Do gravitational time dilation effects cancel out or add up?

[smoothoperator]

If we use one gravitational field in GR and try to desribe it, we will see that from the perspective of the observer on the centre of mass of the object the clock on the hovering observer (farther away from the gravitational field) runs faster, while from the hovering observer's perspective the clock on the centre of mass observer runs slower.

Now what happens if we combine two gravitational fields? For instance if we use the Sun and the Earth and their combined gravitational fields. Does GTD add up, or does it cancel out? For instance, will an observer at the Earth still see that the hovering observer from the Earth's clock runs faster, but as he looks closer to the Sun the time will eventually slow down relative to his clock and the clock on the centre of the Sun will run slowly than his? How does this combination effect gravitational time dilation?

Regards and thanks for your patience.

 

[A.T.]

Does GTD add up, or does it cancel out?

Gravitational time dilation depends on the gravitational potential which doesn't exactly "add up" linearly, but it definitely doesn't cancel.

With two idenetical masses nearby the clocks at their centers will run at the same rates (if that's what you mean by 'cancel gravitational time dilation '). But a distant observer will see both clocks run slower than his, and slower than with just one of the masses (if that's what you mean by 'add up gravitational time dilation ').

 

[smoothoperator]

Gravitational time dilation depends on the gravitational potential which doesn't exactly "add up" linearly, but it definitely doesn't cancel.

With two idenetical masses nearby the clocks at their centers will run at the same rates (if that's what you mean by 'cancel gravitational time dilation '). But a distant observer will see both clocks run slower than his, and slower than with just one of the masses (if that's what you mean by 'add up gravitational time dilation ').

Can you analyze the Earth-Sun problem that I've mentioned before in the same context?

 

[PeterDonis]

what happens if we combine two gravitational fields?

For weak fields, the gravitational potentials add. (In general, GR is nonlinear, so arbitrary potentials do not add--in fact the "gravitational potential" is only really well-defined in certain classes of spacetimes. But for considering the Earth-Sun scenario, weak fields are sufficient.) For a gravitational potential $\phi$, the gravitational time dilation is $\sqrt{1 + 2 \phi / c^2}$, where we have normalized $\phi$ so that it goes to zero at infinity (and is negative at a finite distance from a gravitating mass). So time dilations don't add directly, but you can add potentials and use the final potential (the sum of all the individual ones) to calculate an overall time dilation.

For a single, spherically symmetric mass, in the weak field approximation, we have $\phi = - G M / r$, where $M$ is the mass and $r$ is the radial coordinate (note that this is not exactly equal to the radial distance, but we can ignore the difference here). So the time dilation factor is $\sqrt{ 1 - 2 G M / c^2 r }$.

For two spherically symmetric masses $M_1$ and $M_2$, and an object at a distance $r_1$ from the first mass and $r_2$ from the second mass, the total potential is $\phi = \phi_1 + \phi_2 = - G \left[ (M_1 / r_1) + (M_2 / r_2) \right]$, so the overall time dilation will be $\sqrt{1 - (G / c^2) \left[ (M_1 / r_1) + (M_2 / r_2) \right]}$. (Note that we are ignoring any effects due to the relative motion of the masses, or the motion of the object relative to the masses; i.e., we are not considering kinematic time dilation due to relative motion, only gravitational time dilation. This does not mean kinematic time dilation is negligible; it isn't. But the full formula including it would be more complicated.)

 

[A.T.]

Can you analyze the Earth-Sun problem that I've mentioned before in the same context?

Clock at Sun's center runs slower than at the Earth's center.

 

[PeterDonis]

Clock at Sun's center runs slower than at the Earth's center.

This reminds me that I forgot a key qualification in post #4: the formulas I gave for gravitational potential are only valid in the vacuum region outside the gravitating masses. Inside a mass (for example, inside the Earth or the Sun), the formula is different. However, its qualitative behavior is still the same: the potential due to a single gravitating mass continues to decrease as the center of the mass is approached, i.e., as $r$ goes to zero. (It just doesn't decrease as $1 / r$, which would be singular at $r = 0$. It decreases more slowly, to a finite negative value at the center that is roughly proportional to the mass of the object.)