# A Lorentz-Einstein law of motion for a point particle

1. Dec 15, 2017

### Blackforest

I shall try to expose my question as briefly and clearly as possible.

Within a very old and classical approach (Newton), discussions take place in a three-dimensional Euclidean space and the concept of force is represented by a vector which is proportional to the mass of the object at hand (m: a scalar) and to its acceleration (g: a vector):

f = m.g with g = dv/dt where v is a speed

Within the restricted (first) version of the theory of relativity (end of 19th century), the classical concept of force can be extended and roughly understood as the variation of the kinetic momentum (p = m. v) by respect for the time:

f = d(m.v)/dt

This mental attitude is justified by the fact that, in that context, masses are no more invariant; in extenso, masses are depending on the speed of the particle and it is seemingly allowed to write:

f = dm/dt. v + m.g

Within the theory of relativity (general version in four dimensional spaces – 1916/17), even if the concept “force of gravitation” is ill-defined and must be abandoned in favor of the relation describing the variations of the geodesics (see “Gravitation” written by Misner Thorne and Wheeler, 1973, p. 224), it is not rare to find books where the concept of “force in presence of a gravitational field” is exposed and explained. The following relation is then introduced:

g = Dv/dt

Here, “D” denotes the total derivative or the covariant derivative. This justifies the next relation:

f = m.Dv/dt = m. (dv/dt+ …)

Here the “” depends on the local Levi-Civita connection and may be seen as a kind of deformed tensor product. This way of thinking has a well-known illustration named “the Lorentz-Einstein law of motion” which is nothing but a peculiar case corresponding to situations where f is the EM Lorentz-force. This relation is known since a very long time and is supposedly written:

m.Dv/dt = e.(v + E x B)

On the other side, huge efforts have been recently done (2003 – 2011) concerning the understanding and the validity of that formulation, for example in: “The motion of a point particle in a curved space-time” https://arxiv.org/abs/1102.0529, v3 September 2011.

My questions:
- Who has first introduced that law? When?

- Is there another motivation than the respect of the principle of covariance justifying its formulation?

- Or, said with different words, is there a mathematical demonstration for that Lorentz-Einstein law of motion; a demonstration that would be rooted into pre-existing theories and formula but not only on a principle?

2. Dec 17, 2017

### pervect

Staff Emeritus
I'm afraid I don't know the history. I do believe that the law as you state it is essentially replacing the ordinary derivative with the covariant derivative, which is a technique that Wald discusses in his textbook. This approach usually works, but Wald gives some examples in a different context where this general approach omits some terms.

As the paper you cite mentions, this approach to point particles is an approximation, it doesn't include back-reaction terms if there is significant gravitational wave emission. Usually, this is a good approximation. It also won't find the equations of motion for a spinning mass, as it omits some frame-dragging effects from the magnetic part of the Riemann tensor (the magnetic part being found through the Bel decomposition, https://en.wikipedia.org/wiki/Bel_decomposition.) I believe one needs the Papapatrou equations for a spinning massive object, https://en.wikipedia.org/wiki/Mathisson–Papapetrou–Dixon_equations.

So what the equation does do is find the equations of motion of a non-spinning test particle (a particle with so little mass that it doesn't significantly distort space time.) . Motivationally, the equations aren't very surprising, but addressing the problem rigorously seems like it would demand further literature searching rather than an off-the-cuff post.