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 outlines 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 that the frequency of light changes when entering or leaving a gravitational field - which has been demonstrated 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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"Five or six weeks elapsed between the conception of the idea for the special theory of relativity and the completion of the relevant publication" Einstein to Carl Seeling on March 11, 1952

"Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
David Hilbert

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