# Gravitational Time Dilation

## Main Question or Discussion Point

I'm looking at this equation for gravitational time dilation:

$$T = \frac{T_0}{\sqrt{1 - (2GM / rc^2)}}$$

I understand the relation of time dilation and velocity, and how v must be less than c, but I don't understand what exactly is implied here. At a certain point, M could be great enough such that the square root becomes negative or 0, or r could become small enough that the same thing happens. So what exactly does that mean? Is it possible that M or r could be a size such that you get an imaginary or undefined answer?

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DaveC426913
Gold Member
I'm looking at this equation for gravitational time dilation:

$$T = \frac{T_0}{\sqrt{1 - (2GM / rc^2)}}$$

I understand the relation of time dilation and velocity, and how v must be less than c, but I don't understand what exactly is implied here. At a certain point, M could be great enough such that the square root becomes negative or 0, or r could become small enough that the same thing happens. So what exactly does that mean? Is it possible that M or r could be a size such that you get an imaginary or undefined answer?
Sure. If M were infinite or r were zero, your answer could be 0. But how useful a solution is that in describing anything in the universe?

But no, it could never be negative.

pervect
Staff Emeritus
I'm looking at this equation for gravitational time dilation:

$$T = \frac{T_0}{\sqrt{1 - (2GM / rc^2)}}$$

I understand the relation of time dilation and velocity, and how v must be less than c, but I don't understand what exactly is implied here. At a certain point, M could be great enough such that the square root becomes negative or 0, or r could become small enough that the same thing happens. So what exactly does that mean? Is it possible that M or r could be a size such that you get an imaginary or undefined answer?
The square root becomes zero just at the event horizon of a black hole, where r=2GM/c^2. This is an indication of the fact that one cannot have a stationary observer exactly at the event horizon (one could have a stationary light beam, but a light beam isn't an observer).

One also cannot have a stationary observer inside the event horizon, i.e r < 2GM/c^2.

r here is the schwarzschild r cooridnate, btw.