Hi. The actual universe does not obey time translation symmetry, which means that energy is not conserved.

Wow this blew up. People are really attached to energy conservation!

All right, all right. I'll try to explain. 2/

One of the most important theorems in physics is due to Emmy Noether, who by any measure stands as one of history's greatest mathematicians. 3/

https://en.wikipedia.org/wiki/Emmy_Noether

Noether showed that symmetries in the laws of physics are related to conservation laws. 4/

Symmetry under spatial translation implies conservation of momentum. Symmetry under rotations implies conservation of angular momentum. 5/

Symmetry under time translation t -> t + Δt implies conservation of energy. 6/

Another way to say this is that the Hamiltonian (which is the total energy) is the generator of time translations. 7/

In Special Relativity, spacetime ("Minkowski Space") is static, and infinite in both time and space. 8/

Minkowski Space is highly symmetric. It is not just symmetric under rotations and translations in space and time, it is also symmetric under Lorentz transformations. 9/

The full set of symmetries of Minkowski Space is called the Poincaré Group. 10/

Poincaré symmetry results in a conservation law for a tensorial quantity, the energy-momentum tensor, which includes the usual notions of energy and momentum conservation in any choice of inertial coordinate system. 11/

Poincaré symmetry is a global symmetry, which means symmetry transformations are applied to every point in the spacetime identically. 12/

This means that energy and momentum are likewise conserved globally. 13/

General Relativity promotes this to a local symmetry, in which coordinate transformations can themselves be coordinate-dependent. 14/

This means that energy-momentum is still conserved locally, but there is no longer a well-defined global notion of energy conservation. 15/

In particular, when you look at cosmological solutions, you find that spacetime is no longer static, but is either expanding or contracting. Our universe, of course, is expanding. 16/

The expansion of the universe BREAKS time translation invariance, and the universe is globally no longer symmetric under the Poincaré group. 17/

The universe is no longer invariant under arbitrary coordinate transformations: there is a preferred coordinate system, at rest relative to the expansion, called comoving coordinates. 18/

Comoving coordinates are a cosmic rest frame. A particle in motion relative to a comoving rest frame will lose momentum, and asymptotically come to rest in the comoving frame. 19/

Where does the kinetic energy of the particle go? No place. It just goes away. 20/

Likewise, if I take a comoving box full of dark energy, the volume of the box increases with time, and since the amount of dark energy is proportional to the volume, the amount of dark energy increases. 21/

Where did the extra dark energy come from? No place. 22/

Energy is still conserved locally: the energy flowing across the boundary of the box adds up to zero, even as the total amount of energy inside increases. 23/

This breaking of time translation invariance in cosmological spacetime is a form of spontaneous symmetry breaking, like the Higgs boson. 24/

The standard model is symmetric under electroweak symmetry, SO(3) x U(1). However, the vacuum state of the Higgs boson does not preserve the symmetry, but breaks it to a U(1). This means that conserved currents of the full symmetry are no longer conserved. 25/

Likewise, the laws of General Relativity are invariant under general coordinate transformations, but the spacetime itself is not. 26/

The cosmological spacetime does not globally preserve the full Poincaré symmetry of Special Relativity, which results in a corresponding global violation of energy-momentum conservation. 27/27

SU(2)xU(1)