The gravitational redshift

doug  <sweetser@world.com>

Introduction
The Pound and Rebka experiment
Escape from a gravitational potential
Clocks at different heights in a gravitational field
Implications
References
Initialization functions

Introduction

Gravitational redshift experiments are tests of conservation of energy in a gravitational potential.  A photon lower in a gravitational potential expends energy to climb out, and this energy cost is seen as a redshift.  In this notebook, the difference between weak gravitational potentials will be calculated and shown to be consistent with experiment.  Quaternions are not of much use here because energy is a scalar, the first term of a quaternion that is a scalar multiple of the identity matrix.

The Pound and Rebka experiment

The Pound and Rebka experiment used the Mossbauer effect to measure a redshift between the base and the top of a tower at Harvard University.  The relevant potentials are

\(ϕ\_tower = \ \(G\ M\)\/\(r\  + \ h\); 

    ϕ\_base\  = \ \(G\ M\)\/r; \)

The equivalence principle is used to transform the the gravitational potential to a speed (this only involves dividing phi by the constant c^2).

\(β\_tower = \ \(G\ M\)\/\(\(c\^2\) \((r\  + \ h)\)\); 

    β\_base\  = \ \(G\ M\)\/\(\(c\^2\) r\); \)

Now the problem can be viewed as a relativistic Doppler effect problem.  A redshift in a frequency is given by

ν'  ==  
      (γ[β]  +  β γ[β]) (Subscript[ν, o])

For small velocities, the Doppler effect is

Series[
      γ[β]  +  β γ[β], {β, 0, 1}]

The experiment measured the difference between the two Doppler shifts.

Series[
      ((1  +  (Subscript[β, tower])) - (1 +  (Subscript[β, base]))) 
        (Subscript[ν, o]), {h, 0, 1}]

Or equivalently,

ν'  ==  g h (Subscript[ν, o ])

This was the measured effect.

Escape from a gravitational potential

A photon can escape from a star and travel to infinity ( or to us, which is a good approximation).  The only part of the previous calculation that changes is the limit in the final step.

Limit[
      ((1  +  (Subscript[β, tower])) - (1 +  (Subscript[β, base]))) 
        (Subscript[ν, o]), h -> Infinity]

This shift has been observed in the spectral lines of stars.

Clocks at different heights in a gravitational field

C. O. Alley conducted an experiment which involved flying an atomic clock at high altitude and comparing it with an atomic clock on the ground.  This is like integrating the redshift over the time of the flight.

(∫_0^t)-((G M h)/((c^2) (r^2))) 
        \[DifferentialD]t

This was the measured effect.

Implications

Conservation of energy involves the conservation of a scalar.  Consequently, nothing new will happen by treating it as a quaternion.  The approach used here was not the standard one employed.  The equivalence principle was used to transform the problem into a relativistic Doppler shift effect.  Yet the results are no different.  This is just part of the work to connect quaternions to measurable effects of gravity.

References

For the Pound and Rebka experiment, and escape:
Misner, Thorne, and Wheeler, Gravitation, 1970.

For the clocks at different heights:
Quantum optics, experimental gravitation and measurement theory, Ed. P. Meystre, 1983 (also mentioned in Taylor and Wheeler, Spacetime Physics, section 4.10)

Initialization functions

All that is need is gamma.

γ[β_] := (1/(Sqrt[1  -  (β^2)]))