Einstein Relatively Easy

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By applying the Equivalence Principle, Einstein was able to obtain important results of the general theory of relativity even before he could solve the corresponding field equations.

You can read this demonstration in the 1907 article On the relativity Principle and the conclusions drawn from it". Actually, as outlined in our article 1907 Equivalence Principle first published mention, the equivalence of gravitation and acceleration is mentionned there for the first time by Einstein.

We will try to demonstrate that the gravitational redshift , i.e the fact that the light freqency changes when entering or leaving a gravitational field - which has been derived from the metric tensor in Newtonian limit in the previous article, could also be derived from this principle of Equivalence.

Consider light traveling from the bottom to the top of a rocket undergoing constant acceleration a. Let point A be at the bottom of the spacecraft, and point B at the top.  The separation distance measured in the reference frame of the rocket is H.


When light first leaves point A, the velocity of the rocket is vA with respect to another reference frame (the Earth, for example), and  let's call T the time  for light to travel to point B, so:

  • vA = velocity of the rocket when light is emitted at point A
  • vB = vA + aT = velocity of the rocket when light reaches point B

 

The time T for light to reach B (from the Earth perspective) is:

 

we approximate T to H/c as we consider a << c ('small' acceleration)

Also the change in velocity of the rocket between emission and reception is:

In the Earth referential (external to the rocket), the observer at  point B is moving away from the light at an additional relative velocity of v. In this external frame, we can use therefore the results of Einstein’s Special Theory of Relativity, and particularly the Doppler shift equation for receding velocities:

The gravitational redshift of a light wave
as it moves upwards against a gravitational field

But The Principle of Equivalence tells us that there is no experiment done in a small confined space which can tell the difference between a uniform gravitational field and an equivalent acceleration.

So the exact same redshift phenomenon should be observed on the Earth, if we replace the rocket by a tower of height H and the constant acceleration a of the rocket by the gravitational field g on the Earth.

 Let's try to confirm that the formulation of the gravitationnal shift derived in the previous article from the metric tensor equals the new formulation

Based on the previous article, we have the frequency redshift equals to:

 

 The Equivalence Principle, almost without any calculus but just by using the classical Doppler effect of special  relativity flat spacetime, has led us to exactly the same result!

Note: You can find a scintillating explanation of the derivation of time dilation (equvalent to the gravitationnal redshift) in a gravitationnal field  in the Feynman's lecture dedicated to Curved Space; more particularly, paragraph 42-6 The speed of clocks in a gravitationnal field

Feynman's lecture about curved space

 

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"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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