Einstein Relatively Easy

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Suppose that an observer is standing on the surface of Earth and is pointing a torch in the sky. A pair of events might be the successive peaks of the light wave leaving the torch.

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

From our article Generalisation of the metric tensor in pseudo-Riemannian manifold, we know that in the non inertial Earth's referential frame, the space time line element between the two events can be written as:

where the indices μ and ν run over 0, 1 ,2, 3 for spacetime.

But as the observer is at rest in his own referential, the only non null coordinates is x0, so that the square of the line element can be simplified to:

If our observer is at rest in his own referential, we know also how to express the space time distance with respect to the proper time τ (tau), as explained in Proper time article

But we know from our previous article The Geodesic equation in the Newtonian Limit that at the surface of the Earth, the metric tensor can be expanded in terms of the gravitational potential as follows

 

 which gives then, by expliciting the value of the Newtonian gravitational potential field at a point in a gravitational field

 In the above equation, the infinitesimal intervall dt can be considered as the time interval observed in a referential without gravitational effect, or say in another way in a ideal  referential situated at an infinite distance

We can therefore write:

 where dτrefers to the period of the light wave as measured by a distant observer without gravity and where dτ is the period of the wave measured where it is emitted, ie from the surface of the Earth.

This equation tells us that clocks run slower in a gravitationnal field as seen by a distant observer, this effect is known as gravitationnal time dilation.

As a direct consequence, because frequency is the reciprocal of the period, we can say:

where f is the frequency of the wave as measured by a distant observer and femission is the frequency of the wave measured at the point of emission.

This equation tells us that the frequency of a wave as recorded by a distant observer is less than the frequency recorded by an observer located where the events occured in the gravitationnal field. This phenomenon is known as the gravitationnal redshift, because a reduction in frequency means a shift toward the longer wavelengths or 'red' end of the electromagnetic spectrum.

We can think of the photons losing energy as they climb out of the gravitational field - loss of energy equating to drop in frequency.

Remark1: we can simplify the equation by replacing the quantity 2GM/rc2 by the Schwarzchild radius notation Rs

 Remark2: we have considered so far the case of an distant observer upon which no gravitational field acts. A more realistic scenario is that the second observer is itself under the effect of the gravitational field.

Let's assume that the observer A pointing the torch stands at the surface of the Earth  at a distance ra from the center of the Earth and that the second observer B stands by example at the top of a tower at the distance rb = ra + h

We can then write:

.If we suppose as it is the case on Earth that Rs<<ra (= Earth radius) and so that Rs<<rb,  the redshift can be approximated by a binomial expansion to become:

 

 We have then the confirmation that when observed in a region of a weaker gravitational field (rb > ra), an electromagnetic radiation originating from a source that is in a gravitational field is reduced in frequency (fb < fa) or redshifted.

Remark3: we demonstrate in the article Newtonian limit how this gravitational redshift, i.e. the time curvature is enough to ensure that free-falling bodies follow their Newtonian trajectories. Put in other words, Newtonian gravitation is equivalent to time curvature which is the same as gravitational redshift!

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"Pas plus de cinq ou six semaines s'écoulèrent entre la conception de l'idée de la relativité restreinte et la rédaction de l'article correspondant."
Einstein à Carl Seelig, 11 Mars 1952

"N'importe quel étudiant dans les rues de Göttigen en connaît plus qu'Einstein sur les géométries à 4 dimensions. Et pourtant ce fut Einstein qui accomplit le travail, et non les mathématiciens."
David Hilbert

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