Undergrad Gravitational Time Dilation: Radius & Clock Rate Variation Explained

[Zman]

(Apologies I posted this initially as a conversation. Not familiar with the format)

I used the ‘gravitational time dilation’ equation to see how the clock rate varies with distance from the center of an object. I got the opposite result to what I was expecting.From Wikipedia;

Gravitational time dilation outside a non-rotating sphere

t0 is the proper time between events A and B for a slow-ticking observer within the gravitational field,

tf is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate),

r is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate),

rs is the Schwarzschild radius.I was interested to find out how the radius r varies with the time ratio t0/ tf for a given mass.

I plugged in t0/ tf = 1/10

This is effectively asking what is r when the fast clock is running 10 times faster than the slow clock.

The answer is r = rs X 100/99Then I asked what is r when the fast clock is running 2 times faster than the slow clock.

I plugged in t0/ tf = ½ expecting a smaller radius

I got the answer r = rs X 4/3 which is a bigger radius than the previous case.Clocks tick more slowly at the center than higher up. The higher up (the greater the radius) the faster a clock ticks relative to the center clock.Looking to clear up my confusion

 

[Ibix]

Clocks tick more slowly at the center than higher up. The higher up (the greater the radius) the faster a clock ticks relative to the center clock

Yes. Incidentally, the analysis you are doing assumes the clock is hovering (or sitting on a solid surface), not orbiting.

The numbers you give seem consistent with this - the slowest clock is at about 1.01$r_S$, the next fastest is higher up at 1.33$r_S$, and the fastest clock is at infinity.

Edit: so $r$ is the "altitude" of the lower clock. This is being compared to a clock at infinity.

Edit2: $t_0/t_f=1/2$ means that the clock at infinity ticks twice in the time it takes the lower clock to tick once.

 

[Zman]

Edit: so r is the "altitude" of the lower clock. This is being compared to a clock at infinity.

Yes, the lower clock running at half the rate of the clock at infinity will have a larger radius than the lower clock running at a tenth the rate of the clock at infinity. Obvious. Though I need to make a note of how to correctly interpret the equation.

Thank you for that.