# Resultant time dilation from both gravity and motion

1. Jun 2, 2010

### 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?

2. Jun 2, 2010

### starthaus

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 $$d\theta=dr=0$$

Last edited: Jun 3, 2010
3. Jun 2, 2010

### 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.

4. Jun 2, 2010

### 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.

5. Jun 2, 2010

### espen180

Thank you very much. All replies were very useful.

6. Jun 3, 2010

### starthaus

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.

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

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 $$v=\frac{dr}{dt}$$

Last edited: Jun 3, 2010
7. Jun 3, 2010

### JesseM

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.

8. Jun 3, 2010

### DrGreg

I believe that the equation

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

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" (and repeated in post #46).

9. Jun 3, 2010

### starthaus

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

Last edited: Jun 3, 2010
10. Jun 3, 2010

### starthaus

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)

Last edited: Jun 3, 2010
11. Jun 3, 2010

### 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?

12. Jun 3, 2010

### JesseM

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?

13. Jun 3, 2010

### starthaus

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

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 $$d\theta=d\phi=0$$:

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

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

Last edited: Jun 3, 2010
14. Jun 3, 2010

### starthaus

Yes.

I don't define anything.

15. Jun 3, 2010

### espen180

How do I interpret it then?

16. Jun 3, 2010

### espen180

But it looks from the Schwartzschild metric that it would be

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

?

17. Jun 3, 2010

### 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?

18. Jun 3, 2010

### starthaus

yes, fine

19. Jun 3, 2010

### yuiop

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

Since $$v' = dr'/dt' = (dr/dt)/(1-r_s/r)$$

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

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

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

20. Jun 3, 2010

### starthaus

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.

21. Jun 4, 2010

### yuiop

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.

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:

$$\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{v'^2}{c^2}}$$

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.

22. Jun 4, 2010

### espen180

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

23. Jun 4, 2010

### yuiop

You are very welcome.

The equation I gave

$$\frac{\text{d}\tau}{\text{d}t}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-\frac{v '^2}{c^2}}$$

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:

$$\frac{\text{d}\tau}{\text{d}t} = \sqrt{1-\frac{v'^2}{c^2}}$$

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

Last edited: Jun 4, 2010
24. Jun 4, 2010

### JesseM

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 $$u = v \sqrt{1-\frac{r_s}{r}}$$.

25. Jun 4, 2010

### starthaus

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