# Gravitational Time Dilation

DW
pmb_phy said:
...$$\int \bold g_{inc} \bullet d\bold s = -4\pi G M_{inc}$$...
...$$\Phi(r){<} = -\frac{Mr^2}{2R^3} + C$$...
...$$\Phi(r)_{<} = -\frac{Mr^2}{2R^3} -\frac{3GM}{2R}$$...
Corrections:
$$\int \bold g_{inc} \bullet d\bold a = -4\pi G M_{inc}$$
$$\Phi(r)_{<} = -\frac{GMr^2}{2R^3} + C$$
$$\Phi(r)_{<} = -\frac{GMr^2}{2R^3} -\frac{3GM}{2R}$$

russ_watters
Mentor
Chronos said:
Gack! Remember once you reach the center of earth [or mars] the gravitational potential is exactly.. zero.
That's gravitational potential energy you are talking about. Not the field strength (as I understand it "gravitational potential" is the field strength).

DW
russ_watters said:
That's gravitational potential energy you are talking about. Not the field strength (as I understand it "gravitational potential" is the field strength).
Neither the gravitational potential, nor the gravitational potential energy which is proportional to that are zero at the center when as in this case the zero point is taken to be at infinity.

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pervect
Staff Emeritus
russ_watters said:
That's gravitational potential energy you are talking about. Not the field strength (as I understand it "gravitational potential" is the field strength).
The quantity $$\Phi$$ is referred to in my text as just "The Newtonian Potential". See MTW, "Gravitation", pg 445,1073 and elsewhere. It's equal to the Newtonian gravitational potential energy per unit mass. It has the additional requirement that it is zero at infinity, so it doesn't have the same degree of freedom to add a scalar quantity to it that potential energy does in classical mechanics. Hence the confusion about the value of the potential in the center of a mass. Classically, you could set the potential energy at the center of the mass to zero, and to a positive value at infinity. But here we require the potential at infinity to be zero.

It's useful in general relativity in the weak field limit, in spite of its name.

In the literature of PPN formalism, $$-\Phi$$ is traditionally called U (MTW, pg 1073), though it's called by the more-difficult-to-type $$\Phi$$ elsewhere in the book.

Hi to All,

I want to know the gravitational time dilation on Jupiter. I saw the equation for it but I'm not sure what the value of "to" is. I'm looking for "t" on the surface is "to" the time at the center of jupiter? If I set "to" to 1, "t" is bigger, indicating a gain of time rather than a loss.