Resultant time dilation from both gravity and motion

[espen180]

When a frame is moving in relation to an observer in his rest frame at infinity, and the frame is in a gravitational well, is the resultant time dilation simply the sum of the motional and gravitational dilation, e.g.

t=\tau\left(\gamma^{-1}+\gamma_g^{-1}\right)=\tau\left(\sqrt{1-\frac{v^2}{c^2}}+\sqrt{1-\frac{GM}{c^2r}}\right)

Where \tau is proper time and t is measured by the observer?

If, not what is the correct expression?

 

[starthaus]

When a frame is moving in relation to an observer in his rest frame at infinity, and the frame is in a gravitational well, is the resultant time dilation simply the sum of the motional and gravitational dilation, e.g.

t=\tau\left(\gamma^{-1}+\gamma_g^{-1}\right)=\tau\left(\sqrt{1-\frac{v^2}{c^2}}+\sqrt{1-\frac{GM}{c^2r}}\right)

Where \tau is proper time and t is measured by the observer?

If, not what is the correct expression?

There is no reason why it would be the sum , you can calculate the expression easily from the Schwarzschild metric:

(cd\tau)^2=(1-r_s/r)(cdt)^2-(1-r_s/r)^{-1}(dr)^2-(rd\theta)^2-(rd\phi sin\theta)^2

Make d\theta=dr=0

 

[JesseM]

For an object in a circular orbit, the total time dilation is a product of gravitational and velocity-based time dilation--see kev's post #8 on this thread and post #10 here. But cases other than a circular orbit would probably be more complicated.

 

[Jonathan Scott]

In non-relativistic situations, you can simply fall back on Newtonian theory:

The fractional time dilation (that is, the difference in time rate divided by the original time rate) due to velocity is equal to the ratio of kinetic energy to rest energy.

The fractional time dilation due to gravity is equal to the ratio of potential energy to rest energy.

The combined effect simply adds the fractions together to give the overall fraction (which is equivalent to multiplying the time dilation factors for each of the two effects).

For free fall (including any shape of orbit around a static mass), the sum of kinetic energy and potential energy is constant, so the time dilation is constant (and so is the total energy, as in Newtonian theory).

The relative time rates for different orbits can be compared using Newtonian potential theory.

 

[espen180]

Thank you very much. All replies were very useful.

 

[starthaus]

For an object in a circular orbit, the total time dilation is a product of gravitational and velocity-based time dilation--see kev's post #8 on this thread and post #10 here. But cases other than a circular orbit would probably be more complicated.

Hi Jesse,

I don't think the expressions put down by kev in that post are correct. The correct result is derived from the Schwarzschild metric, the periods of two clocks situated at radiuses r_1 and r_2 respectively is expressed by the ratio:

\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}}\sqrt{\frac{1-(r_1sin\theta_1\omega/\sqrt{1-r_s/r_1})^2}{1-(r_2sin\theta_2\omega/\sqrt{1-r_s/r_2})^2}}

where r_s is the Schwarzschild radius.The above is valid for a uniform density sphere.
Start with the Schwarzschild metric:

(cd\tau)^2=(1-r_s/r)(cdt)^2-(1-r_s/r)^{-1}(dr)^2-(rd\theta)^2-(rd\phi sin\theta)^2

and make d\theta=dr=0 for an object orbiting at r=constant.

If d\theta=d\phi=0 we get the expression for an object moving radially, which is still different from kev's expressions. In kev's notation:

\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{1}{c^2}\frac{dr/dt}{1-r_s/r})^2}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{v/c}{1-r_s/r})^2}

where v=\frac{dr}{dt}

 

[JesseM]

If d\theta=d\phi=0 we get the expression for an object moving radially, which is still different from kev's expressions.

kev wasn't talking about an object moving radially, as I said before he was dealing with the scenario of an object in circular orbit. pervect also found that for this case, the total time dilation was "almost" a product of SR and GR time dilations here...I think the difference was just because pervect was using coordinate velocity in Schwarzschild coordinates in the part of the equation that looked "almost" like SR time dilation, whereas kev was using the local velocity as seen in a freefalling frame for an observer whose coordinate velocity in Schwarzschild coordinates is zero at the moment the orbiting object passes it.

 

[DrGreg]

I believe that the equation

<br /> \frac{dt}{d\tau} = \frac{1}{\sqrt{1-v^2/c^2}\sqrt{1 - 2GM/rc^2}}<br />​

always applies (for radial, tangential or any other motion) where v is speed relative to a local hovering observer using local proper distance and local proper time.

I derived this in posts #9 and #7 of the thread "Speed in general relativity"[/color] (and repeated in post #46).

 

[starthaus]

kev wasn't talking about an object moving radially, as I said before he was dealing with the scenario of an object in circular orbit. pervect also found that for this case, the total time dilation was "almost" a product of SR and GR time dilations here...I think the difference was just because pervect was using coordinate velocity in Schwarzschild coordinates in the part of the equation that looked "almost" like SR time dilation, whereas kev was using the local velocity as seen in a freefalling frame for an observer whose coordinate velocity in Schwarzschild coordinates is zero at the moment the orbiting object passes it.

kev's expression for radial motion is not correct (see post #6 above). It is very easy to obtain the correct expressions.

 

[starthaus]

I believe that the equation

<br /> \frac{dt}{d\tau} = \frac{1}{\sqrt{1-v^2/c^2}\sqrt{1 - 2GM/rc^2}}<br />​

Yes, this is correct, provided "v" in your case is defined as:

\frac{dr/dt}{1-r_s/r}

or as:

\frac{r*sin\theta* d\phi/dt}{\sqrt{1-r_s/r}}=\frac{\omega rsin\theta}{\sqrt{1-r_s/r}}

r_s=\frac{2GM}{c^2}

(see post 6)

 

[espen180]

So the correct expression is

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c\left(1-\frac{r_s}{r}\right)}\right)^2}

, right?

Do you then define r\frac{\text{d}\phi}{\text{d}t} as coordinate velocity?

 

[JesseM]

kev's expression for radial motion is not correct (see post #6 above). It is very easy to obtain the correct expressions.

Your post #6 seems to be addressing a different question than pervect and kev, since you are finding the ratio of ticking rates of two clocks orbiting at finite radius, while pervect and kev were deriving time dilation of an orbiting clock relative to a stationary clock at infinity (as in the commonly-used equation for gravitational time dilation). I suppose your expression would probably have a well-defined limit as r2 approaches infinity though. Anyway, it might be easier to deal with pervect's derivation rather than kev's, since pervect's equation is expressed entirely in Schwarzschild coordinates rather than including a non-Schwarzschild notion of "velocity". Do you disagree with pervect's conclusions here? If so, where's the first line you would dispute?

 

[starthaus]

Your post #6 seems to be addressing a different question than pervect and kev,
since you are finding the ratio of ticking rates of two clocks orbiting at finite radius

Precisely. It addresses the question in the OP. (post 1). That is, what is the difference in rates for atomic clocks on the geoid.

while pervect and kev were deriving time dilation of an orbiting clock relative to a stationary clock at infinity (as in the commonly-used equation for gravitational time dilation). I suppose your expression would probably have a well-defined limit as r2 approaches infinity though.

No, the first formula in post 6 is derived from :

\frac{d\tau_1}{dt}=...

and

\frac{d\tau_2}{dt}=...

where \frac{d\tau}{dt} is derived straight from the metric:

(cd\tau)^2=(1-r_s/r)(cdt)^2-(1-r_s/r)^{-1}(dr)^2-(rd\theta)^2-(rd\phi sin\theta)^2

Make d\theta=d\phi=0:

\frac{d\tau}{dt}=\sqrt{1-r_s/r}\sqrt{...}

Anyway, it might be easier to deal with pervect's derivation rather than kev's, since pervect's equation is expressed entirely in Schwarzschild coordinates rather than including a non-Schwarzschild notion of "velocity". Do you disagree with pervect's conclusions here?

Pervect's formula in the post you linked is identical to mine. So, no dispute.

 

[starthaus]

So the correct expression is

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c\left(1-\frac{r_s}{r}\right)}\right)^2}

, right?

Yes.

Do you then define r\frac{\text{d}\phi}{\text{d}t} as coordinate velocity?

I don't define anything.

 

[espen180]

I don't define anything.

How do I interpret it then?

 

[espen180]

So the correct expression is

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c\left(1-\frac{r_s}{r}\right)}\right)^2}

But it looks from the Schwartzschild metric that it would be

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{1}{1-\frac{r_s}{r}}\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2}

?

 

[espen180]

The Schwartzschild metric for constant r and \theta=\frac{\pi}{2} gives us

c^2\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=c^2\left(1-\frac{r_s}{r}\right) - r^2\left(\frac{\text{d}\phi}{\text{d}t}\right)^2

If we divide both sides with c² we get

\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=\left(1-\frac{r_s}{r}\right) - \left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2

"Factoring out" 1-\frac{r_s}{r} on the right side gives

\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=\left(1-\frac{r_s}{r}\right)\left(1 - \frac{1}{1-\frac{r_s}{r}}\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2\right)

then, taking the square root gives the result in #16;

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{1}{1-\frac{r_s}{r}}\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2}

I don't see where the mistake is. Would you please point it out to me?

 

[starthaus]

The Schwartzschild metric for constant r and \theta=\frac{\pi}{2} gives us

c^2\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=c^2\left(1-\frac{r_s}{r}\right) - r^2\left(\frac{\text{d}\phi}{\text{d}t}\right)^2

If we divide both sides with c² we get

\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=\left(1-\frac{r_s}{r}\right) - \left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2

"Factoring out" 1-\frac{r_s}{r} on the right side gives

\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=\left(1-\frac{r_s}{r}\right)\left(1 - \frac{1}{1-\frac{r_s}{r}}\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2\right)

then, taking the square root gives the result in #16;

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{1}{1-\frac{r_s}{r}}\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2}

I don't see where the mistake is. Would you please point it out to me?

yes, fine

 

[yuiop]

Hi Jesse,

I don't think the expressions put down by kev in that post are correct. The correct result is derived from the Schwarzschild metric, the periods of two clocks situated at radiuses r_1 and r_2 respectively is expressed by the ratio:

\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}}\sqrt{\frac{1-(r_1sin\theta_1\omega/\sqrt{1-r_s/r_1})^2}{1-(r_2sin\theta_2\omega/\sqrt{1-r_s/r_2})^2}}

where r_s is the Schwarzschild radius.The above is valid for a uniform density sphere.
Start with the Schwarzschild metric:

(cd\tau)^2=(1-r_s/r)(cdt)^2-(1-r_s/r)^{-1}(dr)^2-(rd\theta)^2-(rd\phi sin\theta)^2

and make d\theta=dr=0 for an object orbiting at r=constant.

If d\theta=d\phi=0 we get the expression for an object moving radially, which is still different from kev's expressions. In kev's notation:

\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{1}{c^2}\frac{dr/dt}{1-r_s/r})^2}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{v/c}{1-r_s/r})^2}

where v=\frac{dr}{dt}

I was using a notion of local velocity (v' = dr'/dt') as measured by a stationary observer at r.

Since v&#039; = dr&#039;/dt&#039; = (dr/dt)/(1-r_s/r)

the value of v' can be directly substituted into your expression to obtain:

\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{1}{c^2}\frac{dr/dt}{1-r_s/r})^2}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{v&#039;}{c})^2}

The two forms are numerically the same and in agreement with #8 by DrGReg here:

I believe that the equation

<br /> \frac{dt}{d\tau} = \frac{1}{\sqrt{1-v^2/c^2}\sqrt{1 - 2GM/rc^2}}<br />​

always applies (for radial, tangential or any other motion) where v is speed relative to a local hovering observer using local proper distance and local proper time.

I derived this in posts #9 and #7 of the thread "Speed in general relativity"[/color] (and repeated in post #46).

 

[starthaus]

I was using a notion of local velocity (v' = dr'/dt') as measured by a stationary observer at r.

Since v&#039; = dr&#039;/dt&#039; = (dr/dt)/(1-r_s/r)

There is no mention of any such convention in this post. Actually, there is no derivation, the expression is put in by hand, you simply multiplied the kinematic factor by the gravitational factor.

 

[yuiop]

There is no mention of any such convention in this post. Actually, there is no derivation, the expression is put in by hand, you simply multiplied the kinematic factor by the gravitational factor.

It was not meant to be a derivation, just a statement of facts from various references, put into context and interelated to each other. If you want a derivation, Dr Greg has done some perfectly good ones that come to the same conclusion. In the post you linked to, I made it clear in the surrounding text that I was talking about the the local velocity.

But it looks from the Schwartzschild metric that it would be

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{1}{1-\frac{r_s}{r}}\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c}\right)^2}

?

Yes, that equation is correct.

There are two motion/gravity time dilation equations if purely Schwarzschild coordinate measurements are used.

The time dilation ratio for orbital motion is:

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\left(\frac{r\frac{\text{d}\phi}{\text{d}t}}{c\sqrt{1-\frac{r_s}{r}}\right)^2}

The time dilation ratio for radial motion is:

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\left(\frac{\frac{\text{d}r}{\text{d}t}}{c(1-\frac{r_s}{r})\right)^2}

Now if I define v' = dx'/dt' as the local velocity of the moving test particle as measured by a stationary observer at r using local clocks and rulers (where dx' can be a vertical or horizontal distance), then a single equation is obtained:

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{v&#039;^2}{c^2}}

which is equally valid for horizontal or vertical motion of the particle.

To try and make the concept of "local velocity" even clearer, this is the velocity calculated by a local stationary observer orientating a ruler of proper length (dx') parallel to the motion of the test particle and timing the interval (dt') it takes for the test particle to traverse the ruler according to the stationary observers local clock.

 

[espen180]

Thank you very much kev! Everything fits now. :)

 

[yuiop]

Thank you very much kev! Everything fits now. :)

You are very welcome.

The equation I gave

\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{v &#039;^2}{c^2}}

uses an odd mix (something DrGreg alluded to) of velocity measured locally (v') and Schwarzschild coordinate gravitational time dilation.

A more general equation is:

\frac{\text{d}\tau}{\text{d}t}= \sqrt{\frac{1-r_s/r}{1-r_s/r_o}} \sqrt{1- \left (\frac{1-r_s/r_o}{1-r_s/r} \right )^2 \left (\frac{\text{d}r}{c\text{d}t} \right )^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right ) \left(\frac{r \text{d}\theta}{c\text{d}t}\right)^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right) \left(\frac{r \sin \theta \text{d}\phi}{c\text{d}t}\right)^2 }

where r_o is the Schwarzschild radial coordinate of the stationary observer and r is the Schwarzschild radial coordinate of the test particle and dr and dt are understood to be measurements made by the stationary observer at r_o in this particular equation.

For r_o = r the time dilation ratio is:

\frac{\text{d}\tau}{\text{d}t} = \sqrt{1-\frac{v&#039;^2}{c^2}}

in agreement with the generally accepted fact that local measurements made in a gravitational field are Minkowskian.

 

[JesseM]

Pervect's formula in the post you linked is identical to mine. So, no dispute.

Then why did you dispute kev's equations? He explicitly stated in post #8 here (which I linked to earlier) that he was just working from pervect's derivation, but with the substitution of a "local velocity" v for the Schwarzschild coordinate velocity u, related by u = v \sqrt{1-\frac{r_s}{r}}.

 

[starthaus]

Then why did you dispute kev's equations? He explicitly stated in post #8 here (which I linked to earlier) that he was just working from pervect's derivation, but with the substitution of a "local velocity" v for the Schwarzschild coordinate velocity u, related by u = v \sqrt{1-\frac{r_s}{r}}.

Because kev's equations did not apply to the OP. Since then, the threads have been split.

 

[JesseM]

Because kev's equations did not apply to the OP.

Why do you say that? The original post asked "When a frame is moving in relation to an observer in his rest frame at infinity, and the frame is in a gravitational well, is the resultant time dilation simply the sum of the motional and gravitational dilation", it wasn't asking about the ratio between ticks of clocks at different finite radii. I brought up the result kev derived because it gives the time dilation in one special case--for a clock in a perfect circular orbit in a Schwarzschild spacetime--relative to an observer at infinity, which seemed to me to be relevant to the OP.

 

[JesseM]

To try and make the concept of "local velocity" even clearer, this is the velocity calculated by a local stationary observer orientating a ruler of proper length (dx') parallel to the motion of the test particle and timing the interval (dt') it takes for the test particle to traverse the ruler according to the stationary observers local clock.

Would this be the same as the velocity measured in the locally inertial frame of a free-falling observer who happens to be instantaneously at rest (in Schwarzschild coordinates) at the moment the orbiting clock passes him? I assumed this was what was meant by "local" velocity but from your description above I'm not sure if it's the same...

 

[starthaus]

Why do you say that? The original post asked "When a frame is moving in relation to an observer in his rest frame at infinity, and the frame is in a gravitational well, is the resultant time dilation simply the sum of the motional and gravitational dilation", it wasn't asking about the ratio between ticks of clocks at different finite radii.

kev didn't derive any result, kev puts in results by hand.

I brought up the result kev derived because it gives the time dilation in one special case--for a clock in a perfect circular orbit in a Schwarzschild spacetime--relative to an observer at infinity, which seemed to me to be relevant to the OP.

The question came up in the different thread, the one about "Why do all clocks tick at the same rate on the geoid" by Dmitry7. I pointed out repeatedly to you why kev's formulas were not appropiate for answering that thtrad.

 

[starthaus]

Then why did you dispute kev's equations? He explicitly stated in post #8 here (which I linked to earlier) that he was just working from pervect's derivation, but with the substitution of a "local velocity" v for the Schwarzschild coordinate velocity u, related by u = v \sqrt{1-\frac{r_s}{r}}.

Because post #8(and all subsequent posts based on it) by kev contains a glaring mistake. I have corrected it in post 25.
Citing kev's posts does nothing but perpretrate mistakes.

 

[yuiop]

Would this be the same as the velocity measured in the locally inertial frame of a free-falling observer who happens to be instantaneously at rest (in Schwarzschild coordinates) at the moment the orbiting clock passes him? I assumed this was what was meant by "local" velocity but from your description above I'm not sure if it's the same...

My understanding or interpretation (up to now) is that the local velocity is measured by a non inertial accelerating observer that is stationary at r with respect to the Schwarzschild coordinates. i.e. the velocity of this non inertial observer is dr/dt = r d\theta/dt = r d\phi =0. It might be better to think of it terms of inertial observer that happens to be apogee at r when the orbiting particle (with orbital radius r) passes close by. As I understand it, the clock rates and ruler measurements of a non-inertial accelerating observer that is stationary at r are the same as the measurements made by an inertial observer that momentarily happens to be at apogee at r at the same time. I might have to think about it some more. There might be a limitation to how far this "equivalence" (aplication of the clock hypothosis) can be taken when it concerns measurements of acceleration. That is something I am working on.

 

[JesseM]

kev didn't derive any result, kev puts in results by hand.

True, kev didn't derive the relation between local velocity and coordinate velocity, but unless you had definite reason to think the relation he used was incorrect (as opposed to possibly correct but not sufficiently justified in his post), I don't see why you would say "I don't think the expressions put down by kev in that post are correct."

The question came up in the different thread, the one about "Why do all clocks tick at the same rate on the geoid" by Dmitry7. I pointed out repeatedly to you why kev's formulas were not appropiate for answering that thtrad.

What do you mean by "the question"? My post bringing up kev's result was in direct response to espen180's OP on this thread, so it doesn't seem to make any sense to cite some completely different thread in order to back up your claim that "kev's equations did not apply to the OP."

 

[starthaus]

True, kev didn't derive the relation between local velocity and coordinate velocity, but unless you had definite reason to think the relation he used was incorrect (as opposed to possibly correct but not sufficiently justified in his post), I don't see why you would say "I don't think the expressions put down by kev in that post are correct."

What do you mean by "the question"? My post bringing up kev's result was in direct response to espen180's OP on this thread, so it doesn't seem to make any sense to cite some completely different thread in order to back up your claim that "kev's equations did not apply to the OP."

Why don't you read post 29? kev's formulas that you keep citing are wrong, ok?

 

[JesseM]

Because post #8(and all subsequent posts based on it) by kev contains a glaring mistake. I have corrected it in post 25.
Citing kev's posts does nothing but perpretrate mistakes.

In post 25 you say that the mistake was originally pervect's which was just perpetuated by kev, but then earlier in post 13 you said you didn't dispute pervect's results, I guess you changed your mind? It does look like pervect used the wrong formula there.

 

[yuiop]

Because post #8(and all subsequent posts based on it) by kev contains a glaring mistake. I have corrected it in post 25.
Citing kev's posts does nothing but perpretrate mistakes.

The very next line in that post https://www.physicsforums.com/showpost.php?p=2446850&postcount=8 states that I have found and corrected for pervect's rare mistake. I made the edit over a year ago and cleary state that the remaining calculations have been edited to correct for the typo by pervect.

 

[starthaus]

The very next line in that post https://www.physicsforums.com/showpost.php?p=2446850&postcount=8 states that I have found and corrected for pervect's rare mistake. I made the edit over a year ago and cleary state that the remaining calculations have been edited to correct for the typo by pervect.

You are right, you also corrected another error that you made further down in your post.

 

[JesseM]

You are right, you also corrected another error that you made further down in your post. But the derivation in post 8 applies to orbital motion, the equation in this post cited by JesseM). is also for orbital motion and not applicable to this thread.

Why do you think an equation for the time dilation experienced by an orbiting object (an equation which you now agree is correct, I take it?) is "not applicable to this thread"? The OP didn't say anything about the precise state of motion of the object, just that it was in a gravity well and was moving (which would certainly be true for an orbiting object!):

When a frame is moving in relation to an observer in his rest frame at infinity, and the frame is in a gravitational well, is the resultant time dilation simply the sum of the motional and gravitational dilation

 

[starthaus]

Why do you think an equation for the time dilation experienced by an orbiting object (an equation which you now agree is correct, I take it?) is "not applicable to this thread"? The OP didn't say anything about the precise state of motion of the object, just that it was in a gravity well and was moving (which would certainly be true for an orbiting object!):

You are going around in circles. Let's put a stop to this, I gave you the correct expressions , including the derivations for both orbital and radial motion at post 6. My post 6 really belongs in the Dmitry7 thread, whoever split the threads made a mistake.
The reason for all the confusion is that espen180 thread was split from the Dmitry7 thread. The two threads (espen180 and Dmitry7) deal with different situations. The answer I gave you at post 6, stands, the correct answer to Dmitry7 question is not the kev posts you cite but the answer I derived.
Spcifically:

-The correct answer to Dmitry7's question is:

\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}}\sqrt{\frac{1-(r_1sin\theta_1\omega/c\sqrt{1-r_s/r_1})^2}{1-(r_2sin\theta_2\omega/c\sqrt{1-r_s/r_2})^2}}

-The answer to espen180 question is :\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{r\omega sin(\theta)/c}{\sqrt{1-r_s/r}})^2}

 

[JesseM]

You are going around in circles. Let's put a stop to this, I gave you the correct expressions , including the derivations for both orbital and radial motion at post 6.

I don't dispute your expressions, but it seems you also do not dispute that kev's expressions are correct in the case of an orbiting object or that they are relevant to the OP by espen180, correct? So will you acknowledge that all your previous argumentative statements saying that kev was wrong and that I was wrong to cite him were made too hastily?

My post 6 really belongs in the Dmitry7 thread, whoever split the threads made a mistake.
The reason for all the confusion is that espen180 thread was split from the Dmitry7 thread. The two threads (espen180 and Dmitry7) deal with different situations. The answer I gave you at post 6, stands, the correct answer to Dmitry7 question is not the kev posts you cite but the answer I derived.

But I was the one who originally brought up kev's derivation, and I brought it up in response to espen180's post, I never claimed that it was supposed to be relevant to Dmitry7's post. Did you misunderstand who I was responding to? My original post on this thread was made 8 minutes before Dmitry7's first post according to the timestamps on the upper left, so even before the thread split my post should have appeared before his and it should have been clear that I was not responding to his question.

 

[starthaus]

I don't dispute your expressions, but it seems you also do not dispute that kev's expressions are correct in the case of an orbiting object or that they are relevant to the OP by espen180, correct?

Yes, but NOT in the context of the original thread as started by Dmitry7. This is where my objections started. With post 6. Do you now understand what my objection is to your citing the inappropriate material for answering Dmitry7's OP?

So will you acknowledge that all your previous argumentative statements saying that kev was wrong and that I was wrong to cite him were made too hastily?

No. See above.

But I was the one who originally brought up kev's derivation, and I brought it up in response to espen180's post, I never claimed that it was supposed to be relevant to Dmitry7's post.

The thread started as one thread, the Dmitry7 thread. Your citation was inappropriate in the context. It is quite possible that when the split was made, the timestamps were messed up as well. Anyways, I have posted clearly what formula goes with what thread.

-The correct answer to Dmitry7's question is:

\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}}\sqrt{\frac{1-(r_1sin\theta_1\omega/c\sqrt{1-r_s/r_1})^2}{1-(r_2sin\theta_2\omega/c\sqrt{1-r_s/r_2})^2}}

-The correct answer to espen180's question is :\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{r\omega sin(\theta)}{c \sqrt{1-r_s/r}})^2}

I hope that this clarifies things once and for all.

 

[JesseM]

The thread started as one thread, the Dmitry7 thread.

No it didn't, this seems to be your basic misunderstanding. As I already said, you can look at the timestamps in the upper left of each post to see that my post responding to espen180's post was posted 8 minutes before Dmitry7's very first post. The actual time displayed on your browser may depend on your time zone, but on my browser espen180's OP was from Jun2-10, 02:43 PM, my post #3 responding to him (and citing kev's posts) was from Jun2-10, 03:03 PM, while Dmitry7's first post on the split thread was from Jun2-10, 03:11 PM.

It is quite possible that when the split was made, the timestamps were messed up as well.

Isn't it a little more likely that your memory is playing tricks on you? For myself, I remember pretty clearly that espen180's post was in fact the original post when I responded to it.

 

[starthaus]

No it didn't, this seems to be your basic misunderstanding. As I already said, you can look at the timestamps in the upper left of each post to see that my post responding to espen180's post was posted 8 minutes before Dmitry7's very first post. The actual time displayed on your browser may depend on your time zone, but on my browser espen180's OP was from Jun2-10, 02:43 PM, my post #3 responding to him (and citing kev's posts) was from Jun2-10, 03:03 PM, while Dmitry7's first post on the split thread was from Jun2-10, 03:11 PM.

Isn't it a little more likely that your memory is playing tricks on you? For myself, I remember pretty clearly that espen180's post was in fact the original post when I responded to it.

espen180 thread was split from Dmitry7 thread. Besides, if you paid attention to the correct formulas, they both need to contain sin(\theta) and \omega is \frac{d\phi}{dt}, not \frac{d\theta}{dt}. The reason for the error is that kev picked up not only a wrong formula from pervect but also a truncated one. It is the \phi coordinate that describes the complete circle, not \theta. See here. So, kev's post 8 is still wrong becuse he started with the wrong metric and used the wrong definitions all along.

 

[espen180]

A more general equation is:

\frac{\text{d}\tau}{\text{d}t}= \sqrt{\frac{1-r_s/r}{1-r_s/r_o}} \sqrt{1- \left (\frac{1-r_s/r_o}{1-r_s/r} \right )^2 \left (\frac{\text{d}r}{c\text{d}t} \right )^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right ) \left(\frac{r \text{d}\theta}{c\text{d}t}\right)^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right) \left(\frac{r \sin \theta \text{d}\phi}{c\text{d}t}\right)^2 }

where r_o is the Schwarzschild radial coordinate of the stationary observer and r is the Schwarzschild radial coordinate of the test particle and dr and dt are understood to be measurements made by the stationary observer at r_o in this particular equation.

For r_o = r the time dilation ratio is:

\frac{\text{d}\tau}{\text{d}t} = \sqrt{1-\frac{v&#039;^2}{c^2}}

in agreement with the generally accepted fact that local measurements made in a gravitational field are Minkowskian.

Let's first combine \text{d}\theta^2+\sin^2\theta \text{d}\phi^2=\text{d}\Omega^2 and so simplify the equation to

\frac{\text{d}\tau}{\text{d}t}= \sqrt{\frac{1-r_s/r}{1-r_s/r_o}} \sqrt{1- \left (\frac{1-r_s/r_o}{1-r_s/r} \right )^2 \left (\frac{\text{d}r}{c\text{d}t} \right )^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right ) \left(\frac{r \text{d}\Omega}{c\text{d}t}\right)^2 }

Working backwards to get back to the metric gives me

c^2\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=c^2\frac{1-r_s/r}{1-r_s/r_o}-\left(\frac{1-r_s/r}{1-r_s/r_o}\right)^{-1}\left (\frac{\text{d}r}{\text{d}t} \right )^2-r^2\left (\frac{\text{d}\Omega}{\text{d}t} \right )^2

c^2\text{d}\tau^2=c^2\frac{1-r_s/r}{1-r_s/r_o}\text{d}t^2-\left(\frac{1-r_s/r}{1-r_s/r_o}\right)^{-1} \text{d}r^2-r^2\text{d}\Omega^2

I was hoping that doing this would lead me to an explanation as to where the \frac{1-\frac{r_s}{r}}{1-\frac{r_s}{r_0}} came from, but it seems it did not.

I do observe that in modeling this metric the metric coefficients are found by taking the ratio of the coefficients of the particle wrt an observer at infinity to the coefficients of the observer at r_0 to the same observer at infinity, but could I have an explanation of why that works?

 

[JesseM]

espen180, can you settle this? When you originally wrote the OP, were you starting a new thread at the time or were you just responding to a prior thread that had been started by Dmitry67?

 

[espen180]

Isn't it a little more likely that your memory is playing tricks on you? For myself, I remember pretty clearly that espen180's post was in fact the original post when I responded to it.

espen180 thread was split from Dmitry7 thread.

This thread was not split from Dmitry7's thread. I started a new thread with the OP. I hope this settles that dispute.

Besides, if you paid attention to the correct formulas, they both need to contain sin(\theta) and \omega is \frac{d\phi}{dt}, not \frac{d\theta}{dt}. The reason for the error is that kev picked up not only a wrong formula from pervect but also a truncated one. It is the \phi coordinate that describes the complete circle, not \theta. See here. So, kev's post 8 is still wrong becuse he started with the wrong metric and used the wrong definitions all along.

Why not just contract the angle differentials into \text{d}\theta^2+\sin^2\theta\text{d}\phi^2=\text{d}\Omega^2 and avoid the problem alltogether?

Kev's post #8 is in agreement with all the references I can find on the Schwartzschild metric, and the algebra checks out. What, in your opinion, is the right metric and definitions?

 

[starthaus]

Why not just contract the angle differentials into
\text{d}\theta^2+\sin^2\theta\text{d}\phi^2=\text{d}\Omega^2 and avoid the problem alltogether?

Because \phi and \theta are independent coordinates. So your hack is illegal.

Kev's post #8 is in agreement with all the references I can find on the Schwartzschild metric, and the algebra checks out. What, in your opinion, is the right metric and definitions?

Nope, it doesn't. Look it up.

 

[starthaus]

espen180, can you settle this? When you originally wrote the OP, were you starting a new thread at the time or were you just responding to a prior thread that had been started by Dmitry67?

Not relevant. What is relevant is that post 8 by kev is wrong. For a list of errors see here.

 

[espen180]

Because \theta and \phi are independent coordinates.

But you have spherical symmetry, and since the choice of the \theta axis is arbitrary, you can always define a new single coordinate which represents the total angular distance traversed, right?

Nope, it doesn't. Look it up.

I don't have a book handy to look it up in. I can only observe that other PF members like JesseM seem to have given him their support.

 

[starthaus]

But you have spherical symmetry, and since the choice of the \theta axis is arbitrary, you can always define a new single coordinate which represents the total angular distance traversed, right?

Nope. Like I said, you need to read about Schwarzschild metric and Schwarzschild coordinates.
Contrary to your beliefs, \thetaand \phi are not interchangeable.

I don't have a book handy to look it up in.

Google is your friend. Try "Schwarzschild metric", "Schwarzschild coordinates"

I can only observe that other PF members like JesseM seem to have given him their support.

This is not a scientific criterion.I know that you are a big fan of kev's from other encounters but this is not a scientific criterion either. I posted kev's errors in the thread where he did his derivation.

 

[espen180]

Nope. Like I said, you need to read about Schwarzschild metric and Schwarzschild coordinates.
Contrary to your beliefs, thetaand \phi are not interchangeable.

Google is your friend. Try "Schwarzschild metric"
This is not a scientific criterion. I posted kev's errors in the thread where he did his derivation.

I realize that the two independent angle coordinates hav different definitions, but you must also realize that there is no preferred coordinate systems.

Let me elaborate on the contraction. Define new angle coordinates \text{d}\theta^2 + \sin^2\theta\text{d}\phi^2=\text{d}\Sigma^2 + \sin^2\Sigma\text{d}\Omega^2 and define the orientation of this new coordinate system such that \Sigma = \frac{\pi}{2}. Since keeping the same angular orientation of the coordinates is not an issue due to spherical symmetry, there is no problem defining \text{d}\theta^2 + \sin^2\theta\text{d}\phi^2=\text{d}\Omega^2

As for the Schartzschild metric, I know it by heart. Kev is using the correct metric

c^2\text{d}\tau^2=c^2\left(1-\frac{r_s}{r}\right)\text{d}t^2-\frac{1}{1-\frac{r_s}{r}}\text{d}r^2-r^2\text{d}\theta^2-r^2\sin^2\theta\text{d}\phi^2

If you read pervects original derivation you would know that kev was working with a particle in orbit around the equator, where \theta=\frac{\pi}{2}. But he would not need to. He could just have used the nagle contraction explained above to shift the coordinates such that motion around the equator was realized.

 

[starthaus]

As for the Schartzschild metric, I know it by heart. Kev is using the correct metric

c^2\text{d}\tau^2=c^2\left(1-\frac{r_s}{r}\right)\text{d}t^2-\frac{1}{1-\frac{r_s}{r}}\text{d}r^2-r^2\text{d}\theta^2-r^2\sin^2\theta\text{d}\phi^2

Good for you.

If you read pervects original derivation you would know that kev was working with a particle in orbit around the equator, where \theta=\frac{\pi}{2}. But he would not need to. He could just have used the nagle contraction explained above to shift the coordinates such that motion around the equator was realized.

So what? his derivation is wrong just the same.