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I am a newbie but as far as I understand that there are two sources of time dialations, one due to relativistic speeds and the other is due to gravitational field. If I look at these equations they do look pretty similar and if you solve it for v then you get the escape velocity, which seems not random.

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So, you need to match the exact time dilation of earth’s gravitional pull to escape from it.

So, my question is this the cause of escape velocity or something magically occur?

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    $\begingroup$ Also physics.stackexchange.com/q/246133 and physics.stackexchange.com/q/39176 $\endgroup$
    – Vincent Thacker
    Commented Apr 13 at 10:14
  • $\begingroup$ This is essentially asking about the coincidence between Newtonian escape velocity and Schwarzschild radius, which has already been asked many times. $\endgroup$
    – Vincent Thacker
    Commented Apr 13 at 10:27
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    $\begingroup$ In what way do they look similar? That they both contain factors involving $\sqrt{1-x}$ (albeit one is a multiplier and the other a divisor)? $\endgroup$
    – Kyle Kanos
    Commented Apr 13 at 15:31
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    $\begingroup$ The Schwarzschild radius is there in one of the equations, but (a) it's not clear OP is aware of that, and (b) the OP's question is explicitly about the connection between kinematic and gravitational time dilation. This is related indirectly to the Schwarzschild radius but none of the linked questions address that, and it's not obvious to the beginning student. $\endgroup$
    – Eric Smith
    Commented Apr 13 at 19:09
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    $\begingroup$ Hey I finally managed to understand what you mean: If you solve the lover equation for v, you get the escape velocity in a such situation where the t and the t' in the lower equation describe a gravitational time dilation. (The ' seems to be missing) $\endgroup$
    – stuffu
    Commented Apr 14 at 3:36

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Light that is sent up redshifts.

redshift factor = energy factor (because E=hf)

redshift factor = time dilation factor (obviously)

Solving the above system of equations we get:

time dilation factor = energy factor

In the above time dilation meant gravitational time dilation.

Now let us show that: kinetic time dilation factor = energy factor

$E=\gamma mc^2$

$ t=\gamma t'$

Yes, the factor is $\gamma$

So now we understand that:

kinetic time dilation factor = ratio of energies

and

gravitational time dilation factor = ratio of energies

Oh yes, escape velocity is ratio of energies too:

kinetic energy / mass

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