Understanding Time Dilation Equations: Special vs. General Relativity

In summary: In SR, to is the observer's clock and t is the moving body's clock. In GR, as a small body approaches a large body, the small body's clock dilates. 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". Does this sound right?
  • #1
Zman
96
0
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;

[tex]t = \frac{t_0}{\sqrt{1 - v^2/c^2}}[/tex]


And one from general relativity, the Schwarzschild metric;

[tex]t = t_f{\sqrt{1 - 2GM/rc^2}[/tex]

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?
 
Physics news on Phys.org
  • #2
Zman said:
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;

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

[tex]t = t_f{\sqrt{1 - 2GM/rc^2}[/tex]

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.
Zman said:
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.
Zman said:
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.
Zman said:
so in this case it is dealing with ‘relative Time Flow’ as opposed to ‘relative Time Span’.
Both equations deal with time-spans.
 
Last edited:
  • #3
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.
 
  • #4
Can it be confirmed whether I have used Proper and Coordinate Time correctly below;

Lorentz factor;

[tex]t = \frac{t_0}{\sqrt{1 - v^2/c^2}}[/tex]

[tex]CoordinateTime = \frac{ProperTime}{\sqrt{1 - v^2/c^2}}[/tex]


Schwarzschild metric;

[tex]t = t_f{\sqrt{1 - 2GM/rc^2}[/tex]

[tex]ProperTime = CoordinateTime{\sqrt{1 - 2GM/rc^2}[/tex]


Cheers and thanks
Zman
 

1. What is time dilation?

Time dilation is a phenomenon in which time appears to pass at different rates for observers in different frames of reference. This is due to the effects of gravity and relative motion on the flow of time.

2. What is the difference between special and general relativity?

Special relativity is a theory that explains the relationship between time, space, and motion in the absence of gravity. General relativity, on the other hand, is a theory that explains the effect of gravity on the fabric of space-time.

3. How does time dilation occur in special relativity?

In special relativity, time dilation occurs due to the differences in the relative speeds between two observers. As an object approaches the speed of light, time appears to slow down for that object.

4. How does time dilation occur in general relativity?

In general relativity, time dilation occurs due to the effects of gravity on the fabric of space-time. The stronger the gravitational pull, the slower time appears to pass for an observer in that region.

5. How are time dilation equations derived?

The time dilation equations are derived from the principles of special and general relativity, along with the Lorentz transformations. These equations take into account factors such as relative velocity and gravitational potential to calculate the difference in the passage of time between different observers.

Similar threads

  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
17
Views
2K
  • Special and General Relativity
Replies
11
Views
941
  • Special and General Relativity
2
Replies
45
Views
2K
  • Special and General Relativity
4
Replies
115
Views
5K
  • Special and General Relativity
Replies
21
Views
1K
  • Special and General Relativity
Replies
2
Views
996
  • Special and General Relativity
Replies
20
Views
747
  • Special and General Relativity
Replies
32
Views
1K
  • Special and General Relativity
Replies
18
Views
1K
Back
Top