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Special and General Relativity
Understanding 4-Momentum in General Relativity
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[QUOTE="PeterDonis, post: 5451962, member: 197831"] Because angular momentum is not ##m d \phi / d\tau## in general, it just happens to be equal to that in special cases (the same ones where the metric is simple enough that ##p_\phi## is equal to ##p^\phi##). First of all, there is the same issue here as with energy at infinity: ##p_\phi## is only a constant of geodesic motion if the spacetime has an axial Killing vector field and the coordinates are chosen such that the ##\phi## basis vector is tangent to orbits of that axial Killing vector field. The first condition is necessary for there to be an "angular" constant of motion at all; the second is necessary for the ##\phi## component of ##p## to be the only relevant one for that constant of motion. Given those conditions, the constant ##p_\phi## works like the usual constant ##L## for geodesic orbits in Newtonian mechanics, so it is called "angular momentum" (or more precisely "orbital angular momentum") for the same reasons that ##L## is in Newtonian mechanics. Second is understanding the correct expression for the angular momentum constant of the motion (and indeed for energy at infinity as well). See further comments below. We need to be more precise here about what "conservation" means. Strictly speaking, what is conserved is a scalar quantity, not a "component" of a vector or 1-form; that scalar only looks like a particular component of a vector or 1-form because we chose particular coordinates. But the conservation laws in question are independent of any choice of coordinates, so to really understand them we need to express them in a form that is independent of coordinates. The way to do that is to use the fact that I've already referred to, that each conserved quantity corresponds to a Killing vector field. So if energy is conserved, that means there is a timelike Killing vector field, which we can call ##T^\mu##. The conserved quantity "energy at infinity" is then simply the contraction ##T^\mu p_\mu##. Similarly, if angular momentum is conserved, that means there is an axial Killing vector field ##\Phi^\mu##, and the conserved quantity "angular momentum" is the contraction ##\Phi^\mu p_\mu##. These expressions are scalar invariants (like any contraction of a vector and a 1-form), so they are true independently of any choice of coordinates. If we choose coordinates such that ##T^\mu = (1, 0, 0, 0)## (i.e., the "0" basis vector is tangent to the KVF), then we find that ##T^\mu p_\mu = - p_0## (assuming we are using the -+++ metric sign convention). Similarly, if we choose coordinates such that ##\Phi^\mu = (0, 0, 0, 1)## (where the "3" component is the ##\phi## component), then we find that ##\Phi^\mu p_\mu = p_3 = p_\phi##. But those expressions are only valid in the chosen coordinates; they aren't valid generally. The above should also answer the question of why the conserved quantities are usually written with a lower index, i.e., as 1-form components: because in order to obtain the coordinate-independent scalar invariant, we have to contract the object's 4-momentum with the appropriate Killing vector field. The latter is a vector field (at least in its "natural" formulation), so we have to use the 4-momentum 1-form. However, it is perfectly possible to swap the indexes using the metric, and write the conserved quantities as ##T_\mu p^\mu## and ##\Phi_\mu p^\mu##. These must be the [I]same[/I] conserved quantities--i.e., we must have ##T_\mu p^\mu = T^\mu p_\mu## and similarly for the other one--regardless of what the metric is. This is true of [I]any[/I] scalar formed by contracting tensors: swapping the positions of contracted indexes leaves the scalar invariant. That is why I said it doesn't really matter which one you consider to be a vector and which you consider to be a 1-form. Indeed, with the metric, you can contract two vectors, like so: ##g_{\mu \nu} T^\mu p^\nu##. This is still the same scalar as the other two expressions. (Similarly, you can use the inverse metric to contract two 1-forms.) All of these expressions represent the same physics, so there's no reason to worry too much about index placement. [/QUOTE]
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