Time dilation in gravitational field

In summary, the conversation discusses a difficult task involving GPS-satellites and an expression that needs to be proved. The equations used include \tau = \sqrt{1-\frac{2\gamma M}{c^2r}}t and \frac{{t_e }}{{\sqrt {1 - \frac{{2\gamma M}}{{c^2 r}}} }} = \frac{{t_s }}{{\sqrt{1 - \frac{{2\gamma M}}{{c^2 (r + h)}}} }}. The speaker has attempted to solve the task but has been unsuccessful and is unsure of how to proceed. They also mention encountering issues with the LaTex-codes.
  • #1
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Homework Statement


I recently had a test in the theory of relativity, and there was one task which I could not solve. This one has bothered me since the day I had the test.
It was a compound task about GPS-satellites.
There is no tricky calculations, or anything like that. The task aims at proving an expression.

[tex]\frac{{t_e }}{{\sqrt {1 - \frac{{2\gamma M}}{{c^2 r}}} }} = \frac{{t_s }}{{\sqrt{1 - \frac{{2\gamma M}}{{c^2 (r + h)}}} }}\[/tex]

Where [tex]{t_e }\[/tex] is the time on Earth's surface, [tex]{t_s }\[/tex] is the time for the satellite, r is the radius from the center of Earth to the surface.

Homework Equations



[tex]\tau = \sqrt {1 - \frac{{2\gamma M}}{{c^2 r}}} t\[/tex]

The Attempt at a Solution


I've tried a lot of editing these expressions, but it didn't get me anywhere. I really do not now how to prove the expression. I do imagine that there are some theory I need to figure, so that I can do something quirky with the expressions or something.
 
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  • #2
Oh. It looks like something happened to the LaTex-codes. Any ideas of what to do?
 
  • #3


Dear student,

Thank you for reaching out for help with your test question. Time dilation in a gravitational field is a fundamental concept in the theory of relativity. It refers to the idea that time moves slower in a strong gravitational field compared to a weaker one. This is due to the curvature of spacetime caused by massive objects.

In the case of GPS satellites, they are orbiting the Earth at a distance of about 20,000 km. This means they are in a weaker gravitational field compared to the Earth's surface. As a result, time moves slightly faster for the satellites compared to the surface of the Earth.

The expression you have been given is known as the gravitational time dilation formula. It relates the time experienced by an observer in a gravitational field (t) to the time experienced by an observer in a weak gravitational field (τ). In this case, the observer on the Earth's surface is experiencing time (t_e) and the observer on the satellite is experiencing time (t_s).

To prove this expression, you can use the equation you have provided:

\tau = \sqrt {1 - \frac{{2\gamma M}}{{c^2 r}}} t\

First, we need to define the variables in the equation:

- \tau is the proper time experienced by the observer in a weak gravitational field (in this case, the satellite)
- \gamma is the gravitational constant
- M is the mass of the Earth
- c is the speed of light
- r is the radius from the center of the Earth to the observer in a weak gravitational field (in this case, the satellite)

Next, we can substitute in the values for the satellite and the Earth's surface into the equation:

\tau_s = \sqrt {1 - \frac{{2\gamma M}}{{c^2 (r + h)}}} t_s\

\tau_e = \sqrt {1 - \frac{{2\gamma M}}{{c^2 r}}} t_e\

Where h is the altitude of the satellite above the Earth's surface.

Now we can rearrange the equation to get the expression you were given:

\frac{{\tau_e }}{{\tau_s }} = \frac{{t_e }}{{t_s }}\frac{\sqrt {1 - \frac{{2\gamma M}}{{c^2 (r + h)}}} }{{\sqrt {1 - \frac{{2\gamma M}}{{c^2 r}}}
 

1. What is time dilation in a gravitational field?

Time dilation in a gravitational field refers to the phenomenon where time runs slower for an object that is in a stronger gravitational field compared to an object in a weaker gravitational field. This is due to the curvature of space-time caused by the presence of massive objects, as predicted by Einstein's theory of general relativity.

2. How does time dilation in a gravitational field occur?

Time dilation in a gravitational field occurs because massive objects, such as planets or stars, create a curvature in space-time. This curvature causes time to pass at a slower rate for objects that are closer to the massive object. The closer an object is to the massive object, the stronger the gravitational field and the slower time will pass for that object.

3. Does time dilation in a gravitational field only occur near massive objects?

Yes, time dilation in a gravitational field only occurs in the presence of massive objects. The strength of the gravitational field is directly proportional to the mass of the object, so the larger the object, the stronger the gravitational field and the greater the time dilation.

4. How is time dilation in a gravitational field measured?

Time dilation in a gravitational field can be measured using highly accurate clocks. One clock is placed on Earth's surface, while the other is placed in orbit around Earth. Due to the difference in gravitational potential between the two locations, the clock in orbit will run slightly faster than the clock on Earth's surface. This difference can be measured and used to calculate the amount of time dilation that is occurring.

5. Can time dilation in a gravitational field affect everyday life?

Yes, time dilation in a gravitational field can have small but measurable effects on everyday life. For example, GPS satellites have to take into account the time dilation caused by their orbit around Earth in order to accurately transmit location data. Additionally, the time dilation near black holes can have extreme effects on the perception of time for objects or people that are close to them.

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