Understanding Time Dilation Equations: Special vs. General Relativity

[Zman]

I would like to know if I have understood the following or not;

There are two time dilation equations that I am using;
One from special relativity, involving the Lorentz factor;

$$t = \frac{t_0}{\sqrt{1 - v^2/c^2}}$$


And one from general relativity, the Schwarzschild metric;

$$t = t_f{\sqrt{1 - 2GM/rc^2}$$

In the SR scenario, to is the observer's clock and t is the moving body’s clock.
As v gets bigger (approaches c), t becomes bigger.
But as this indicates time dilation t’ and to must represent time spans.

In the GR scenario, as a small body approaches a large body, the small body’s clock dilates. The reference clock in this case is at infinity and is represented by the symbol tf. This clock is analogous to the observer’s clock in the SR scenario.
But as r gets smaller t gets smaller, so in this case it is dealing with ‘relative Time Flow’ as opposed to ‘relative Time Span’.

Where Time Span = 1/ Time Flow


Does this sound right?

 

[JesseM]

I would like to know if I have understood the following or not;

There are two time dilation equations that I am using;
One from special relativity, involving the Lorentz factor;

$$t = \frac{t_0}{\sqrt{1 - v^2/c^2}}$$And one from general relativity, the Schwarzschild metric;

$$t = t_f{\sqrt{1 - 2GM/rc^2}$$

In the SR scenario, to is the observer's clock and t is the moving body’s clock.

You've got it backwards there--if we have two events which occur on the worldline of a clock, then t₀ is the time between events as measured by the clock itself, while t is the time between the same events as measured in the observer's frame where the clock is moving.

As v gets bigger (approaches c), t becomes bigger.
But as this indicates time dilation t’ and to must represent time spans.

Yes. For example, if a clock is moving at 0.6c relative to an observer, and between two events on its worldline it ticks forward by t₀ = 20 seconds, then in the frame of the observer the time interval between these events is a larger t = 25 seconds, meaning the observer perceives the moving clock to have been running slow during those 25 seconds.

In the GR scenario, as a small body approaches a large body, the small body’s clock dilates. The reference clock in this case is at infinity and is represented by the symbol tf. This clock is analogous to the observer’s clock in the SR scenario.
But as r gets smaller t gets smaller

Yes, the clock at some finite r will tick less time between two events on its worldline than the time between these events as measured in Schwarzschild coordinates, a coordinate system where the clock at infinity keeps pace with coordinate time. In other words, clocks closer to the source of gravity run slower in Schwarzschild coordinates. See also the outside a non-rotating sphere section of wikipedia's gravitational time dilation article.

so in this case it is dealing with ‘relative Time Flow’ as opposed to ‘relative Time Span’.

Both equations deal with time-spans.

 

[Zman]

Yes I did get the SR equation backwards.
Completely fundamental and I had it back to front.

Thanks for your help.

I had assumed that the subscript ‘o’ into stood for the (stationary) observer’s clock and not the time recorded on the moving clock which is time dilated relative to the observer’s clock.

 

[Zman]

Can it be confirmed whether I have used Proper and Coordinate Time correctly below;

Lorentz factor;

$$t = \frac{t_0}{\sqrt{1 - v^2/c^2}}$$

$$CoordinateTime = \frac{ProperTime}{\sqrt{1 - v^2/c^2}}$$


Schwarzschild metric;

$$t = t_f{\sqrt{1 - 2GM/rc^2}$$

$$ProperTime = CoordinateTime{\sqrt{1 - 2GM/rc^2}$$


Cheers and thanks
Zman