Undergrad Conserved Quantities in GR: Explained

[Silviu]

Hello! I am reading about spherical geometry and for a static system and based on the metric, $p_0$ and $p_\phi$ are constant of motion. I am not sure I understand in which sense are they constant? The energy of a particle measured by an observer depends on the metric (so on its position) in space. So $p_0$ is not the energy of the particle (can't even be measured I think). Can someone explain this to me a bit more? Thank you!

 

[PeterDonis]

I am reading about spherical geometry

From what source? Please give a specific reference.

based on the metric, $p_0$ and $p_\phi$ are constant of motion

They are constants of geodesic motion.

I am not sure I understand in which sense are they constant?

"Constant" means "does not change".

The energy of a particle measured by an observer depends on the metric (so on its position) in space

No, it doesn't. It depends on the 4-momentum of the particle and the 4-velocity of the observer.

So $p_0$ is not the energy of the particle

Not the energy measured by an observer that is (momentarily) co-located with the particle, no--at least, assuming the observer and the particle are not at infinity.

(can't even be measured I think)

Yes, it can, but it's not as straightforward as an observer co-located just measuring the particle's energy.

Can someone explain this to me a bit more?

It would help greatly if you would give a specific reference and show some math. You should particularly focus on the actual mathematical expression for $p_0$. What quantities occur in it? What do those quantities mean?