I suppose my language was too imprecise here. I should have asked “Why does c have the value it does (namely, 299,792,458 m/s or 186,000 mi/s or whatever you prefer) rather than some other value?”
Ok thanks everyone for helping me out. what i take from this is that we use C squared because it works , when you input all the other numbers that what you come up with each time and i am guess either we don’t know exatcly why it work or that its to complex to understand on a message board…cool ;j
Take the antidervative of the momentum, mv. 
It’s not in E = mc^2 because how that term is derived is quit a bit different (since it includes relativism). Also, note that E = mc^2 only when an object isn’t moving; the full form is like E = y(v)mc^2, where y(v) is 1/sqrt(1 - (v/c)^2).
Anyway, this all stems from what the fact that light travels at the same speed “no matter what,” which is a really weird thing. If you take that idea to some logical conclusions, you end up with this bizarre speed limit on the universe. If you mix that idea up with some physics - which is a logical thing to do, since physics is quite interested with moving things! - you end up relating mass and energy.
What exactly is energy here? The dictionary says in physics energy is the ability to do work and work is transfer of energy from one physical system to another. But I thought we were trying to figure out the “energy” of one system? Are we measuring the force of the total internal movemnets of atoms of an object or what? IF so where do you draw the line between object and non-object? Obviously I am confused. Is energy here purely a matter of movement such as kinetic energy?
No, ‘c’ is short for “celerity” (in English) or “celeritas” (or something like that, in Latin), which means “speed.” The speed of a wave in a fluid is its celerity, and that’s why ‘c’ was chosen.
the energy we are talking about is the total energy a non-moving mass would have if all its matter were turned into energy. The internal movements of atoms, bonds, etc are trivial and can be ignored when compared to the conversion of matter into energy as seen in an atomic bomb.
Not directly. In units, {energy} is {mass}{distance}[sup]2[/sup]/{time}[sup]2[/sup]. But the geometry of spacetime says that distance and time are the same thing (with a conversion factor (whose value depends on the units) called c). A distance unit is c times a time unit, so the energy unit is c[sup]2[/sup] times the mass unit. The formula is just a conversion of units exactly analogous to yard = 3*foot.
Also, the structure of a message board (as opposed to a personal interaction) really hobbles the sort of interchange I’d need to learn what this particular non-expert doesn’t know. Often enough people get very annoyed if I low-ball them, so I tend to high-ball them and dial it back through the exchange until I know exactly where they stand.
Also, if I could use proper symbology (though I avoid it with non-experts it can still be helpful to write a basic equation and refer back to it) or draw something it’d be a lot clearer.
That explains it perfectly, but I should have been more clear about what I was getting at.
I was thinking purely from an abstract mathematical viewpoint. I found your statement that I quoted before to be quite intriguing and profound. For example, time and position have clear physical meanings but mathematicaly they are just ordinary dimensions in a geometrical space. Distance (interpreted as how much I have to travel in my everyday life to get from A to B) can be defined by a metric, which has mathematical significance outside of our physical universe. In other words it’s part of geometry, an abstract theory. I was wondering if such an abstract concept existed for “energy”. That is if the function E(m) = c[sup]2[/sup] x m could be given mathematical significance.
After some thought I think the answer is it couldn’t. Energy (and mass) only has meaning/interest/usefulness when studied in relation to the physical universe.
Ah, I thought you meant that as a clarification to my answer to the OP. Well, the short answer is that there actually is a geometric description of energy. I’d advise fastening your brain’s seatbelt, though, 'cause it’s gonna be a bumpy ride and I won’t be able to finish in a single post due to time constraints.
The first step is to forget about mass as anything separate from energy. Time is just just distance measured in seconds instead of centimeters. Mass is just energy measured in grams rather than ergs.
The next step is to throw away discrete classical mechanics and move to a field-theoretic picture, which we’d have to do eventually in our pursuit of a theory of everything anyhow. Don’t think of bodies with momentum and mass (er, “rest energy”) and kinetic energy and so on. It’s all fields like the electromagnetic field for photons and the electron field for electrons. If you really need to think about a cannonball, think of its field as the composite of all the fields of each constituent particle. If you can’t keep that in your head, don’t worry and just think about abstract fields.
Now, at every point of spacetime the field takes some value in some vector space. In general you can’t really think of this as a function from spacetime to the vector space because of the way the geometry is set up. What we need are “vector bundles”.
VECTOR BUNDLES
A vector bundle is a map of topological spaces
p: E —> B
such that the fiber p[sup]-1/sup over each point b of the “base space” B is homeomorphic to a topological vector space V (let’s just say over C to keep things simple), and such that B has a covering by open sets U[sub]i[/sub] such that
p: p[sup]-1/sup —> U[sub]i[/sub]
is isomorphic to the projection
p[sub]1[/sub]: U[sub]i[/sub] x V —> U[sub]i[/sub]
That is, there is a homeomorphism
h: p[sup]-1/sup —> U[sub]i[/sub] x V
such that p = p[sub]1[/sub]h.
The classic reference is Steenrod’s “The Topology of Fibre Bundles”. More recently, Hermann’s “Vector Bundles in Mathematical Physics” is more on-point. Anyhow, a good nontrivial example is the “Möbius bundle”. Think of a regular Möbius strip but rather than have the cross-section be an interval imagine that it’s the whole real line. Anyhow, you can think of it in two halves, one twisted and one not. Each one looks like an interval crossed with R[sup]1[/sup], but they paste together into a bundle with a circle as the base space that isn’t homeomorphic to that circle crossed with the real line.
Over a “trivializable” set U (one like the U[sub]i[/sub] above) you can think of functions from U to V. Each such function has a graph in U x V and we’ll generally think of the two as the same thing. This is a map s from U to U x V of the form
s(u) = (u,f(u))
Note that p[sub]1/sub = u, and we’ll take this as the definition of a more general concept.
A “section” of a vector bundle is a continuous map
s: B —> E
such that p(s(b)) = b for each b in the base space – it sends b to a point in the fiber above b. This is not the same thing as a function from B to V, even though each fiber looks exactly like V; the copies of V can be sewn together in a topologically nontrivial way.
As an example, the trivial real line bundle over the circle has all functions from the circle to R as sections since it’s, well, trivial. On the other hand, it isn’t hard to see that there are no sections of the Möbius bundle with no zeroes: as you go around the band positive numbers flip over into negative numbers so you must pass through zero somewhere. In fact, I’m being a little fast and loose with the notion of “function” here, and thinking of sections as functions within small enough regions that you can trivialize the bundle over the region. Best just think of sections as maps s as defined above.
Anyhow, ask any questions you want and I’ll be back later to push ahead with this description of energy.
That’s fascinating Mathochist, thank you! I get the gist of what you’re saying but it’s really over my head. I have to mull over it to see if I can come up with any intelligent questions. Unfortunately my background is in engineering and I never studied topology.
Tomorrow I’ll check if they have Steenrod or Hermann available at my college library.
This has become interesting topic to me, and if I may I propose a question that may help me and the OP understand this better.
If I know the mass(m if I’m not mistaken) of an object and it’s speed, how do I apply it to the formula to determine it’s energy?
I thought c was velocity, but am confused by the discussion of this being the speed of light.
For example. a 400 lb object is traveling at 1/100 the speed of spped.
In my mind I would have created the equation like thus. E=400x100/(speed of light)^2 to determine how many pounds of energy the rock would impact with.
The equation E = mc[sup]2[/sup] applies only to an object at rest. c is a constant, so you just multiply the mass by that, and you’re done.
In natural units, c has the value 1. You only get some other number when you use units other than natural ones, and the value will depend on what units you choose. The question then becomes, why did humans choose meters and seconds as our normal units of space-length and time-length. Of course, when those units were defined, the speed of light was not well-known, much less the fact that that was the Universe’s one “special” speed, so it’s hardly a surprise that the speed of light is such an awkward number in those units. And the reason that those units were chosen in the first place is that they’re convenient for human-scale things: A human’s arm is about a meter long, and a person’s pulse rate is about one per second. So any further inquiry into the question becomes one of biology, not of physics: Why did humans evolve to have such a size, and why do we have such characteristic times?
Randomy, if you’re just throwing a big rock, then you’re probably not going to be converting any rock atoms into other forms of energy, and you’re probably throwing it at much less than the speed of light. In that case, there’s a very good approximation you can use to find the energy available in the impact, and it does not involve the speed of light. For such objects, you can just use E = 1/2 m*v[sup]2[/sup], where v is whatever the object’s speed is. This is the equation Newton would have used, or anyone else before Einstein, and it’s very accurate for most objects you’ll ever encounter. You could use the relativistic equations (which involve the speed of light), but that’s a lot more work, and unless your object is moving close to the speed of light, it’s made of antimatter, or you need insane levels of accuracy, that extra work isn’t worth it.
I aced the AP Physics class, so I should know this. Of course that was 17 years ago, and I barely remember F=ma.
Here’s my follow-up: Everyone keeps repeating that E=mc^2 for objects at rest ONLY. But isn’t “at rest” subjective? I mean, the car COULD be moving. Or the car is still and the earth around it is moving. So I’m not sure why “at rest” matters.
You have to choose your inertial reference frame first. Inertial means not accelerated, so that Newton’s first law remains valid on that frame. Only then will “at rest” be properly defined.
If you define your reference frame to be centered on the car, then it is the earth that is moving, and the car is at rest. If you define your reference frame to be the earth, then it’s the car that is not at rest (assuming it’s traveling of course). Physically both frames give equivalent results.
I apologize for my ignorance, but I need to ask another follow-up, since I still don’t get it. Maybe a real-life example would be easier:
Let’s say I’m driving my car and there is a can of Pepsi in the cup holder (We can’t use Coke in this example because it’s so vile that it obviously doesn’t follow the laws of the universe). Relative to me, the can is stationary and its energy is its mass multiplied by the square of the speed of light. This remains true whether or not I’m stopped at a traffic light or flying down the highway. Even if I’m accelerating, the can is stationary relative to me. But to the pedestrian I just missed hitting because he was in my blind spot, the can was definitely not stationary. In fact, the can was moving at very high speed. Based on what I’ve learned from you guys, for the pedestrian the can’s E does Not equal mc^2. But once I screech to a halt, it does. What am I missing here?
E = mc[sup]2[/sup] only refers to the at-rest energy of a given mass. More generally, the total energy of a mass (one possibly in motion in your reference frame) is given by Equation 8.3b on this page.
You and the pedestrian have different inertial frames of reference.
Also, note that acceleration is not relative motion, so you might want to leave that out of the picture as much as possible.
For those confused by all this, just wait until you try to cope with Quantum Mechanics and the Supeerstrign theory. :