Gravitational redshift 


A thoughtexperiment demonstrating the existence of gravitational redshift.  


Gravitational redshift.  
Gravitational redshift  that is, the redshifting of photons that
climb out of a gravity well  occurs for fundamentally the same reason
that projectiles slow down when rising  because they have to transfer
kinetic energy (their speed) into potential energy (their height).
Projectiles, such as a cannon ball, do this by slowing down. Photons, however, cannot slow down  they are constrained to always travel at exactly c, the speed of light, and no faster. So how does a photon shed this required kinetic energy? By lowering in energy and thus frequency. Since a lower frequency means a longer, or "redder," wavelength, this process is called gravitational redshifting. (A similar process occurs when a photon is falling into a gravitational well; it trades potential energy for kinetic energy and gains in frequency, and thus gets a shorter, "bluer," wavelength; this is called a blueshift.) One can use simple Newtonian gravitation and the Einsteinian equation E = m c^{2} 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 massenergy of 

E = m c^{2}.  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 c^{2}, plus the kinetic energy K that it has gained while falling:  
E' = E + K  equation 5  
E' = m c^{2} + 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'/c^{2}.  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. We can even determine the Newtonian form of gravitational redshift by analyzing this more closely. Redshift is measured with the redshift parameter, z, defined as 

z == deltalambda/lambda  equation 8  
where lambda is the original wavelength and deltalambda is the change in wavelength, defined as lambda'  lambda, where lambda' is the final wavelength. Thus we can rewrite z as  
z = deltalambda/lambda  equation 9  
z = (lambda'  lambda)/lambda  equation 10  
z = lambda'/lambda  1  equation 11  
1 + z = lambda'/lambda.  equation 12  
Wavelength lambda and frequency nu are related by the wave equation  
c = lambda nu  equation 13  
so we can rewrite 1 + z in terms of the final frequency nu and the final frequency nu' (we reverse the ordering here because the photon is going up against the gravitational field, not down like the object we analyzed earlier):  
1 + z = lambda/lambda'  equation 14  
1 + z = nu'/nu.  equation 15  
Furthermore, the energy of a photon is h nu, so we can write this in terms of the original energy E and the final energy E':  
1 + z = E'/E  equation 16  
1 + z = (m c^{2} + m g h)/(m c^{2})  equation 17  
1 + z = 1 + g h/c^{2}  equation 18  
z = g h/c^{2}.  equation 19  
This redshift has been verified to 1% by Pound and Snider in
experiments concerning nuclear resonance and gamma radiation in 1964
and 1965.


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