BMS on my mind

I thought that I knew the Minkowski spacetime solution fairly well. But recently, to my surprise, I found that there is much more physics in that solution, especially at the null infinity. There is a set of certain symmetry transformations acting asymptotically at the null infinity which preserve some boundary conditions. This group of symmetries is known as BMS (Bondi-Metzner-Sachs) symmetry group. I will make more precise statements below in this post. Another, more surprising fact I found is that general relativity may not be a truly diffeomorphism invariant theory. In fact these BMS transformations map one asymptotically flat spacetime solution of constraints to another physically inequivalent asymptotically flat solution ( But it is a topic for later post (when I am older and wiser).

Before progressing further, let me try to explain why physicists are interested in this symmetry group. In the well established Standard Model, the particles are the unitary irreducible representations of the isometry group of the flat Minkowski spacetime (known as the Poincaré Group), and, that of internal symmetries. For curved and dynamical spacetime, a part of the isometry group breaks down and the rest gets gauged (I plan to write a blog post describing this phenomenon in the near future). The Standard Model is formulated within the framework of Quantum Field Theory which works only on the flat spacetimes or curved spacetimes with fixed geometry or curved dynamical spacetimes with classical graviton.  Gravity on the other hand is the theory of the dynamics of the spacetime. And, therefore, the Standard Model can not include the gravitons, the quantized form of gravity (like photons are quantized form of electromagnetic fields).

Now, this is the most interesting part, if we study physics on the asymptotically flat dynamical spacetime solutions, then we do have the asymptotic symmetry group of this gravitational space (it is essentially BMS). And the irreducible representation of this group will give the usual particles and some extra multiplets. These extra multiplets are termed as soft gravitons which have gained much attention recently in the quantum gravity community.

Again, in the hope of using these notes for my future reference, and to save a great fraction of my energy, I will take the liberty to be mathematically intense. But, I will try to maintain the rigor of topology, differential geometry and group theory so that my mathematician friends don’t get annoyed.

Now there are two ways to fish the BMS group of Minkowski space. One is by Sachs ( and other is by Penrose. Let us start with the first one.

Consider a normal-hyperbolic Riemann manifold (\mathcal{M},g^{\mu\nu}) and a chart (u(=t-r),r,\theta,\phi) in the neighborhood of point P with the following properties

  1. the hypersurface u=\text{constant} are tangent to the local lightcone everywhere.
  2. r is the corresponding luminosity distance.
  3. the scalars  \theta, \phi are constant along each ray defined by the tangent vector k^a=- g^{ab}\partial_b u.

For such manifold \mathcal{M}, the line element ansatz can be written as ds^2=e^{2\beta}V\frac{du^2}{r}-2e^{2\beta}dudr+r^2h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu) where A,B \in (\theta, \phi) and V, \beta, U^A \text{and} h_{AB} are the functions of coordinates and determinant of h_{AB}=b.

After feeding this ansatz into the Einstein’s field equations one obtains the following asymptotic behavior of the functions

  •  V=-r+2M+\mathcal{O}(\frac{1}{r})
  • \beta=\frac{cc^*}{(2r)^2}+\mathcal{O}(\frac{1}{r^4})
  • h_{AB}dx^Adx^B=(d\theta^2+\sin^2\theta d\phi^2)+\mathcal{O}(\frac{1}{r})
  • U^A=\mathcal{O}(\frac{1}{r^2})

A spacetime is said to be asymptotically flat if

  1. there exist the chart u,r,\theta,\phi with the properties mentioned above
  2. the line element equation and the asymptotic behavior mentioned above hold

Also at large r the line element \lim_{r\to\infty}{ds^2}=-du^2-2dudr+r^2d\Omega^2, which is what we would expect.

At this point Bondi and Metzner studied all the set of coordinate transformations which preserve the line element ds^2 and the asymptotic behavior of the functions. Their considerations were subsequently generalized and the following group was obtained.

So consider the spacetime smoothly covered by the coordinates 0\leq r<\infty, 0\leq\theta\leq\pi, 0\leq\phi\leq 2\pi \text{ and} -\infty<u<\infty. The BMS transformations are given by (\alpha, \Lambda)

  • \bar{u}=(K_\Lambda(x))^{-1}(u+\alpha(x))+\mathcal{O}(\frac{1}{r})
  • \bar{r}=K_\Lambda(x)r+J(x,u)+\mathcal{O}(\frac{1}{r})
  • \bar{\theta}=(\Lambda x)_\theta+H_\theta(r,u)r^{-1}+\mathcal{O}(\frac{1}{r})
  • \bar{\phi}=(\Lambda x)_\phi+H_\phi(r,u)r^{-1}+\mathcal{O}(\frac{1}{r})

Here x is the coordinate on S^2 given by (\theta,\phi)\Lambda is the Lorentz transformation acting as conformal transformation on S^2 and on K_\Lambda(x) is the corresponding conformal factor. \alpha is a scalar function on S^2 related to the supertranslation subgroup. Rest of the functions are uniquely determined by imposing the following constraints of composition

  • (\alpha_1,\Lambda_1)(\alpha_2,\Lambda_2)=(\alpha_1+\Lambda_1\alpha_2,\Lambda_1\Lambda_2)
  • (\Lambda_1\alpha_2)(x)=(K_\Lambda(x))^{-1}\alpha_2(\Lambda_1^{-1}x)

One can immediately notice the semi-direct product structure of the BMS group here. B=N\ltimes L where N is the infinite dimensional group of supertranslations and L is the connected component of the homogeneous Lorentz group.

Penrose’s derivation coming soon!


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