Why c2 (speed of light squared)?

Muphrid

I meant what I said. Every non-massless object has a four-velocity with magnitude [itex]c[/itex]. Hence, it travels through spacetime (space and time together, not separately) at the speed of light. I included the "in mathematical language" part to make that statement precise; I wouldn't expect it to be used among laymen. Still, I feel that understanding this point and how all four-velocities for massive objects lie on a hyperbola (and hence, all of them have magnitude [itex]c[/itex]) is critical in the chain of reasoning here.

AnotherDave

Ok, that makes sense.

But I still have to find a way of explaining this to someone who majored in Art Philosophy *facepalm*

russ_watters

Sorry, but the only possible answer here is that this is a wrong question. The framing of the question implies that he thinks C^2 is a speed. It isn't. It is just a piece of a chopped-up equation that is meaningless on its own. Just look at the units.

The fact that the equation includes C^2 doesn't have anything to do with why objects can't exceed the speed of light.

TheEtherWind

I would refer you to the "Minute Physics" video on youtube I believe to be titled "Derivation of E=mc2" in which the narrator derives the equation in a very similar fashion to the way Einstein did in "Does the Inertia of a Body Depend upon its Energy-Content." Except he makes it more animated by making the body a cat and simplifying generalizations if I remember correctly. You'll see exactly why c^2 appears.

PeterDonis

A minor but IMO important point: this applies to timelike objects, but *not* to lightlike objects. A photon, for example, has a null four-momentum, so it can't be described by any 4-vector with nonzero length.

WonderWatcher

How about this....
The speed of light is squared in E=MC^2 because we are dealing with a three dimensional area, much the same way as when we calculate the divergence of EM radiation we used the surface area of a sphere to solve the inverse square law. The surface area of a sphere is calculated as "4*pi*r^2". Therefore, asking why C is squared in E=MC^2 is the same as asking why r is squared in finding the surface area of a sphere in "4*pi*r^2".
Is this on the right track of thinking?

Mark M

No, it's just the way it shows up in the equation when you derive. There isn't any particular reason why. In special relativity, $$E^{2} = p^{2}c^{2} + m^{2}c{4}$$ Since photons have no mass, it follows that $$E = pc$$ So, since p = mv, E = mc^2. There isn't any real 'reason'.

AnotherDave

I like the spirit of WonderWatcher's response. "Why C2" is really not a meaningful question as Russ Walters correctly points out. But I don't want to be insulting by telling an old man his question isn't meaningful. Using the 4×∏×R2 which is something most people remember, I can respectfully make a comparison without being disrespectful or sounding like and arrogant pr*ck.

tryme83

I think this is a fair question that I have asked myself aswell. According to E=mc2, c2 tells us how much energy is retained in a unit of mass. But why is this c2 and not c or c3?

Besides (to me unfulfilling) mathematical explainations, C-squared makes sense to me after watching 'the car example' in this popular video:
http://youtu.be/xvZfx7iwq94?t=2m40s

A lightspeed particle moves at max speed (C) only through the space dimension, because time is theoretically frozen at this speed. C-squared may represent a particle moving through both the space AND time direction at lightspeed. This is practically impossible and therefore we can maximally measure C. However, I think this is theoretically possible if effects of space contraction and time dilation are somehow overcome? Or from another perspective, at C2 speed through both space and time, both space and time would be frozen, eliminating any reference frame. Anyway, rather then C, C-squared may be the asymptote of the universal energetic speed limit, directly relating to mass and energy.

StationZero

That would be 2c, not c squared, wouldn't it? Maybe not.

Space and time are already frozen for a photon traveling at c, we do not need c squared for this effect.

StationZero

I wouldn't. Those minute physics videos give me a headache and I leave more perplexed and anxious than I did going into it. I think they should be "retired" from service. Mgmt.

StationZero

Willem, you got the chain rule wrong here in the 5th step, it should be dt dx dx dt.

Last_Exile

AnotherDave Hi,

I think the simplest explanation for grandpa is that is "all due to the way the maths works"

For example, we all now velocity is distance/time and has uits of metres per second (m/s).

Acceleration, however, is the rate of change of velocity and has the units of metres per second per second (or metres per second squared).

Your original question is the same, in this example, as asking "how can we have time squared??"

We aren't actually "squaring time" itself - it's just the way the maths works.

Sometimes you have to know when not to take some things too literally!

HallsofIvy

You are conflating two completely different questions. I presume that "grandpa" here is complaining that c² is larger than c- violating "nothing can travel faster than the speed of light". But the "c²" in E= mc² is NOT the speed of anything- as pointed out before it doesn't even have the correct units for a speed. And, it is not necessarily true that "c² is larger than c"- that depends upon the units used. In papers about relativity, it is quite common to choose units so that c= 1. And then we have c²= c. In fact, in such units, E= mc²= m.

A perfectly reasonable answer to the different question "why is the E= mc² true?" is that we know, from relativity that energy and mass are proportional. That is, Energy is some constant times mass. We know, by comparing units (dimensional analysis) that the constant must have units of "speed". And the only fixed speed in the universe is the speed of light. That's very much a "hand waving" explanation but a more accurate explanation would have to be how the equation was derived in the first place and a link to that has already been given.

danR

You've thrown red meat before the pack haven't you?

Fortunately, you've asked the easy question: why is the velocity of light squared, rather than:

what the heck's light's velocity got to do with it?

If you may recall from school math/science/physics, they built up the measures of various properties rather carefully from distinctive units: length, time, 'weight' (mass), etc.

So, the area of an equal-sided rectangle is the square of one of its sides: length X length. Of course the figure in question is called a 'square', so that one is pretty clear.

What we call 'velocity' is defined (not merely 'found', or 'figured out') as Length/Time (and direction, but we won't worry about that right here).

Length and Time are the fundamental units of 'velocity'.

To cut to the chase, 'Energy' is not just some airy, vague concept like the 'whizzing around of atoms', or the great, big powerfulness of a supernova explosion, etc. In later school, it was probably defined as something, and also determined by, some specific units. By the time the physics teacher had explained why those particular units were relevant, most of the students heads were aching, but the nerdy types were really getting it, and packing it down.

But in what units was Energy measured. Let me direct you to a great Wikipedia article on 'Energy'.
http://en.wikipedia.org/wiki/Classical_mechanics

You will a see a long table there near the top, with things like 'position', 'velocity', 'acceleration', etc. Down the list you will see the units that 'Energy' is derived from:

E (is derived from)...kg (mass, m) X (meters²/ seconds²)

(the symbol something^-2 we see in the Wikipedia table is just shorthand for dividing: putting something in the denominator of a fraction.)

Now, if you remember from early algebra, a-squared is just shorthand for a x a, and our energy derivation has an a-squared, a x a, and a b-squared b x b, and following our elementary rules of algebra, we can pull the whole meters² / seconds² thingy apart and recombine it as follows:

(meters x meters)/(seconds x seconds)= meters/seconds x meters/seconds

Now, look at our table again; what is meters/seconds (or meters x seconds^-2, as they put it) ?

RIGHT!!

VELOCITY!

Now the whole thing becomes clear: Energy is derived from a MASS times a pair of VELOCITIES multiplied together: E = m times v².

Nevertheless, for people in high school who couldn't follow the derivation of simple measures like v (length/time, eg. miles per second (in a particular direction) ) up into more complicated measures such a acceleration, momentum, they got pretty confused by the time Energy and Power were derived. The problem is that past 'velocity' (or speed with direction), the derivations become increasing divorced for most people from anything we can immediately picture in our heads, or a have a feel for. 'Energy' is a mathematical/physical abstraction best left just as it is: something defined by its units of mass, length, and time.

If you look around various Wikipedia articles, and in basic physics books, you'll see this velocity-squared item popping up all the time with respect to energy. Now you know the reason. It has less to do with relativity than common old-fashioned physics definitions and derivations. Sometimes you won't see the v², because it's deeply buried in some other expression that has to be un-packed to discover it. But it's probably lurking in there somewhere.

danR

Keep in mind neoweb, if you're disappointed by the answers given, that physics forums does not guarantee layman's answers, and that most of the respondents got their degrees by gearing their answers up to their profs and dissertation inquisitors. Their training is entirely antithetical to the giving of layperson's explanations.

Also this is a relativity section, but you've framed your emphasis such that it is more easily explained in a classical physics or high-school physics homework type of forum. The squaring of the velocity is not specific to relativity (although the squaring of light's velocity is). It is old-school stuff from hundreds of years ago.

However, had you asked in another forum, it probably would've been moved here anyway.

Here's a good article that will explain the beginnings and fundamentals of deriving Energy (and other things):

http://www.nmsea.org/Curriculum/Prim...discovered.htm

AnotherDave

For my purposes, "Grandpa" has been satisfied and this question has been thoroughly answered. I sincerely thank everyone who has responded. To the "admins" I may recommend that the best of this thread be archived and put up on a FAQ page, as I'm sure this isn't the last time someone will ask this question :-)