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The Equation of the Geodetic line (Extract from the Manuscript "The Foundation of the General Relativity of Relativity §9 1916)

 

As per the considerations of the Equivalence Principle, if we were to describe the movement of an object in the Earth's gravitational field, we would then have to follow the following steps:

  1. Describe the movement in a local inertial free falling referential
  2. Operate a coordinate transformation from this local inertial referential to the Earth referential, this one seen as accelerated upwards
 Step 1: Describe the movement in a free falling referential

Note: You can refer to this article The Lorentz transformations Part IV - Lorentz transformation matrix (tab index notation) for a gentle introduction to the index notation and Einstein summation convention if you are not familiar with these concepts.

In the free falling referential, let's name the spacetime coordinates ξα  in index notation, i.e:

  • α ∈ {0,1,2,3}
  • and ξ0 = ct, ξ1 = x, ξ2 = y, ξ3 = z

 

The movement of the free particule is given by the four-acceleration vector's magnitude equal to 0 so using the index notation:

With τ refers to the time as measured by an observer at rest  in his own rest referential also called the Proper time for a non massless particle.

 
Step 2: Describe the movement in Earth referential (accelerated towards the upper direction)

Let's name xμ the coordinates in the new arbitrary referential (non inertial)

So applying the chain rule to our initial free-falling equation we get:

 

or Kronecker delta is defined as 1 only if β = μ and 0 otherwise, that means we can replace the μ indice  by β in the last term

 

The geodesic equation in the (accelerating) laboratory's referential shows that the particule's motion is no more a straight line, because some kind of 'inertial force'  represented by the term with the Christoffel Symbol or Connection coefficient is now acting on it.

At this point, the coordinate transformation does not seem to have anything to do with the gravitational force. However, if our referential in ξ is free falling in a gravitational field, then in the fixed laboratory referential, the inertial force fβ is the gravitational force. And the movement of a body in the laboratory can be determined if we know the Christoffel symbol components. That's precisely one  of the goals of general relativity.

This equation generalizes the notion of a "straight line" to curved spacetime. Actually, we will see in Christoffel symbols in terms of the metric tensor how the Christoffel symbol (at the heart of the gravitational force) can be calculated from the space time metric.

Even better, from the same geodesic equation, we will show later in The Geodesic equation in the Newtonian Limit how, under specific conditions called as Newtonian limit, we can derive the direct relation between the metric tensor and the gravitational potential.

Finally, in a later article Geodesic equation from the principle of least action, we will demonstrate how to derive the same geodesic equation from the principle of least action and the Euler-Lagrange equation.

 

 

 

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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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