I've been trying to see how relativity shows that the quantity gamma*m*c^2 (total energy) is conserved. I assumed that this would proceed from the conservation of momentum. So I researched momentum in relativity, and noticed that it has a time-component: gamma*m*c (which is E/c). So this partially answers my question. If four-momentum is conserved so is E. But how and why does this fourth component get introduced? The spacelike components of momentum have been changed from m*v (classical physics) to gamma*m*v. This part I understand. But can someone show me how this fourth time component (gamma*m*c) is necessary from the first principles of relativity? Thanks a bunch!