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I was reading "The young center of the Earth" by U.I. Uggerhøj et al., lately, and the standard gravitational redshift equation given in the paper is: enter image description here The link to the paper: https://arxiv.org/abs/1604.05507

But the standard gravitational redshift equation given by J. B. Hartle in his book Gravity : an introduction to Einstein's general relativity is: enter image description here The link to the e-book: https://birmingham-primo.hosted.exlibrisgroup.com/permalink/f/14feaum/44BIR_ALMA_DS51174597710004871 The standard gravitational redshift equation was discussed in Chapter 6: Gravity as Geometry, pp. 113-117.

In order to understand the equation in the paper by Professor U.I. Uggerhøj et al., I searched up reference 3 given in the paper, "On the universality of free fall, the equivalence principle, and the gravitational redshift" by Professor A.M. Nobili et al., in which the equation for gravitational time dilation is given as follows: enter image description here this perhaps explains why the sign of equation (4) in Uggerhøj et al.'s paper is negative, since the sign would indeed be negative after applying binomial expansion to equation (5) in order to get the frequency.

But equation (5) is different from the equation given by Professor J. B. Hartle, which is: enter image description here

And that brings my question, why are the signs different? I can see that their derivations/methods of the equations are different, but I couldn't think intuitively which one should be used to calculate the age difference between the Earth's surface and center and why. Maybe the sign difference is a question of different perspectives/viewpoints/observers that were used in the derivation?

Side note: just for the hell of it, I used J. B. Hartle's equation instead of U.I. Uggerhøj's to calculate the age difference between the Earth's centre and surface, and got a difference in days instead of years, which interestingly fits Richard Feynman's prediction, which Uggerhøj believes is wrong. enter image description here

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  • $\begingroup$ It depends how you define $\Delta \Phi$, $\omega$ and $\omega_0$ doesn't it? $\endgroup$
    – ProfRob
    May 19, 2022 at 8:51
  • $\begingroup$ Related: physics.stackexchange.com/a/69048/123208 $\endgroup$
    – PM 2Ring
    May 19, 2022 at 9:20
  • $\begingroup$ Note that for small $x$, $(1+x)^{-1}\approx(1-x)$. Also, $-gh$ gives the gravitational potential outside the sphere (where $h$ is measured from the centre of the sphere). It's not valid inside the sphere. $\endgroup$
    – PM 2Ring
    May 19, 2022 at 9:29

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In your first equation, $\omega$ is defined to to be the frequency (of a clock) at the centre of the Earth and $\omega_0$ is the frequency of a similar clock at the surface, both of which are defined with respect to some observer at infinity. $\Delta \phi$ is defined (in the mathematics, though not in the text!) as the difference between the potential at the surface and the potential at the centre, so that it is a positive quantity.

Thus if we let the surface be A and the centre be B $$ \omega_B = \omega_A\left( 1 - \frac{(\phi_A - \phi_B)}{c^2}\right)\ . $$ i.e. According to some universal observer at infinity, the clock at B has a lower frequency than the clock at A.

In Hartle's equation the frequency "received at B" would correspond to a signal received at the centre from the surface, while the "rate emitted at A" would correspond to the frequency measured at the surface of the Earth. Since $\phi_A-\phi_B>0$, then the signals received at the centre of the Earth would be observed to have a higher frequency. It isn't saying that the clock frequency is higher at the centre of the Earth - quite the opposite.

The two equations aren't really describing the same thing - Uggerhøj et al. is describing the ratio of clock frequencies in difference places as judged by a hypothetical observer at infinity, whereas the Hartle equation refers to the ratio of the clock frequencies for an observer at the position of one of the clocks. They are entirely consistent with each other, as I show now.

Suppose I was to use Hartle's equation to prove Uggerhøj et al.'s equation. I use the rate of receipt of signals in Hartle's equation for an observer at infinity to represent the apparent clock rates in Uggerhøj et al.'s equation; and at infinity, I can use $\phi_B = 0$.

For the two clocks, both of which have the same intrinsic rate $\omega_i$, then an observer at infinity observes the one at the surface of the Earth to be running at a rate $\omega_0$ (from Hartle's equation) $$ \omega_0 = \omega_i\left( 1 + \frac{\phi(R)}{c^2}\right)\ , $$ where $\phi(R)$ is the gravitational potential at the surface of the Earth and we recall that this is negative. The clock at the centre of the Earth emits signals at the same intrinsic frequency but they are received at a frequency (from Hartle's equation) $$\omega = \omega_i\left( 1 + \frac{\phi(0)}{c^2}\right)\ . $$ Dividing out these equations, we have $$\omega = \omega_0 \left(1+ \frac{\phi(R)}{c^2}\right)\left( 1 + \frac{\phi(0)}{c^2}\right)^{-1}\ .$$ Expanding the second bracket as a binomial, ignoring higher order terms and using Uggerhøj et al.'s definition of $\Delta \phi$ - $$\omega \simeq \omega_0 \left(1 + \frac{(\phi(0) - \phi(R))}{c^2}\right) = \omega_0 \left(1 - \frac{\Delta \phi}{c^2}\right)\ .$$

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