Geodesic Equations: Newtonian vs Einstein

[GRstudent]

$\dfrac{d^2 x}{dt^2}=-\nabla \Phi$

$\dfrac{d^2 x^\mu}{d\tau^2}$$= -\Gamma^{\mu}_{\alpha \beta}{}$$\dfrac{dx^\alpha}{d\tau}\dfrac{dx^\beta}{d\tau}$

These two equations, to be true, the way they are written should ring a bell. They are similar yet not identical. What is the meaning behind them?

I guess first is Newtonian; second, is Einstein.

 

[bcrowell]

Newtonian gravity can be represented using curved spacetime, so if that was the physics behind the equations, then both equations could be representing exactly the same thing.

However, the physical context of the second equation could be much more general than that of the first. For instance, the second equation could be used to define the great-circle course of a passenger jet from Europe to America.

 

[GRstudent]

For instance, the second equation could be used to define the great-circle course of a passenger jet from Europe to America.

Ok, nice! I will be flying soon from Germany to Houston, what inputs should I put into second equation?

 

[alialice]

See pages 76-77 of Wald's General Relativity.
The first equation, which describes the motion of a particle in the gravitational field phi, can be derived from the second, the geodesic equation for a point particle in a curved spacetime, in the Newtonian limit.