Four Momentum in General Relativity

In summary, the author defines p_0=-E when discussing trajectories in the Schwarzschild metric, but changes the definition later to include metrics where the 00 component is -1.
  • #1
MrBillyShears
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Alright, I'm rather new to General Relativity, and I'm getting confused with four momentum. Back in SR, [itex]p^α=mU^α[/itex], but, this relationship doesn't hold in curved space, does it? Because, now I'm seeing that four momentum is somehow a covector in GR, and [itex]p_0=-E[/itex], so the time component of the contravariant component can't be E, since the 00 component of the metric isn't always -1. So, is four momentum now redefined in GR, and if the SR relationship between four velocity isn't valid, what is the new relationship to U? Is four momentum naturally a covariant thing?
 
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  • #2
MrBillyShears said:
Back in SR, [itex]p^α=mU^α[/itex], but, this relationship doesn't hold in curved space, does it? Because, now I'm seeing that four momentum is somehow a covector in GR,
I am not sure if this is your concern, but you would probably call both ##p^α=mU^α## and ##p_α=mg_{\mu α}U^{\mu}## "the four momentum". If you need to distinguish between the two verbally you would call one "the four momentum vector" and the other "the four momentum covector".

MrBillyShears said:
and [itex]p_0=-E[/itex], so the time component of the contravariant component can't be E, since the 00 component of the metric isn't always -1.
##p_0=-E## would only hold in metrics where the 00 component is -1. In GR that won't always be the case. The 00 component may not be the time-time component. In fact, there may not even be a time-time component. Regardless, the definition above will hold, even if the components don't have such a clear interpretation.
 
  • #3
That's ironic. I'm confused now too. In the introductory book I'm reading on GR, the author defines [itex]p_0=-E[/itex] when discussing trajectories in the Schwarzschild metric, and then he goes to define [itex]p^0=g^{00}p_0=(1-{2M}/r)^{-1}E[/itex]. Why does he do this then? Is this a matter of convention, defining the vector momentum from the covector momentum?
 
  • #4
Sorry, I was indeed being a little ambiguous (actually a little wrong). What I was thinking and what I should have said is that ##p_0=\pm p^0=\pm E## only in metrics where the 00 component is ##\pm 1##, and even then only if the 0i components are 0 too.

The vector and covector momentum are always related as ##p_{\mu}=g_{\mu\nu}p^{\nu}##, regardless of the metric. The vector and the covector are in some ways fundamentally different quantities, but nevertheless it is a very strong convention to think of them as the same quantity just with raised or lowered indexes.
 
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  • #5

1. What is four momentum in general relativity?

Four momentum in general relativity is a mathematical quantity that describes the total energy and momentum of a particle or system in the context of Einstein's theory of general relativity. It takes into account both the mass and the motion of the particle, and is an important concept in understanding the behavior of objects in the presence of strong gravitational fields.

2. How is four momentum calculated in general relativity?

In general relativity, four momentum is calculated using a mathematical expression called the covariant derivative. This involves taking into account the curvature of spacetime and the energy-momentum tensor, which describes the distribution of energy and momentum in the universe. The resulting four momentum vector has four components: three for momentum in the x, y, and z directions, and one for energy.

3. What is the significance of four momentum in general relativity?

Four momentum is significant because it is a conserved quantity in general relativity, meaning that it remains constant in any inertial reference frame. This allows us to make predictions about the behavior of objects in the presence of strong gravitational fields, such as those near black holes. Additionally, four momentum is used in many important equations in general relativity, including the Einstein field equations.

4. Can four momentum be used to describe the motion of particles in curved spacetime?

Yes, four momentum can be used to describe the motion of particles in curved spacetime in general relativity. This is because the covariant derivative takes into account the curvature of spacetime, which affects the path of a particle as it moves through it. Four momentum allows us to track the energy and momentum of the particle as it follows this curved path.

5. How does four momentum differ from three momentum?

Four momentum differs from three momentum in that it takes into account the energy of a particle in addition to its momentum. Three momentum only describes the motion of a particle in space, while four momentum describes both its motion and its total energy, including its rest mass. Additionally, four momentum is conserved in all inertial reference frames, while three momentum is only conserved in a specific reference frame.

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