# Acceleration/gravitation vs velocity and the Twin Paradox

I see in Professor Pogge’s explanation http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html dec. 2020. that

“Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion [2].

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away. As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these two relativistic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)!”

I realize this is about clocks in Earth orbit and not starry stravel.

Nevertheless, I note that the special relativity (velocity) vs general relativity (acceleration/gravity) effects seem to have different signs from the resting twin's point of view..

Does this imply that the travelling twin’s age at arrival back to the other twin can be adjusted by a choice of times for straight line velocity vs. acceleration?

• FWIW, Wikipedia has a nice diagram showing both kinds of time dilation for a body orbiting Earth: en.wikipedia.org/wiki/… Commented Dec 4, 2020 at 15:29

## 1 Answer

Yes, there will be both Special Relativity time dilation due to velocity and General Relativity time dilation due to gravitational potential. However, the amount of GR time dilation will be tiny. On Earth's surface it's approximately 2.19 seconds per century.

Wikipedia has several formulae for GR time dilation in the neighbourhood of a massive (non-rotating) sphere. This one parallels the SR formula:

$$t_0 = t_f \sqrt{1 - \frac{v_e^2}{c^2}}$$

$$t_0$$  is the proper time between two events for an observer close to the massive sphere, i.e. deep within the gravitational field.

$$t_f$$ is the coordinate time between the events for an observer at an arbitrarily large distance from the massive object [using Schwarzschild coordinates]

$$v_e$$ is the escape velocity [at that particular distance from the centre of the sphere].

So GR time dilation is only significant in extremely strong gravitational potentials, where the escape velocity is a significant fraction of c. There's also a rotational component to GR time dilation, but once again, it's negligible unless the spin is approaching relativistic speeds (like in the vicinity of a rotating black hole).

Presumably, you don't want to subject the travelling twin to such high accelerations. ;)

So typical treatments of the twin paradox can safely ignore GR time dilation and only deal with SR effects due to velocity and changing of frames. In SR, acceleration per se has no effect on time dilation. As I said here,

It's that change of reference frame that makes the big difference, not the acceleration. The acceleration is merely the mechanism whereby the change of reference frame is performed.

Of course, you can't change your velocity without acceleration, but in SR the acceleration doesn't cause time dilation separately from or additionally to the associated velocity. Your velocity (relative to some inertial frame) is the slope of your worldline, and your proper time is the length of your worldline.

• FWIW, here's another recent answer of mine on time dilation: physics.stackexchange.com/a/590047/123208 And here I show how to calculate velocity (& a couple of other things) in SR with constant acceleration: physics.stackexchange.com/a/345492/123208 Commented Dec 4, 2020 at 16:56
• It looks like - in Prof Pogge's quote - regarding a GPS sattelite, the GR effekt 45 us/d is much bigger than the SR effekt 7 us/d. Still the gravity effekt is relatively small for a hypothetical human passenger, possibly a twin. What is the difference here? Commented Dec 8, 2020 at 13:47
• @MikaelJensen The difference is that the satellite has a speed of $v_e/\sqrt{2}$, where $v_e$ is the escape velocity at that altitude. That's pretty fast compared to a car or plane, but it's very slow compared to the speed of the spaceship in a typical twin paradox scenario, where the traveller is going at a significant fraction of lightspeed. Even, at $v=0.28c$ the time dilation factor $\gamma$ is only $25/24$. Commented Dec 8, 2020 at 13:57