Volume forms on the mass-shell

The setting for dynamics is the cotangent bundle \mathcal{T}^*\mathcal{M} of a manifold \mathcal{M} with pseudo-Riemannian metric g_{\mu\nu}; relevant observables can be functions of both position and momentum. For example, the distribution function f(x,p), which is the number density of particles in phase space (x^\mu and p^\mu are coordinates on \mathcal{M} and \mathcal{T}^*_x\mathcal{M} respectively).

The spatial volume form (integral measure) is \mathrm{vol}(\mathcal{N}) \equiv \sqrt{g(x)} dx^1 \wedge \ldots \wedge dx^n, where g(x) is the determinant of the metric evaluated at x \in M; and the volume form for the full phase space is dx^1 \wedge \ldots \wedge dx^n \wedge dp_1 \wedge \ldots \wedge dp_n. If we want to integrate out the momentum-dependence of some observable, we need just the momentum-part of the volume form. From the two expressions above we can see that this is \mathrm{vol}(\mathcal{T}^*_x\mathcal{M}) = \frac{1}{\sqrt{g(x)}} dp^1 \wedge \ldots \wedge dp^n.

However, paths that obey the classical equations of motion are constrained to lie on the mass-shell: in flat Lorentzian spacetime this is a hyperboloid in the momentum cotangent space \left(\mathbb{R}^{1,3}\right)^*, given by -p_0^2 + p_i^2 = -m^2, where p_0 is the energy, p_i the spatial 3-momentum and m the rest mass of a particular particle. However, in fully general curved spacetime the corresponding condition gives a hypersurface \mathcal{N} \equiv \{(x,p) \in \mathcal{T}^*\mathcal{M} \: | \: p^\mu p_\mu = -m^2 \}.

Because the permitted region of phase space has been restricted, we can eliminate one momentum component from the integral, treating it as a function of the other coordinates. Conventionally we pick p_0 to be this unwanted component, writing p_0 = p_0(p_i), and note that the equation for \mathcal{N} can be written p_\mu p^\mu = p_0 p^0 + p_i p^i = -m^2, which allows us to solve for p_0.

We then find the volume form induced on \mathcal{N}. Let n be the unit vector field normal to \mathcal{N}:

n = \left.\frac{d(p_\mu p^\mu)}{\left\|d(p_\mu p^\mu)\right\|}\right|_\mathcal{N}

The denominator works out to be -4m^2. Noting that the metric is a function of x alone, we perform the derivative on the numerator and so find that

n = \frac{2p_\mu dp^\mu}{-4m^2} = \frac{-p_\mu dp^\mu}{2m^2}

(the normal index convention for vectors and covectors is swapped round, as we are working on \mathcal{T}^*\mathcal{M}, so the coordinates p_\mu are covariant to begin with).

In general, the volume form induced from a manifold \mathcal{M} onto a submanifold \mathcal{N} with normal VF n is

\mathrm{vol}(\mathcal{N}) = \left. n \:\lrcorner\: \mathrm{vol}(\mathcal{M}) \right|_\mathcal{N}.

For the present purposes we therefore have

\sqrt{g(x)} \mathrm{vol}(\mathcal{N}) = n \:\lrcorner\: \left( dp_0 \wedge dp_1 \wedge dp_2 \wedge dp_3 \right) \\ \:\: = (n \:\lrcorner\: dp_0) dp_1 \wedge dp_2 \wedge dp_3 - (n \:\lrcorner\: dp_1) dp_0 \wedge dp_2 \wedge dp_3 \\ \:\:\:\:\:\: + (n \:\lrcorner\: dp_2) dp_0 \wedge dp_1 \wedge dp_3 - (n \:\lrcorner\: dp_3) dp_0 \wedge dp_1 \wedge dp_2

We differentiate the condition p^\mu p_\mu = -m^2 with respect to a spatial component p_i and rearrange, giving (note the positions of the indices):

\frac{\partial p_0}{\partial p_i} = \frac{-p^i}{p^0} \\ dp_0 = \frac{\partial p_0}{\partial p_i}dp_i = \frac{-p^i}{p^0}dp_i.

Expressions of the form dp_0 \wedge dp_i \wedge dp_j can be simplified to involve only the k-component of dp_0 ( i \neq k \neq j ), as repetition of a form in a wedge product sets the entire expression to zero. We also have n \:\lrcorner\: dp_\mu = n_\mu = \frac{-p_\mu}{2m^2}.

So, putting it all together,

\sqrt{g(x)} \mathrm{vol}(\mathcal{N}) = \\ \:\:  \left( \frac{-p_0}{2m^2} \right) dp_1 \wedge dp_2 \wedge dp_3 - \left( \frac{-p_1}{2m^2} \frac{-p^1}{p^0} \right) dp_1 \wedge dp_2 \wedge dp_3 \\ \:\:\:\:\:\: + \left( \frac{-p_2}{2m^2} \frac{-p^2}{p^0} \right) dp_2 \wedge dp_1 \wedge dp_3 - \left( \frac{-p_3}{2m^2} \frac{-p^3}{p^0} \right) dp_3 \wedge dp_1 \wedge dp_2 \\ \:\: = \frac{-1}{2 m^2 p^0}\left(p_0p^0 + p_ip^i\right) dp_1 \wedge dp_2 \wedge dp_3 \\ \:\: = \frac{-(-m^2)}{2 m^2 p^0} dp_1 \wedge dp_2 \wedge dp_3

And so the final result is

\mathrm{vol}(\mathcal{N}) = \frac{dp_1 \wedge dp_2 \wedge dp_3}{2 p^0 \sqrt{g(x)}} \\

which has the expected form. Integrals over momentum space therefore look like

\int_\mathcal{N} f(p_i) \frac{dp_1 \wedge dp_2 \wedge dp_3}{2 E(p_i) \sqrt{g(x)}}.

Everything we wrote down was manifestly covariant, so this volume form transforms in the correct way under general coordinate transformations. The rest mass m does not appear in the final volume form, so we are free to set m = 0 if we choose, as is the case with photons.

About ejlflop

Intrepid explorer of music, mathematics, computer programming. physics (an unordered list). Enthusiastic semi-lay-person.
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