# Does the recent Gravity Probe - B mission mean both the mass of an object and the spin of an object affect time?

Is it correct that this means that both the rotation of the earth and the size affect time? Would a result be that a person living on an earth-sized planet spinning faster make that person's perception of time relative to an earth-bound person different, and if so, in what way? Also, how massive must an object be and and how fast must an object rotate to have noticable effect?

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the Einstein field equations, describing the gravitational field are given by $$R_{\mu\nu}-\frac12 g_{\mu\nu}R = -\frac{8\pi G}{c^4}T_{\mu\nu}\ .$$

They relate in a complicated manner the gravitational field that can be seen as the metric $g_{\mu\nu}$ itself to the stress-energy tensor $T_{\mu\nu}$ that might also depend on $g$.

## Newtonian gravitation

If now velocities are small compared to the speed of light $c$, the stress-energy tensor approximately only consists of its time-component, $$T_{tt} \approx \rho c^2$$ and the metric is flat with the exception of $$g_{tt} \approx 1 + \frac{2 U}{c^2}$$ where we can find, directly from the Einstein equations, that the Newtonian $$\Delta U = 4\pi G \rho$$ holds. This is a static approach and we see that there is no dependence on any flux term $j_i \propto T_{ti}$.

## Velocity matters: rotating disc of dust

Considering the whole theory, we find that of course the metric will depend on contributions of all components of $T$ but only have a meaningful effect if associated characteristic velocities are approaching the speed of light.

A prominent analytic solution is that of a rigidly rotating disc of dust. Taking this solution, you can get an idea of how relativistic effects are important for the theory calculating the multipole moments $Q_n$ with respect to some relativistic parameter $\mu$ (corresponding also to the angular frequency $\Omega$ of the disc).
In the following picture you can see that the Kerr-spacetime is approached (from above!) for all moments $Q_n(\mu)$ for $\mu \rightarrow \infty$. This means that there is some $\mu$ where the effects of rotation dominate those of the mass itself.

(Picture taken from here.)

So, to conclude, there will only be some measurable time change for a person living on a rotating planet if it is extremely fastly rotating. It is hence needless to say that this person would have some other difficulties than to measure this deviation from an almost flat metric.

Sincerely

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