The discussion centers on the conservation of energy in General Relativity (GR), specifically addressing the implications of the Schwarzschild metric and the covariant divergence of the stress-energy tensor. Participants clarify that while classical conservation laws apply in flat spacetime, in GR, energy conservation is more complex and can only be approximated due to the influence of gravitational fields. The relevant conservation relation in GR is expressed as T^{\mu\nu}_{:\nu}=0, contrasting with the classical form T^{\mu\nu}_{,\nu}=0, which does not hold in curved spacetime. The concept of "energy at infinity" is introduced as a conserved quantity in stationary spacetimes, providing a framework for understanding gravitational potential energy.
PREREQUISITESPhysicists, students of General Relativity, and anyone interested in the complexities of energy conservation in curved spacetime.
sweet springs said:is wrong?
sweet springs said:Normality of 4-velosity
sweet springs said:tells us about speed, doesn't it ?
Because at the top v=0 so it is easier to calculate. Remember, the energy is a constant over the whole trajectory, so we can pick any point to calculate it. We may as well pick a point that makes the math easiersweet springs said:Why this top position in trajectory was chosen to give the value ?
It's independent of the motion and the position. Even ##\gamma## at infinity is disregarded. It is equal to rest mass in local Minkowsky space where the body is at rest. I am afraid that might be all that GR says about energy rigorously.stevendaryl said:Second, using the effective Lagrangian above, we could form the corresponding Hamiltonian, which is a conserved quantity. However, as I said, as a conserved quantity, it's kind of boring, since its value is just 12mc2\frac{1}{2} mc^2. So it's independent of the motion of the test particle.
sweet springs said:I am afraid that might be all that GR says about energy rigorously.