Connection between kinetic and gravitational time dilation

Question

I am learning about relativity and am interested in time dilation. I now exactly understand where the formula of kinetic time dilation comes from and wondered about gravitational time dilation. I have some questions about this

1. The formulas of both resemble each other. I think v in the formula of kinetic time dilation is changed into $GM/r$ (don't really know what it means). I believe this is because gravitational time dilation is the same as kinetic but with an acceleration corresponding to a certain g-force. Is this correct and can someone explain a bit more? I have also read that you could calculate gravitational time dilation with kinetic but instead of using a speed using an acceleration with the same g-force that corresponds to the gravitational field, how does this work? Acceleration isn't the same as speed right, so...?

2. If I would calculate the time dilation between mars and earth do I need to do this with the gravitational fields of both, the velocity of both and with the gravity of the sun that works on both, then add everything and done? or do I need more/fewer factors and not add them but do something else?

these are the formulas that I refer to

-kinetic:

$Δt' = Δt/(1-v^2/c^2)$

-gravitational:

$Δt' = Δt/(1-2GM/rc^2)$ OR

$Δt' = Δt\times(1-2GM/rc^2)$ OR

$Δt' = Δt/(1-2gR/c^2)$

3. Which of those is correct/can I use to calculate the earth - Mars time dilation?

Sorry if the question is confusing, trying my best to understand it. Pls answer simply enough, I am not a physics university student (yet maybe :)) or something like that, just interested.

Answer 1

As a start, it's probably easier if you use standard formulas. You can find the kinetic time dilation calculator here: https://www.omnicalculator.com/physics/time-dilation

and the gravitational time dilation calculator here: https://www.omnicalculator.com/physics/gravitational-time-dilation

Both of these show the formulas.

In those formulas,

G= Newtons Gravitational Constant: 6.67 x 10^-11

M = Mass of the planet

c = speed of light

r = radius of the planet

In order to solve your question for Earth - Mars time dilation differential, it would be best if you use the gravitational time dilation calculator to find the Earth time dilation, then the Mars time dilation, and then compare them. The formula is based on empty space far from any gravitational force.

Let's take a close look at the gravitational time dilation formula:

G is Newtons gravitational constant. It was actually discovered by experiment, where they took small weighted balls suspended next to large weights and measured the gravitational attraction. This was answered in more detail here So it was not derived mathematically, but experimentally.

We think of "c" as the speed of light. But you can also think of it as a proxy for the rate of flow of time. In your question we are looking for how time flows on Earth compared to how it flows in deep space. So it is the rate of flow of time that we are interested in.

For "r" or the radius, if you hold a light bulb up close to your eye and then pull it away, you will notice that the light gets dimmer by the distance squared. This is known as the r-squared rule. The same thing holds for gravitational time dilation. It dissipates at the same r-squared rate. (Remember that r is measured from the centre of the planet, not the surface.)

"M" is the mass of the planet. It was Newton who discovered that gravity is directly related to Mass, and not to some other thing like size of the planet.

The "1-" part is because we are looking for the change in rate. It's like saying what is Bob's height compared to Sam's height? We don't care about the actual measurement, just the difference between them.

And the whole thing is put inside a square root sign because, again, we are dealing with squared distance.

We can ignore the delta-t because we can just set that at 1

Answer 2

The manifold of the universe has some intrinsic properties and these properties cause the kinetic time dillation.But stuff inside the manifold influence the manifold itself giving it new properties.These new properties cause the gravitational time dillation.