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I have two questions - the first is about an approximation of a central gravitational force on a particle (of small mass) based on special relativity, and the second is about the legitimacy of a Lagrangian I'm using to calculate the motion of a particle in the Schwarzchild metric.

First of all, I should mention I have not yet properly studied GR, but have simply had some ideas about how to calculate trajectories of particles using different metrics.

1. A couple of sources, including Susskind's theoretical minimum, show that the Lagrangian of a free relativistic particle (in flat spacetime) is: [tex]L=-m{ c }^{ 2 }\sqrt { 1-\frac { { v }^{ 2 } }{ { c }^{ 2 } } } [/tex] I noticed that, in spherical coordinates, this is equal to: [tex]L=-m{ c }^{ 2 }\sqrt { \frac { -{ \eta }_{ \mu \nu }{ \dot { x } }^{ \mu }\dot { x } ^{ \nu } }{ { c }^{ 2 } } } [/tex] where [tex]\eta =\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & { r }^{ 2 } & 0 \\ 0 & 0 & 0 & { r }^{ 2 }\sin ^{ 2 }{ \theta } \end{pmatrix}[/tex] is the metric for flat spacetime.

Now, I can add a potential energy term to this Lagrangian, so could I not simply add a gravitational potential energy term [tex]L=-m{ c }^{ 2 }\sqrt { \frac { -{ \eta }_{ \mu \nu }{ \dot { x } }^{ \mu }\dot { x } ^{ \nu } }{ { c }^{ 2 } } } +\frac { GMm }{ r } [/tex] and solve the Euler-Lagrange equations?

When I do this, and set the initial conditions such that the particle is in an elliptical orbit, I actually do get an apsidal precession, but the rate of this precession is much smaller than that I get from what I did below.

2. Having noticed the form of the Lagrangian above, I decided to change the flat metric to the Schwarzchild one, and setθto be π/2 so that I could simplify the problem: [tex]\eta \rightarrow g=\begin{pmatrix} -\left( 1-\frac { { r }_{ s } }{ r } \right) & 0 & 0 & 0 \\ 0 & { \left( 1-\frac { { r }_{ s } }{ r } \right) }^{ -1 } & 0 & 0 \\ 0 & 0 & { r }^{ 2 } & 0 \\ 0 & 0 & 0 & { r }^{ 2 } \end{pmatrix}[/tex] This gives me the Lagrangian: [tex]L=-m{ c }^{ 2 }\sqrt { \frac { -{ g }_{ \mu \nu }{ \dot { x } }^{ \mu }{ \dot { x } }^{ \nu } }{ { c }^{ 2 } } } =-m{ c }^{ 2 }\sqrt { 1-\frac { { r }_{ s } }{ r } -\frac { { \left( 1-\frac { { r }_{ s } }{ r } \right) }^{ -1 }{ \dot { r } }^{ 2 }+{ r }^{ 2 }{ \dot { \varphi } }^{ 2 } }{ { c }^{ 2 } } } [/tex] Solving the Euler-Lagrange equations of this Lagrangian gives me an apsidal precession much larger than the one before.

My two questions boil down to this: First, what are the main issues with the SR approximation in part 1, and is the Lagrangian in part 2 (using the Schwarzchild metric) legitimate?

As an extra, perhaps, why do the two Lagrangians both give rise to an apsidal precession, but with different rates of precession?

Thanks in advance :)

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# I Special Relativity Approximation of Gravitation

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