General Relativity by Leonard Susskind

Author:Leonard Susskind
Language: eng
Format: epub
Publisher: Basic Books
Published: 2022-06-15T00:00:00+00:00

Inside the integral, we have the Lagrangian when the speed of the particle is small and we can make a non-relativistic approximation.

If we use this Lagrangian in the Euler–Lagrange equations, the constant term – mc2 has no effect. The only thing we do with a Lagrangian is differentiate it. When we differentiate a constant, we get zero. So – mc2 can be disregarded.

The other two terms in the Lagrangian are a conventional kinetic energy, , minus a potential energy, mU(X), which actually depends on X. For the potential energy we can choose the function we like. Incidentally, in a gravitational problem the potential energy of a particle is always proportional to its mass.

Finally, when we use this Lagrangian, the Euler–Lagrange equations will of course simply produce Newton’s equation for a particle in a gravitational field U(X), just as it did when we carried out exactly the same calculations in volume 1.

We arrive at

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