Oops, sorry, no spell checker here. Try: “Superstring.”
Klondike
I thought the OP was asking something like this (or at least this is something that always bugged me):
If the c^2 is a constant, why can’t we just say E = mK and just change the numbers on the E or the m to accomodate it?
At some point, we still use those formulas for calculating something. . .that is, in Einstein’s equations, I can tell you the energy of a particle is 50 “einsteins”, and you go, “all right, then the mass is 50/c^2.”
But, in my system, I tell you the energy of the same particle is 100 “trunks”, and you go, “all right, then the mass of the particle is 100/K.”
Succinctly: It seems as if the relationship between the units we use for mass and energy was just arbitrarily defined through c^2 but the actual physical relationship between mass and energy has nothing to do with the speed of light at all.
Energy depends on your reference frame. So to that pedestrian, your Pepsi can has more energy than it does for you in the car.
And energy was originally defined long before Einstein, and used in equations such as kinetic energy = 1/2 m*v[sup]2[/sup]. Once you’ve defined energy in this way, when you discover relativity, you discover that the constant used for rest energy is c[sup]2[/sup].
The main use of E=mc^2 is a simple conversation from mass to energy, which are of course different forms of the same thing. This mostly comes up in reactions between subatomic particles, during which particles are routinely converted into photons and vice verca. Since you always get out the same amount of energy you started with, but you might get a flash of light instead of more particles as a result, it’s easier just to work these calculations all in terms of energy. So instead of an electron being 9.11E-31 kg, it’s .511MeV/C^2, units of energy. Then when the electron and positron (another particle) collide and completely annihilate, it’s just a matter of adding their two energies to see how much energy the photons resulting in this collision will have.
Since macroscopic objects like us don’t routinely turn into photons then it’s not really necessary to do Newtonian mass calculations with energy units.
I’d intended to get back to this earlier, but I had a long drive today and am just getting caught up.
So, the naïve view on fields is that they’re functions which assign a vector to each point of spacetime. More accurately, a field is a section of some vector bundle over spacetime. If spacetime’s geometry is complicated, these bundles can be nontrivial, which means that we can only think of a field as a function in small enough patches that we can trivialize.
Another example of a vector bundle is now in order, possibly the single most important object in all of geometry: the tangent bundle. There are very technical details involved in setting it up properly rigorously, but I’ll try to avoid them.
Imagine a sphere S. At any point p there is a well-defined tangent plane T[sub]p/sub, which is a two-dimensional real vector space. The tangent bundle is the (disjoint) union of all these planes, but it’s not S x R[sup]2[/sup]. Instead the tangent planes to nearby points are sewn together with “as little rotation as possible”. As p moves around the sphere the tangent planes need to rotate a bit, and so the bundle gets twisted up sort of like the Möbius bundle got twisted as we moved around the circle.
We can do the same thing for any n-manifold (something that looks locally like R[sup]n[/sup] to get a family of n-dimensional real vector spaces (the tangent spaces at each point), which sew together to give the tangent bundle, and whose nontriviality reflects topological information about the manifold itself In general it’s the set of pairs (p,v), where p is a point of the manifold and v is a tangent vector at p, and this set is given an appropriate topology which in general is not M x R[sup]2[/sup]. Really, there’s a lot I’m leaving out here, but this is the basic idea. The books I referred to last time also talk about tangent bundles, as does the Phone Book (Misner, Thorne, and Wheeler’s Gravitation. This is THE BIBLE for anyone who wants to learn the geometric underpinnings of General Relativity, as far as I’m concerned).
Now, an aside on linear algebra. I’m also going to speed up and give more references because the further we go the more the details cloud the picture. If you’ve got a vector space V, you can consider 2-forms: the set of functions f on V x V satisfying
bilinear: f is linear in each variable
alternating: f(u,v) = -f(v,u)
In particular, we can do this on the tangent space at a point p of spacetime. What we get is a six-dimensional vector space at every point, and the way tangent spaces sew together into the tangent bundle gives us a way of sewing these into a “bundle of 2-forms”. This should be worked out in detail in Steenrod. Also highly recommended is John Baez’ and Javier Munian’s Gauge Fields, Knots, and Gravity. Baez is an amazing writer, and this text is particularly on-point. Anyhow, to recap: the bundle of 2-forms on M is basically the set of pairs (p,f), where f is an alternating bilinear function on T[sub]p/sub.
So, why do we care? Because now we can define our first familiar field. The electromagnetic field is a section of the bundle of 2-forms on spacetime. If we pick four basic directions at a point (one for time and three for space) we can read off the six components of E and B fields from basic electromagnetism at that point. The choice of directions corresponds to picking an observer and asking what values he will measure for the electric and magnetic fields. In particular, if we think in a three-dimensional “space” slice of spacetime, a photon is a section whose value is zero (f is the function which takes the value zero for all u and v) everywhere but a very small region. In particular, the region is technically what mathematicians call “compact”, and the section is said to “have compact support”. What this means in detail isn’t really important, but it is important for something we’ll want to do later. I’ll point that out as when it comes.
And, with the objects of study defined (fields, interpreted as sections of vector bundles) and with a concrete example in mind (the electromagnetic field as a section of a six-dimensional vector bundle whose components are essentially those of the electric and magnetic “vector fields” from basic physics), we can finally define the geometric notion of energy.
Rather than thinking of sections over all of M, we can just consider sections over subsets of M. Remember in calculus you hardly ever used functions on the whole real line, but just on intervals. So, given a bundle E —> M and a subset U of M, we can consider the part of E sitting above U, which is a bundle in its own right, and has a set of sections. What we want to do is assign a value to each section over U so that if we break U up into a bunch of pieces (and break up the section with it) then the value of the section over U is the sum of the values of the sections over each piece. Also, it should satisfy some natural rules – the value of the sum of two sections should be the sum of their values, for instance. Such an assignment is called a “functional” (not a “function”, mainly because the set of inputs is pretty hairy). The energy of a field is the value of an appropriate functional.
Now, I’ll admit that the geometry has gotten lost in the details here, and a lot of the time it seems (at least to a mathematician like myself) that the functional is picked “because that’s what works” and not to reflect underlying geometrical structure. Still, the energy functional of the electromagnetic field is pure geometry.
Remember I’ve gone this whole time without choosing a metric. For the moment, a metric is a way of measuring lengths of curves in a manifold, specifically given by defining a “dot product” in each tangent space and integrating the length (square root of v-dot-v) of the tangent vector to the curve along the curve. For more depth: Baez&Munian or the Phone Book. The presence of a metric is what separates differential geometry from differential topology. All we’ve been talking about is shape, but now we can talk about size. The metric is the fundamental ingredient in general relativity. The gravitational field is the metric.
So, since we’re in a GR spacetime we’ve got a metric: a “dot product” for each tangent space. The way we constructed the space of 2-forms at each point from the tangent space there allows us to construct a unique dot product for that space (of 2-forms) from the metric. In particular, if we interpret the six components of a 2-form at p as the components of E and B there, we get exactly the same formula as for the energy density from electromagnetism! All that’s left to do is integrate this density and get the total energy of the field. And yes, it turns out that taking the dot product of the value of a section with itself and integrating this number-valued function over U does give a functional. In fact, for most simple fields the energy is exactly of this form.
Just to stick the landing: intepreting a particle-view photon as a compactly-supported section of the electromagnetic field and calculating this energy goes as follows: Since the support of the section is compact, we can show that the integral in question converges. This gives us a finite value (even though spacetime may be infinite, “most places” the function will be zero), which is the classical “kinetic energy” of the photon considered as a particle.
The last few paragraphs are very superficial, I know. It’s all worked out nicely in B&M and the Phone Book, and looking through Rubakov’s The Classical Theory of Gauge Fields might not be a bad idea either. Also, as a preemptive disclaimer: I’m sure I oversimplified something in the physics to the point of falsity, but the end result of the argument is correct as far as I know. Corrections from those who know are welcome, as are questions for those who don’t. For the moment I will go pass out, though.
This might be the explanation that your looking for. Light comes into it only because it is the thing that can achieve maximum speed in the universe, so let’s omit it.
Think of it as E=M(moving at the fastest rate possible) squared. So when M (mass) travels at the conversion factor, C (the fastest rate of speed possible in the universe), it is transformed into E (energy).
It is interesting to note that the change of an object (M) was gradual. if I remember correctly, as it approached “the fastest possible speed” it’s properties changed. In addition to it’s time reference shortening (the twins thought experiment), its length diminished, and weight increased. I think that’s right, it’s been a while.
This really isn’t it at all. E = mc[sup]2[/sup] is an equation which holds in a comoving reference frame. That is, when the object in question is at rest. Further, there is an additional increase in the mass as measured by an observer who isn’t comoving with the object.
Nope. Any object of non-zero mass traveling at the speed of light would have infinite energy.
I certainly don’t know enough about this to argue, but can the above be right? In the equation it seems that the value of E is dependent and will change based on what is on the right side of the equal sign. And since C is a constant, E will increase as M increases. Please tell me where my thinking is wrong. Thanks.
m (not M) increases as a function of m[sub]0[/sub]/sqrt(1-v[sup]2[/sup]/c[sup]2[/sup]).
E=mc[sup]2[/sup] only applies to masses at unaccelerated and at rest relative to an inertial coordinate frame.
Stranger
Yes, E will increase as m increases. you were talking before about varying speed, while now you assert that the speed term is constant.
Now, go back and read the thread and see all the times I (and others) said that E = mc[sup]2[/sup] is using c[sup]2[/sup] as a conversion factor, just like distance (in yards) = distance (in feet) * 3.
I get it now. thanks that make so much more sence to me. it not the speed of light squared its just the only thing that can reach that speed. now if i can ask another question why is it squared? if light is the cosmic speed limit why do we square it?
Simplified, off the cuff derivation:
Imagine a system, a box with mirrors at both ends we’ll say, with a mass M and a velocity v. Within the system, there is a photon with energy E and velocity c. Now (assuming that M is large and v is small, such that Lorentz effects can be ignored) we have a resulting conservation of momentum such that:
M*v = E/c (1)
Reordering (1), we get
v = E/M****c (2)
We’ll define Δt as being the time that it takes light to travel a distance L, such that
Δt = L / c (3)
Multiplying (2) by (3) gets
v*Δt = EL/Mc[sup]2[/sup] (4)
(4) implies movement
Δx = v*Δt (5)
but by Newton’s Second Law the box, being a closed system unimpinged by external forces can’t move its center of mass without undergoing acceleration, e.g. a change in energy. Since this isn’t the case, the photon must have a mass of its own (m) that counteracts the displacement of the box, so
mL = MΔx (6)
Momentum is thus conserved, and there is much rejoicing. Substituting (5) into (6) and solving for m gives:
m = M*Δx/L = M/L * EL/Mc[sup]2[/sup] (7)
Canceling M and L gives
m = E/c[sup]2[/sup]
or
E = m****c[sup]2[/sup]
This is the classical derivation, but there are other, more complex and rigorous explainations as well. (And I hope to Og I didn’t fark this up, 'cause I took my physics book home last night and my Mark’s doesn’t cover relativity dynamics. I’m sure if I did, Chronos and Mathochist will take turns busting my kneecaps, Casino-style.)
Stranger
More simply, it’s squared because that’s what the units demand. Energy has units of mass times length squared divided by time squared. Any equation that gives you energy must have those units. For instance, kinetic energy (at speeds much less than the speed of light) is one half times mass times speed squared. The “one half” doesn’t have any units at all, mass has units of mass (of course), and speed has units of length divided by time, so speed squared has units of length squared divided by time squared. Put it all together, and you get mass times length squared divided by time squared. Likewise, gravitational potential energy is mgh, where g is the acceleration due to gravity (9.8 meters per second squared) with units of length divided by time sqared; if we multiply that by another length (h, the height of the object) and a mass, then we have those same units.
Stranger, your post looks correct, but you’re using the fact that the momentum of a photon is E/c, which is correct but not necessarily obvious.