Travel time of light in gravity field of sun.

[iloveannaw]

Homework Statement

Given the distance from the sun to Earth x_E, the distance from the sun to venus x_V and the mass of the sun m calculate the time delay due to gravity in receiving a radio signal sent from Earth to venus (and reflected back). Ignore the gravity fields of venus and earth. Venus lies directly between Earth and Sun!

Homework Equations

from lectures:

[itex]t' = (1 + \frac{\Delta\phi}{c^{2}}) t[/itex]

[itex]\Delta\phi = \frac{-Gm}{r}[/itex]

The Attempt at a Solution

proper time as recorded by distant observer =
[itex] t_{0} = \frac{x_{E}-x_{V}}{c}[/itex],
this is the time for a one way trip from Earth to Venus.

I think that the time delay is caused by the slowing of time due to change in grav. potential but I don't know how to calculate this.

This isn't homework just self study, thanks

 

[mark726]

Thank you for your interesting question. I would approach this problem by first considering the basic principles of general relativity and how it relates to the concept of time dilation. According to general relativity, time is not absolute but is instead relative to the observer's frame of reference and the gravitational potential they are experiencing.

In this scenario, we have two observers - one on Earth and one on Venus - both measuring the same event (the transmission and reception of a radio signal) from their own reference frames. The observer on Earth is located at a distance xE from the sun and the observer on Venus is located at a distance xV from the sun, with Venus lying directly between Earth and the sun.

To calculate the time delay due to gravity, we need to consider the difference in gravitational potential between the two observers. The general formula for time dilation due to gravity is:

Δt' = Δt √(1 + 2Φ/c^2)

Where Δt is the time interval measured by the observer in the weaker gravitational potential, and Φ is the difference in gravitational potential between the two observers. In this case, Δt is the time for a one-way trip from Earth to Venus, which we can calculate using the distance xE-xV and the speed of light c. We also know that the difference in gravitational potential between Earth and Venus is given by:

Φ = -Gm/r

Where G is the gravitational constant, m is the mass of the sun, and r is the distance between the two observers (in this case, the distance between Earth and Venus).

Therefore, we can calculate the time delay as follows:

Δt' = Δt √(1 + 2(-Gm/r)/c^2)

= Δt √(1 - 2GM/rc^2)

= Δt √(1 - 2xV/c^2)

= Δt (1 - xV/c^2)

= (xE-xV)/c (1 - xV/c^2)

= (xE-xV)/c - (xV^2/c^3)

= (xE-xV)/c - (xE-xV)^2/c^3

Therefore, the time delay due to gravity is given by (xE-xV)/c - (xE-xV)^2/c^3. This is the time difference between the two observers, with the