How is energy conserved in General Relativity?

[PeterDonis]

is wrong?

It's ok as far as it goes. The errors in your previous posts came after that equation.

Normality of 4-velosity

The 4-velocity is a unit vector in GR, yes.

 

[PeterDonis]

tells us about speed, doesn't it ?

It's true that $u^1$ tells you something about speed; but it's not equal to speed. (Also, you need to be careful how you're defining "speed". Speed relative to what?)

 

[Dale]

Why this top position in trajectory was chosen to give the value ?

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 easier

 

[sweet springs]

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.

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.

 

[PeterDonis]

I am afraid that might be all that GR says about energy rigorously.

No, it isn't. An object's energy at infinity is not in general the same as its rest mass. That is obvious from the equation for energy at infinity that we have already derived.

Also, as I have already pointed out, the term "energy" has multiple meanings. Several of them have been covered in this thread, and energy at infinity is only one of them.

At this point I am closing the thread. I think you need to consider more specifically what concept of energy you want to ask about, so that future threads can have a more focused and useful discussion.