I'm kind of new to special relativity, I mean beyond what they tell you in survey courses. In any case, I've heard there was a relationship between energy and time in special relativity (I've actually heard someone say the "time component" of energy), and I've always been fascinated with E = mc2 (who isn't?), so I've spent some time trying to derive it. My question arises from a questionable application of the definition of work that I use to get the total energy equation. I'm not sure it's a legal move, since to get it I integrate over time instead of over distance. Usually when I see it "derived" it's by showing how the special relativity kinetic energy equation is the same as the classical kinetic energy equation when c >> u. Otherwise I've never really seen it derived, hence the reason I wanted to explore this. The thing is, I get correct results for both kinetic energy and total energy, so I'm wondering if there is something to it, or if it is merely coincidence and my own lack of understanding. In any case, here is my methodology (using only two dimensions for simplicity): (1) Start with a 2-dimensional space time vector: (dx, cdt) because c makes the time unit a distance unit (I've seen this done repeatedly in special relativity texts and tutorials, so I'm assuming this is allowed) (2) Divide both components by proper time (to make it compatible with a lorentz transformation): (γu, γc) (3) Multiply both components by mass to get momentum: (γmu, γmc) (4) Take the derivative with respect to time of both components to get force: (γm du/dt + mu dγ/dt, mc dγ/dt) (5) Integrate over distance... HERE IS THE QUESTIONABLE PART. With the spatial component of this "vector," it is straight forward Calculus II. I divide it into two integrals to make it easier, then use trig substitution and voila , out pops the CORRECT term for kinetic energy in special relativity: mc2 (γ - 1) (I made it a definite integral from 0 to u). This is definitely the right answer (I am willing to show my math if needs be, but it would be kind of tedious). It sits neatly in the spatial component of my new "vector." However, when I tried this with time, I ended up with γmcu, Since energy is conserved with respect to time, I expected to get TOTAL energy in the time component: γmc2, which means my answer is right for only one value of u, one that the object can never achieve. So then I adjusted my strategy, and did the one thing I have a question about: Since I started off with cdt as my time component, I figured, what will happen if instead of at step (5) integrating with dx, I integrate with cdt? The result was a shockingly easy integral. After doing everything to get force, my time component of force was: mc dγ/dt so I had mc ∫dγ/dt*cdt and of course dt/dt cancels to 1, leaving only mc2 ∫dγ from 0 to γ which is just γmc2. That's the right answer that I expected, because it is total energy. Essentially all I did was multiply dt by a scalar of c and use that instead of dx for my integration. But I'm still not sure if it is a legal move. I feel like I cheated. Here's why: First of all, what the heck does "work" mean in this context? Purely mathematically it seems to make sense to me, but in my experience work only applies to space, unless you're taking about dW/dt, which I am definitely NOT talking about. Did I just sort of pull that out of thin air? Or is this legal because we're talking about a four-dimensional space where time is treated similar to a spatial direction? Second, I kind of feel like the constant c is used arbitrarily here just to make the vector. I know c is special, but it does feel kind of like an arbitrary construction. What would you guys say about the validity of this method of deriving the total energy by treating it as sort of a "time" component of a displacement vector? Is it compatible with physics or just a waste of time?