# Gravitational Time Dilation: Radius & Clock Rate Variation Explained

• I
• Zman
In summary, 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.
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

Zman said:
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.

Last edited:
Ibix said:
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.

## 1. What is gravitational time dilation?

Gravitational time dilation is a phenomenon in which time appears to pass slower in regions with stronger gravitational fields. This is due to the curvature of spacetime caused by massive objects, which affects the flow of time.

## 2. How does the radius of an object affect gravitational time dilation?

The radius of an object does not directly affect gravitational time dilation. However, the mass of an object, which is closely related to its radius, does have an impact on the strength of its gravitational field and therefore the amount of time dilation experienced.

## 3. What is clock rate variation in relation to gravitational time dilation?

Clock rate variation refers to the difference in the rate at which time passes between two clocks in different gravitational fields. The closer a clock is to a massive object, the slower it will appear to tick compared to a clock in a weaker gravitational field.

## 4. Can gravitational time dilation be observed in everyday life?

Yes, gravitational time dilation has been observed and measured in various experiments and observations. For example, GPS satellites have to account for the effects of time dilation in order to accurately calculate the positions of objects on Earth.

## 5. Is gravitational time dilation the same as time travel?

No, gravitational time dilation does not allow for time travel. It simply means that time appears to pass at a different rate in different gravitational fields. Time travel, on the other hand, involves physically moving through time to the past or future, which is not possible according to our current understanding of physics.

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