Some easier equotation for gravitational time dilation?

[SpiderET]

Im looking for some easy equotation which would help me to make graph of time dilation depending on gravity expressed not in relation to Schwarzschild radius but simply and directly to gravitational acceleration. I would like to get some easy graph similar to this:
http://frigg.physastro.mnsu.edu/~eskridge/astr101/timedilat.gif

But instead of time dilation / speed there would be time dilation / gravitational accelaration in m/s².
Is something like this possible or I can't get rid of this radius in the equotation?

 

[A.T.]

Im looking for some easy equotation which would help me to make graph of time dilation depending on gravity expressed not in relation to Schwarzschild radius but simply and directly to gravitational acceleration.

The coordinate acceleration of a free faller in a hovering frame, is related to the radial derivative of gravitational time dilation. But note that you cannot have gravitational time dilation as an unambiguous function of that acceleration, because there are radii with the same acceleration but different gravitational time dilation (inside and outside of the mass).

 

[SpiderET]

The coordinate acceleration of a free faller in a hovering frame, is related to the radial derivative of gravitational time dilation. But note that you cannot have gravitational time dilation as an unambiguous function of that acceleration, because there are radii with the same acceleration but different gravitational time dilation (inside and outside of the mass).

You mean that for example if we have the same gravitational acceleration on the surface of Saturn as on Earth, then there is different time dilation because there is different radius? I am not an expert in GR, but my guess would be that the difference in time dilation would be miniscule and it is still a possibile to create some easy function between gravitational acceleration and time dilation which would give at least some approximation of the exact function which includes radius.
To my surprise it seems nobody has tried something similar so I have to use my poor math abilities and create it.

 

[stevendaryl]

Im looking for some easy equotation which would help me to make graph of time dilation depending on gravity expressed not in relation to Schwarzschild radius but simply and directly to gravitational acceleration.

You can approximate proper time, in the case of weak (constant) gravitational fields and slow velocities by using the gravitational potential $\Phi$ and the velocity $v$:

$\delta \tau \approx \delta t (1 + \frac{\Phi}{c^2} - \frac{v^2}{2 c^2})$

So the two kinds of "energy" in Newtonian physics--gravitational potential energy and kinetic energy--work in opposite ways to affect time dilation. Clocks with more kinetic energy (larger $v$) tend to run slower, while clocks with larger potential energy (larger $\Phi$) tend to run faster.

Note that what comes into the time dilation formula is gravitational potential, not gravitational acceleration. The two are related by:

$\vec{g} = - \nabla \Phi$

where $\nabla$ is the gradient operator: $\nabla = \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z}$

 

[A.T.]

You mean that for example if we have the same gravitational acceleration on the surface of Saturn as on Earth, then there is different time dilation because there is different radius?

Or the center of the mass where gravitational acceleration is zero (like far away from the mass), but the gravitational time dilation is maximal (relative to far away from the mass).

 

[SpiderET]

Or the center of the mass where gravitational acceleration is zero (like far away from the mass), but the gravitational time dilation is maximal (relative to far away from the mass).

My humble opinion is that this is a wrong interpretation of the equotation, because there is difference if you calculate time dilation for example for the center of Sun and time dilation on the surface of the Sun. If you have radius zero, than you have also M zero, because you can't calculate the M of Sun which is above the radius. So in the center of the Sun there is zero time dilation.

 

[A.T.]

So in the center of the Sun there is zero time dilation.

Nope, see Steven's post. Minimal potential, minimal clock rate.

 

[Nugatory]

My humble opinion is that this is a wrong interpretation of the equotation, because there is difference if you calculate time dilation for example for the center of Sun and time dilation on the surface of the Sun. If you have radius zero, than you have also M zero, because you can't calculate the M of Sun which is above the radius. So in the center of the Sun there is zero time dilation.

You are confusing gravitational force with gravitational potential. You are right that the force is zero at radius zero, but the potential is not (this is standard classical mechanics, no relativity needed) and it's the potential that goes into the time dilation equation that stevendaryl posted.

 

[SpiderET]

Nope, see Steven's post. Minimal potential, minimal clock rate.

I have checked it and to my surprise you are right. Its completely contraintuitive for me.
Better said I still have hard time to believe it. There were several experiments with atomic clocks but all were above surface, not a single one was under surface of Earth. Was there any experiment which would confirm it?

Measurements in which the only effect was gravitational have been conducted by Iijima et al. between 1975 and 1977. They carried a commercial cesium clock back and forth from the National Astronomical Observatory of Japan in Mitaka, at 58 m above sea level, to Norikura corona station, at 2876 m above sea level, corresponding to an altitude difference of 2818 m. During the times when the clock stayed at Mitaka, it was compared with another cesium clock. The measured change in rate was (29±1.5)×10−14, consistent with the result of 30.7×10−14 predicted by general relativity.[9]
In 1976, Briatore and Leschiutta compared the rates of two cesium clocks, one in Turin 250 m above sea level, the other at Plateau Rosa 3500 m above sea level. The comparison was conducted by evaluating the arrival times of VHF television synchronization pulses and of a LORAN-C chain. The predicted difference was 30.6 ns/d. Using two different operating criteria, they found differences of 33.8±6.8 ns/d and 36.5±5.8 ns/d, respectively, in agreement with general relativity.[10] Environmental factors were controlled far more precisely than in the Iijima experiment, in which many complicated corrections had to be applied.
In 2010, Chou et al. performed tests in which both gravitational and velocity effects were measured at velocities and gravitational potentials much smaller than those used in the mountain-valley experiments of the 1970s. It was possible to confirm velocity time dilation at the 10−16 level at speeds below 36 km/h. Also, gravitational time dilation was measured from a difference in elevation between two clocks of only 33 cm.[11][12]
Nowadays both gravitational and velocity effects are, for example, routinely incorporated into the calculations used for the Global Positioning System.[13]

 

[A.T.]

I have checked it and to my surprise you are right. Its completely contraintuitive for me.
Better said I still have hard time to believe it. There were several experiments with atomic clocks but all were above surface, not a single one was under surface of Earth. Was there any experiment which would confirm it?

You would have to dig really deep, to reach a region where the gravitation acceleration decreases, to confirm that gravitation time dilation still increases (if that's what you want to confirm)

But quantitative experiments already confirm that gravitation time dilation is a function of graviational potential, not of graviational acceleration.

 

[SpiderET]

You would have to dig really deep, to reach a region where the gravitation acceleration decreases, to confirm that gravitation time dilation still increases (if that's what you want to confirm)
But quantitative experiments already confirm that gravitation time dilation is a function of graviational potential, not of graviational acceleration.

Thanks for clarification, I wonder which experiments confirmed that time dilation is a function of gravitational potential, not of gravitational acceleration.

 

[A.T.]

Thanks for clarification, I wonder which experiments confirmed that time dilation is a function of gravitational potential, not of gravitational acceleration.

The ones you quoted, by being in quantitative agreement with General Relativity.

 

[Nugatory]

Thanks for clarification, I wonder which experiments confirmed that time dilation is a function of gravitational potential, not of gravitational acceleration.

Pretty much all of them. It's easy to see the difference because the acceleration goes as $1/r^2$ while the potential goes as $1/r$ outside the surface of the gravitating body. GPS wouldn't work if we had gotten this wrong.

You will sometimes see gravitational time dilation explained as the result of a light signal having to "climb out" of the potential well. This explanation isn't great, but it does provide an intuitive basis for why the time dilation depends on the total difference of potential between the observers, and not just the local strength of the gravitational field at the location of one of the observers.

 

[pervect]

Thanks for clarification, I wonder which experiments confirmed that time dilation is a function of gravitational potential, not of gravitational acceleration.

The simplest experiment is to check clock rate vs height. Experimentally, one can regard this as equivalent to measuring the change in frequency of a falling photon, (or the classical equivalent, such as a maser beam). If the clock rate was proportional to only the gravitational field, the change in clock rate (or alternatively, the measured frequency of a falling photon) would not change with height, it would only change insofar as the gravitational field changed. If clock rate is proportional to potential energy you expect the fractional change in frequency to be proportional to the product gh , where g is the gravitational acceleration and h is the height that the photon falls. Taking units into account , you get a fractional frequency change of
(1 + gh / c^2), as per the hyperphysics webpage link http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html.

The best experiment that comes to mind is on the same webpage as above, the Scout Rocket experiment, which I believe measured the frequency shift of a hydrogen maser as the rocket fell, and very accurately confirmed the GR formula. The Harvard Tower experiment (mentioned on the same page) is another experiment that confirms the formula, though it's not as good confirmation as the former.

Note that one way you'll find in textbooks of deriving the gravitational redshift formula (which as mentioned is the same experimentally as gravitational time dilation) is via energy conservation - more technically, energy conservation in the Newtonian limit. Suppose you have a particle-anti particle pair. If you let the pair fall in a gravitational field g, you expect each particle to gain an energy of mgh in a uniform gravitational field, where m is the mass of the particle, g is the gravitational acceleration, and h is the height that the particle fell.

If you convert the particle - antiparticle pair into photons by letting them annhilate each other, you expect the falling photon to gain or loose the same amount of energy that the particle would, This again corresponds to a time dilation / frequency change proportioanl to (1 + gh / c^2) as long as g is constant, i.e. you expect it to be proportional to energy, not the acceleration, if energy is to be conserved.