The Mass-Shell Condition and de Sitter Space

String Theory without the Baggage, Episode 2.

Howdy!

Last time we reviewed the concept of phase space for a nonrelativistic point particle. Before doing the same for the relativistic case, it’s worth exploring why that construction won’t be as straightforward.

Sean

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Special Relativity augments the kinetic sector of physical theories. Specifically, it imposes constraints on velocity that impact the relationship between energy and speed. This constraint is often called the mass-shell condition¹.

\(\begin{equation}p^{\mu}p_{\mu} = -E^{2}/c^{2} + p_{x}^{2} + p_{y}^{2} + p_{z}^{2} = -m^{2}c^{2}.\end{equation}\)

Geometrically, this means that space of possible the momenta - which we used to think of as vectors - exist in a curved, three-dimensional space.

Now consider de Sitter space, which is a four-dimensional spacetime of constant, positive curvature. The standard construction begins with Minkowski space subjected to the constraint:

\(\begin{equation} -c^{2}t^{2} + x^{2} + y^{2} + z^{2} - w^{2} = -\ell^{2}.\end{equation}\)

The result is a four-dimensional curved space with a line element (in radially symmetric coordinates):

\(ds^{2} = - \left( 1 - \frac{r^{2}}{\ell^{2}} \right) c^{2}dt^{2} + \left( 1 - \frac{r^{2}}{\ell^{2}} \right)^{-1}dr^{2} + r^{2} \left( d\theta^{2} + \sin^{2}\theta d\phi^{2}\right).\)

Where l is interpreted as a characteristic length scale. This corresponds to a spacetime manifold with constant Ricci scalar curvature:

\(R = \frac{12}{\ell^{2}}.\)

Famously, this is associated with a positive cosmological constant,

\(\Lambda = \frac{3}{\ell^{2}}.\)

Comparing the mass-shell condition and the de Sitter constraint, it's easy to see that the momentum hyperboloid of a massive particle is a space akin² to a kind of three-dimensional de Sitter space, with mass playing the rôle of l.

Last time we modeled the classical phase space of a nonrelativistic point particle as the six-dimensional space R^6:

Phase Space in Classical Mechanics

In that nonrelativistic case, momenta belonged to the tangent space of R^3, which itself was a copy of R^3.

As we have just discussed, relativistically the mass-shell condition deforms that vector space of momenta into a curved manifold, somewhat like the hyperboloid of de Sitter space. As you might expect, this makes the description of relativistic phase space somewhat tricky. We’ll discuss that next time.

1. We will be extremely chaotic about metric sign conventions, as is our nature. Please stay on your toes!

2. Akin, but be mindful of the minus signs!