Deriving Gravitational Time Dilation as a Function of Potential

[modest]

Using U, you can calculate the gravitational time dilation inside and outside the sphere:

[math]T = \frac{T_0}{\sqrt{1-U/c^2}}[/math]

 

I had this wrong in a previous thread. The normal equation for gravitational time dilation is:

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

Substituting U = GM/r actually gives:

[math]T = \frac{T_0}{\sqrt{1 - (2U / c^2)}} [/math]

 

so I forgot the 2. My question is, does anyone know how to derive this time dilation as a function of potential equation? I was able to find two sources listing it:

 

Relativity: An Introduction to Space-time Physics, Steve Adams p.257

Relativity: An Introduction to Space ... - Google Book Search

 

Theoretical Concepts in Physics, M. S. Longair p.441 eq. 17.19

Theoretical Concepts in Physics: An ... - Google Book Search

 

But, nothing deriving it from GR. The reason I ask is I’m thinking of rewriting the "inside a sphere" part of:Gravitational time dilation - Wikipedia, the free encyclopedia which Craig and I proved was wrong in the thread quoted above. The idea is to find potential U using the equation for potential in a solid sphere and plug it into the equation above - that’s the right way to do it (I believe). But without deriving the equation above that has U in it, or a source explaining why it’s correct, I’d feel uncomfortable changing wikipedia

 

I would add that Wiki being wrong there has caused problems not only on this forum, but others as well.

 

I thought I’d throw this out there in case anyone has any ideas.

 

-modest

[modest]

I have successfully found some sources. Here's a fun one:

 

http://www.phys.unsw.edu.au/einsteinlight/jw/2006AJP.pdf

 

So, I will set about changing the article...

 

~modest

[modest]

In case anyone is interested, I've taken to rewriting the entire article. My version so far is here:

User:JModest/Sandbox - Wikipedia, the free encyclopedia

 

The original (which has multiple errors and poorly written language) is here:

 

Gravitational time dilation - Wikipedia, the free encyclopedia

 

I haven't yet started sections on definition, history, experimental confirmation... etc.

 

Any suggestions / corrections are most welcome. Layout, formulas, language, concept.. any suggestions you've got.

 

~modest

[jedaisoul]

Equivalence Principle

 

...In 1915 Einstein published a completely relativistic theory of gravity (general relativity) consequent the equivalence principle which is now the most common tool for calculating gravitational time dilation.

Just to note that my copy of Einstein - The Principle of Relativity dates Einstein's paper "The Foundation of the General Theory of Relativity" to 1916, not 1915. The full credit was:

Translated from "Die Grundlage der allgemeinen Relativitatstheorie," Annalen der Physik, 49, 1916.

 

It should be relatively simple to check the date of that publication (no pun intended).

[modest]

Thank you jedaisoul. I've changed that section significantly which probably hasn't updated yet. Einstein published two papers in November 1915 on GR, the second of which had the completed field equations and is often cited. He also published more than one in 1916 that dealt with GR, so that is probably what you have.

 

List of scientific publications by Albert Einstein - Wikipedia, the free encyclopedia

 

I appreciate the proof.

 

~modest

[Little Bang]

modest, you write, "A clock on the top of a building will have greater gravitational potential and thus run a small fraction slower than a clock at the base of the building.". The clock at the top will run faster than the one at the base.

[modest]

Thank you. Quite right you are :rolleyes:

 

~modest