This might get a tiny bit complicated. I’ve tried to keep the maths simple but towards the end, where I specifically address potential concerns that might be raised by physics students, it does get a bit tricky.
Again, a warning for people studying relativity in a formal setting, please use the derivation methodologies recommended by your teacher or lecturer. I don’t believe that what follows is wrong, but it’s not entirely standard.
We are quite used to the idea of movement with respect to time, although we normally express it in terms of time passing by us, rather than thinking in terms of us passing through time (minutes pass, the years pass by and so on). When we raise the idea of movement, there is a linked idea of the rate of that movement. In other words we could ponder the question “at what speed do we move with respect to time?”
There is actually an answer to this question. In a recent post I talked about how the expansion of the universe can be considered to be time as opposed to the universe expanding with time or over time. Another way of expressing this is to say that if you were to be at rest (in spatial terms), then you’d still have a speed “through” time – the rate at which you travel from one temporal location to another. That speed would be c.
But what happens when you are not at rest? (Note that for the sake of simplicity, I am only talking about uniform rectilinear motion here.)
Here’s an illustration with some simple mathematics (note that we have kept the “good twin” and “evil twin” notation, where the “good twin” is notionally at rest, see the Lightness of Fine-Tuning articles in which this concept was introduced - Part 1 and Part 2):
What this illustration is showing is the idea that our speed through space-time is conserved (or invariant). We can change the direction of our travel and thereby experience some transition through space at the expense of transition through time.
This result should not be a surprise. As Einstein said; “Energy cannot be created or destroyed, it can only be changed from one form to another.” This is a rephrasing of and expansion on the first law of thermodynamics, per Rudolf Clausius in 1850; “In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.”
Both statements express the conservation of energy principle and, from the most celebrated of all equations formulated in the 20th Century, we know that E = mc2. This is, strictly speaking, saying that the energy of a body at rest is the mass times the square of the speed of light.
What I am suggesting is that a body can be said to have an invariant space-time type of "kinetic" energy, related to an invariant space-time “speed” of that body. In other words:
E = m.vspace-time2
Hopefully, people reading this now understand that we have an invariant "space-time speed" (at least to the extent that the maths is accessible to them).
Please note that this is not a standard way of dealing with these concepts. I don’t think it’s wrong, but if you might not want to spring this on your physics teacher without prior warning, certainly not if he or she is teaching you a different way of looking at relativity and the consequences thereof.
Neither m.vspace-time2 nor m.vspace2 is intended to represent classic Newtonian kinetic energy which is given by Ek ≈ ½mv2.
For completeness (and for the sake of pesky physics students who will probably be utterly convinced that the above must be wrong), I’ll now show how we can arrive at this kinetic energy equation.
If you don’t like maths, or physics, and for some reason you have nevertheless made it thus far … turn back now!
The equation E = mc2 tells us that mass and energy are equivalent and for this reason we should be careful about talking about the mass within a system or the energy within a system, since it is “mass-energy” that is conserved, rather than mass or energy separately.
Extending this concept slightly further (and again going slightly beyond the standard conceptionalisation), what we know as mass can be considered as equivalent to the concentration of mass-energy in a body – the more energy in a body of a fixed size, the more mass it has and, perhaps less intuitively, a body with the same amount of energy in a size reduced by the effects of relativity will also have more mass. This will be difficult for a lot of people to stomach. Surely if you have more mass in a smaller space that just means it’s denser mass, but the same amount of mass? Well, yes and no. If there is more energy in a given volume at rest, then that equates to more mass, and the same energy in a smaller volume at rest would be the same mass, but denser. However, if you have length contraction acting on a volume, then as the volume decreases due to relativity the relativistic mass increases.
So, (where V is volume)
Mrel ∝ 1 / Vrel(As an aside, another way to look at this is to say that as the concentration of mass-energy increases, the more difficult it becomes to accelerate that mass-energy and change its inertia. This is the fundamental understanding of mass.)
According to an observer at rest, the volume of a concentration of mass in motion is foreshortened in the direction of the motion (due to length contraction). Since the volume has gone down (according to the observer at rest), the relativistic mass goes up in inverse proportion. I've returned to the good twin/evil twin notation, sub-G and sub-E.
VG = LG . A
VE = LE . A
LE = LG . √(1 – v2/c2)
VE = VG . √(1 – v2/c2)
Therefore (since mG is inversely proportional to VG)
mE = mG / √(1 – v2/c2) = mG . (1 – v2/c2)-½
This is the standard equation for relativistic mass, where mG is the rest mass (the good twin is notionally at rest) and mE is the mass in motion.
(For those keeping track, I should mention that we are now completely back inside the scope of standard method, it’s only that I have slightly different notation. While making an aside, may I add that if you follow the link above, note the objection that Einstein had to the concept of “relativistic mass”.)
If the total energy of the body is considered to be the sum of the kinetic energy of that body and the inherent energy of that body (that is the rest energy), then:
Etotal = Ek + Eo
EE = Ek + EG
Ek = EE – EG
Ek = mE.c2 – mG.c2
Ek = mG / √(1 – v2/c2).c2 – mG.c2
However, where v is sufficiently small (feel free to test out this approximation using a spreadsheet, or if you are old school, a pen and paper)
1 / √(1 – v2/c2) ≈ (1 + ½v2/c2)
Ek ≈ mG.(1 + ½v2/c2).c2 – mG.c2
Ek ≈ mG.c2 + ½mG.v2 – mG.c2
Ek ≈ ½mG.v2