Question

In an universe where the expression for the gravitational redshift is

5R2c2 where Go is equal to 2.00× 10-30cm4 g-2 s-2 and the speed of light is the same as that in our universe. what is the approximate expression for the gravitational redshift in the presence of a weak gravitational field? What is your Schwarzschild radius in this universe?

i got it ty

Answer 1

One can use simple Newtonian gravitation and the Einsteinian equation E = m c2 in an easy calculation to demonstrate that there must be gravitational redshift to satisfy conservation of energy. Begin with an object of mass m at rest in a uniform gravitational field g at a point P. Because of special relativity, this object has a total mass-energy of
E = m c2.	equation 1
Now let the object fall in the gravitational field a distance h to come to a point Q. At this point, its potential energy has changed by an amount
U = -m g h.	equation 2
To compensate for this, the kinetic energy of the object goes up (the object is falling, after all). The change in kinetic energy K is equal and opposite to the change in potential energy, U:
K = -U	equation 3
K = m g h.	equation 4
So now the object has a total energy of E = m c2, plus the kinetic energy K that it has gained while falling:
E' = E + K	equation 5
E' = m c2 + m g h.	equation 6
Now we can conceptually convert this mass completely into energy, we'll put into the form of a photon (or a group of photons) with energy E', given by the above equation. We can let this photon rise through the gravitational field back up to point P. Now, if the energy of this photon is unaffected by the gravitational field -- i.e., if there is no gravitational redshift, then at point P we can turn the photon back into a solid mass, but this time with mass m' given by
m' = E'/c2.	equation 7
m' > m, due to additional kinetic energy K. Now after the cycle is completed with a photon back at point P, we have gained energy. This cycle can be repeated over and over again, with the continuous accumulation of free energy available to do work. Clearly this cannot be the case since it directly violates conservation of energy -- thus photons must be affected by the gravitational field; that is, there must be a gravitational redshift.