Einstein Relatively Easy

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               We saw in our previous articles Geodesic equation and Christoffel symbols and Geodesic that geodesic equations describe the paths of free falling particles in spacetime.

In order to understand how objects move in Schwarzschild spacetime, we therefore require the geodesic equations defined by the Schwarzschild metric.

We have shown that those equations are in the form of parameterised curves [1]

Replacing α by the four variables t, r, φ and θ gives us four complicated-looking differential Schwarzschild equations

For α = t, looking at the values of Γtμν from our previous article, we see that only Γttr and Γtrt are not null, leading to

Looking at α = r, we know that the four diagonal elements of the matrix Γrμν are not null

 so that we can write

 

 If we replace now α by θ, and reminding us the Γθμν matrix, we can write

Finally, we retrieve the last geodesic equation by replacing α by Φ

 

 

Simplify the equations by choosing Θ = Π/2

We can be clever and try to describe the motion of a free falling particle of mass m in the constant plane θ=π/2 (because the field is spherically symmetric, we can understand all the orbits by studying only these equatorial oribts, and there is no loss of generality), so we are left with only three equations (the third equation reduces to 0 + 0 = 0...)

Let's consider the first equation and start by multiplying each member by A to finally find out that this expression reduces to the derivative of d(A(dt/dλ))/dλ=0 so that A(dt/dλ) =cste

 

 If we consider the equation with Φ, we realize equivalently that it can be expressed as a derivative:

We are now left with the third equation with r, which should be equivalent to a derivative as well:

which has the advantage to be dependant on r variable only

As before, we have to find out the expression from which this equation derives.

For that, we recall the spacetime interval expression

At this point, we have to consider two cases.

If we consider a massive particule, we know that the proper time τ between  two distinct events is defined and in terms of the spacetime line element is given by

ds2 = c2 x dτ2

which is the time measured by a stationary clock at the same position as the two events.

and replacing again dt/dτ and dφ/dτ by their respective values

 Deriving this equation with respect to proper times gives

 

 

 which is exactly the third equation with respect to r that we have deduced aboce.

In case of a photon (without mass), we have ds2 = 0, so that finally our final equation can be written as

 We can interpret this expression as being strictly equivalent to the conservation of energy, with the radial cinetic energy per mass unit equals to Ec = 1/2(dr/dλ)2, E the total energy per mass unity and a potential energy per mass unity V(r) in the form of:

 

Comparison with Newtonian mechanics

We can compare this expression with what can be found in newtonian mechanics for a radial potential energy per mass, which only depends on r:

We notice that the two expressions are almost equivalent, except for the presence of the extra term in (1/h3) in the context of the Schwarzschild equation.

 

Newtonian approximation in case of radial free fall

Radial free fall implies the object is moving 'straight down', ie Φ is constant, therefore dΦ/dλ = 0 and so h = r2(dΦ/dλ) = 0

And the last equation thus reduces to

 

We can differentiate this equation with respect to τ, giving

 

 

which should remind you something very familiar in Newtonian mechanics ;-)

 

[1] To be more precise, time-like geodesics (where ds2 > 0 and proper time dτ ≠ 0)  describe the paths of massive objects  and can use proper time as a affine parameter. Null geodesics (where ds2 = 0 and proper time dτ = 0) describe the paths of (massless) photons and need another parameter. In the rest of the article, we use λ for either.

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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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