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I have a mess in interpretation of constants in description of movement in GR.

First of all I define Lagrangian ##l=1/2g_{\mu\nu}u^{\mu}u^{\nu}##, and I would like to talk about axial smyetric spacetime (for example Kerr black hole) ##l(r,\theta)##. l is independent from ##t## and ##\phi## i.e. ##\frac{dl}{d\dot{t}}=const.=-E=u_t## and ##\frac{dl}{d\dot{\phi}}=const.=L=u_{\phi}##.

One can say, ##E## is energy of test particle measured by static observer at infinity and ##L## is angular momentum of test particle measured by static observer at infinity.

Then I may define a localy non rotating frame (LNRF) as frame at some ##r## and has ##L=0## (this frame rotates only thanks a dragging by rotating central body).

1) Are my definition and everything above alright?

2) If I won't be a static observer at infinity and I will measure the energy (##u_t##) of a test particle. Is true that I do not find a value ##-E##? For what and why it will be different?

3) Will I find the value ##-E## for ##u_t## if I will be in LNRF? Or is there some conection with LNRF and measuring ##u_t=-E##?

4) Does it have some conection with killing vectors? For example if I move in direction of some killing vector I will measure same things as static observer at infinity or something like that?

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# I Meaning of constants of motion in General Relativity

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