If the universe has a fixed amount of mass, does it mean it has a fixed amount of energy or not? Because of the whole $E=mc^2$ thing?

Question

$E=mc^2$ states that Energy equals matter times the speed of light. So, can't we just turn all the energy into mass and it would come out to a fixed amount of matter?

Answer 1

When dealing with physics on a curved background (e.g. when we work with standard general relativity and the einstein field equations: $R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi G T_{\mu\nu}$), stress-energy is the term that is conserved. When considering the universe as a whole (as in your question), we should use general relativity. Stress-energy conservation is written as \begin{equation} \nabla_\mu T^{\mu\nu}=0 \end{equation}

$T_{\mu\nu}$ is a tensor (represented by a 4x4 matrix) that describes the distribution of energy density, pressure, etc. Here is the wikipedia link that discusses $T_{\mu\nu}$. So the short answer to your question is no. In some specific (and very useful) cases we may consider energy as a conserved quantity, but in general energy is not conserved.

Answer 2

The universe does not have a fixed amount of mass (particles). Gamma rays (massless) can be converted to electron-positron pairs (massive). Similarly, particle-antiparticle pairs can annihilate to produce gamma rays.