2
$\begingroup$

Relativistically, when two frames, say A & B are moving apart inertially, then the clock of B will look slow compared to that of A; and the same holds for the clock of A from B.

What does the comparable set up look like in a gravity well, with A at the bottom and B at the top?

My guess would be not, because the situation isn't symmetrical; whereas the SR setup is.

I'm not interested in a quantitative calculation; but simply whether they're slower or faster; and simply a confirmation or refutation of the guess.

$\endgroup$
6
  • 1
    $\begingroup$ en.wikipedia.org/wiki/Gravitational_time_dilation $\endgroup$ Dec 14, 2015 at 20:17
  • $\begingroup$ I'm marking this question for close as 'Too Broad' --- as it is more appropriate for, and well answered as, a google or wikipedia search. $\endgroup$ Dec 14, 2015 at 20:18
  • $\begingroup$ @dilthiummatrix: I've just scanned the Wikipedia entry that you suggested and it's not answered there - it simply states that there is a dilation effect, as there is in the SR setup - but that situation is symmetrical; I'm asking whether the same setup in a gravity well is symmetrical or not; my guess would be that it's not - because the situation is not; and I'm just looking for a confirmation or refutation of that guess. $\endgroup$ Dec 14, 2015 at 20:25
  • 2
    $\begingroup$ The asnwer is that the situation is indeed asymmetric, cf. this question $\endgroup$
    – ACuriousMind
    Dec 14, 2015 at 20:33
  • $\begingroup$ I've added this detail to the question. $\endgroup$ Dec 14, 2015 at 20:33

1 Answer 1

2
$\begingroup$

This is not an answer but just a mathematical reformulation, if I get you correctly.

So you're in a 2D world with coordinates $(t,r)$ and a metric such as

$g=\mathrm{diag}(-\kappa(r),\dfrac{1}{\kappa(r)}),$

e.g. $\kappa(r)=1-\frac{R}{r}$ for the Schwarzschild metric.

You want to solve the geodesic equation for the 2-trajectory $x(s)=(t(s),r(s))$ starting at an event $x=(0,r)$ and velocity $w=(w_0,w_1)$ with $g(w,w)=-1$. The solution is some $s$-parametrized line in spacetime that curves towards one direction, eventually ending up at $r=R$. And you want to know how the lightlight grid looks in this world. Here's a link to a talk on the Schwarzschild case.

You consider two guys at $x_A=(0,r_A)$ and $x_B=(0,r_B)$ with $r_B\gg r_A\gg R$ and velocities $v=(1,0)$ (the closer one is at rest, just free falling) and the other guy has some impulse away from $R$, say. These initial conditions give two curves who bend towards $r=R$. Each event in spacetime of course has two outgoing lightlike deformed cones. You're interested in how an the arc-lenght of one guys (say you choose a point and look at a distance of 1 along that guys trajectory) gets "projected" via the light-grid onto the other.

This link (on Gravitational time dilation) might also be relevant. And I now see this SE question.

enter image description here

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.