Einstein Relatively Easy

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Newtonian gravity consists of two equations: one tells us how matter responds to gravity, and the other tells us how matter produces gravity.

The first equation, derived from Newton's second law of motion, says

where a vector is the acceleration through space, nambla  is the Euclidean gradient operator and  Φ is the gravitational potential.

In this article, we will focus on this first equation, and we will try to derive an approximation of the Newtonian gravitational equation with the mathematics of general relativity.

In the Newtonian limit, we make three assumptions:

  • - The particle is moving relatively slowly (compared to the speed of light).
  • - The gravitational field is weak.
  • - The field does not change with time, ie it is static.

 

As the geodesic equation describes the worldline of a particle acted only upon only by gravity, our goal is therefore to show that in the context of the Newtonian limit, the geodesic equation reduces to the first Newton's gravity equation, as expressed above.

We recall from our previous article Geodesic equation and Christoffel symbols that the geodesic equation, using proper time as the parameter of the worldline is:

The second term hides a sum in μ and ν over all indices (16 terms). But because the particle in question is moving slowly (first assumption of the Newtonian limit), the time-component (ie the 0th component of the particle's vector) dominates the other (spatial) components, and every term containing one or two spatial four-velocity components  will be then dwarfed by the term containing two time components.

 We can therefore take the approximation:

If we restrict ourselves to the Newtonian 3-D space, meaning that we assign β to spatial dimensions only, we can then replace β by the latin letter i (i = x, y, z), giving:

Equation 1

 From our article Christoffel symbols in terms of the metric tensor, we know how to calculate the Christoffel symbol with respect to the components of a given metric:

 

But because the field is supposed to be static (second assumption of the Newtonian limit), the time derivative gj0,0 is zero, so that the Christoffel symbol can be simplified to:

Now, if the gravitational field is weak enough, then spacetime will be only slightly deformed from the gravity-free Minkowski space of special relativity, and we can consider the spacetime metric as a small perturbation from the Minkowski metric ημν

At this step, Equation 1 becomes:

 If we now define gij = ηij - hij, we find that gμσ gσν = δμν to within first order of hij, defining an inverse metric.

We obtain then

But as ηij is not null only if j=i, for which ηii = -1 (i refers to the spatial components)

We now need to change the derivative on the left hand side from τ (tau) to t.

For this, let's first replace i by 0 in the above equation:

With this result in the hands, we still need to play around with the partial derivative with respect to tau:

 

 So finally, expressing this in vectorial form:

 

 which is another way of writing the Newtonian gravitational Equation from the beginning.

Remark 1

 By writing the metric component g00 as:

we can see the direct link between the metric tensor (component 00) on the left side and the gravitational potential Φ on the right side.

Remark 2

We can calculate the h00 value on the Earth and check that its value is infinitisimal, meaning that the deviation relative to the Minkowski metric due to the gravitational field is negligible.

In the same way, we would calculate a correction to the Minkowski metric of -10-6 at the surface of the sun and of -10-4 at the surface of a white dwarf. We conclude that the weak-field limit is an excellent approximation.

 

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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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