# How can I calculate time difference of things moving at different speeds?

Satellites orbiting earth are faster than it, and hence from our perspective they age slower, or, for us time moves faster.

Is there a way to calculate this difference?

Even though earth is fast, can I set the speed of earth to 0, since the satellite also moves at the same speed + his?

If any of you by any chance know the factor of how much slower time is moving for the satellites I'd appreciate this, too, but I do care more about the formula.

• – PM 2Ring May 15 at 15:51
• Your question asks about the kinematic time dilation (effect due to speed) for a satellite, but for a satellite there is also a gravitational effect, which has the opposite sign. Either effect can dominate, depending on the altitude. – Ben Crowell May 15 at 17:21
• en.wikipedia.org/wiki/Gravitational_time_dilation#/media/… – safesphere May 15 at 17:46

Speed is relative, that means it has meaning only in reference to something. So in this case the best reference to choose is the Earth itself and the speed of Earth relative to Earth is always zero.

You cannot measure speed of some object without reference point. The speed is distance that is traveled in given time. But the distance measured depends on the reference point. F.e. when you sit in a train the distance you traveled relative to the train is zero. But if the train moves, relatively to earth your distance traveled will not be zero.

Then the question naturally arises, if there is some nice reference point that is the most convenient to use. The answer is, there is - the inertial one. That is the system in which space and time are homogenous and isotropic - meaning same everywhere and in every direction.

But there are infinitely many such systems all of them moving with constant velocity in respect to each other. So the second natural question arises, which inerial system is the best? Now the answer is in general there is none.That is called relativity principle, in its first form formulated by Galileo (at least i think, it is called galileo relativity principle) and later generalized by Einstein.

The relativity principle states that you cannot distinguish two inertial reference frames by physics alone. That means all that can happen in one system, can also happen in another. F.e. playing ping-pong is governed by some physics. But the game will feel the same wheter you are playing on earth, or on train moving with constant velocity in respect to Earth. If the velocity of train wouldn't be constant, then you would feel something different, like you do when train is slowing down or turning.

Now, here is the celebrated formula for time dilatation: $$\Delta t'=\gamma\Delta t$$ where lorentz factor is given by the relative speed (v) of the two frames: $$\gamma=1/\sqrt{1-v^2/c^2}$$ where c is the speed of light.

But there is one further physics involved. Because sattelite is on the orbit there is additional time dilatation caused by curvature of spacetime (gravitation): $$\Delta t'=\Delta t \sqrt{1-\frac{3r_s}{2r_2}}/\sqrt{1-\frac{3r_s}{2r_1}}$$ where $$r_1$$ and $$r_2$$ are radial distances of the observer and sattelite from the centre of earth and $$r_s=2GM/c^2=9mm$$ is schwarschild radius given by gravitational constat ($$G$$), mass of Earth ($$M$$) and speed of light. This radius tells you how small would Earth need to be to become a black hole.

So the final formula is: $$\Delta t'=\frac{\sqrt{1-\frac{3r_s}{2r_2}}}{\sqrt{1-v^2/c^2}\sqrt{1-\frac{3r_s}{2r_1}}}\Delta t$$

But the gravitational time dilatation is not as simple as dilatation due to relative motion. In relative motion, the time dilatation is always proportial with proportionality factor $$\gamma$$, but gravitational dilatation is in general much more complicated.

• I'd like to upvote your reply, it certainly deserves many more of those, yet I can not cause karma too low. However, you explained everything as easily as possible, thank you for that! – bv_Martn May 16 at 5:08