Tagged Questions
0
votes
1answer
76 views
Proper time of circular motion under Schwarzschild metric
I'm trying to calculate the proper time of a massive particle circulating Schwarzschild black hole, using EL equation of the following Lagrangian:
...
1
vote
0answers
45 views
Trouble with calculating Christoffel symbols of FLRW metric using Lagrangian method
The FLRW metric which I am using is $$ds^2 = dt^2 - \frac{a(t)^2}{c^2} \left( dx^2 + dy^2 + dz^2 \right)$$
where $a(t)$ is the so-called 'scale factor'. I did not want to calculate the Christoffel ...
2
votes
2answers
118 views
Different approaches to calculating the Christoffel symbols
I would be very grateful to whoever can debug the following calculations...
We have the metric for static spacetime: $$ds^2 = -\exp(2U(\vec x))dt^2+h_{ij}(\vec x) d x^i d x^j$$
I want to find the ...
1
vote
1answer
131 views
Symmetries of spacetime and objects over it
I guess according to mathematical didactic, we first think of spacetime as a set and we reason about elements of its topology and then it's furthermore equipped with a metric. Appearently it is this ...
12
votes
4answers
514 views
Why does no physical energy-momentum tensor exist for the gravitational field?
Starting with the Einstein-Hilbert Lagrangian
$$ L_{EH} = -\frac{1}{2}(R + 2\Lambda)$$
one can formally calculate a gravitational energy-momentum tensor
$$ T_{EH}^{\mu\nu} = -2 \frac{\delta ...
0
votes
2answers
248 views
How to think of the harmonic oscillator equation in terms of “acceleration = gradient”
This is related to another question I just asked where I learned that the equation of motion of a harmonic oscillator is expressed as:
$$\ddot{x}+kx=0$$
What little physics I grasp centers on ...
1
vote
1answer
131 views
How does General Relativity Emerge from Brans-Dicke Gravity with an Infinite Omega Parameter?
The action for the Brans-Dicke-Jordan theory of gravity is
$$
\\S =\int d^4x\sqrt{-g} \;
\left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + ...
0
votes
0answers
153 views
Comparing Lagrangian in Special Relativity vs General Relativity for a weak gravitational field
This is a sequel to this question.
Who knows a difference between the Lagrangian in SR and GR for a weak gravitational field in non-relativistic case? What is the reason of this difference?
11
votes
1answer
214 views
Lagrangian for Euler Equations in general relativity
The stress energy tensor for relativistic dust
$$
T_{\mu\nu} = \rho v_\mu v_\nu
$$
follows from the action
$$
S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ -g } d^4 x
= -\int c \sqrt{p_\mu ...
