errors. You may want to continue copying from post 38, it is all correct there.

BTW , your new definition (of proper acceleration) is still wrong, though you copied from post 38.

I simply took my definition from http://en.wikipedia.org/wiki/Proper_acceleration#In_curved_spacetime". If this is called copy-paste, people don't use Einstein's field equations because then will a man with a long torch in his hand come to you while whispering to your ears the phrase All rights reserved. You're so funny, boy!

Nope, you still have quite a few. Some are typos but the others are serious conceptual errors.

AS a way to escape from being criticized about your nonsense, I agree with the second part of your theories "Altabeh has conceptual errors". So I'm waiting to know what they are!

Well, it wasn't that obvious to me when I started developing the alternate approach since I was trying to avoid covariant derivatives. Otherwise the approach would not be original, would it? Besides, I was interested in getting an alternate definition for proper acceleration. To date, none of the definitions (see my exchange with Rolfe2) has been satisfactory. This may mean that the variational mechanics method is good for deriving coordinate acceleration (something that the covariant derivative method can't do) and the covariant derivative method is good at deriving the proper acceleration for the hovering particle only. So, we need both methods.

Okay, I see that you finally admitted to being wrong about what is called a "mistake". I doubt that you even know what the variational methods of general relativity are all about; the essence of those methods in GR is found to be meaningful when one knows that they all have to be consistent with covariant methods. Talking about one being good and the other bad for one special purpose gives me this feeling that you've taken shelter in logics in order to hide your BIG mistake. After going through 21 pages, I have to say "what a mess "!

AB

 

[starthaus]

for the colinear vertical (radial) case.

I have already declared that I no mathematician and that my methods are informal.

There is no such thing as "informal" in math. If you redo the last step in your derivation, you will find out the error, it is quite obvious.

 

[starthaus]

Don't play with words.

I don't. You can't even get the definition right after reading the posts that make use of it.
Your errors are staring you in the face.

 

[Altabeh]

I don't. You can't even get the definition right after reading the posts that make use of it.

Play with words again to maybe have a hope to survive in the zone of physics. What's next? Einstein had mistakes or what?! Lool!

AB