Why c2 (speed of light squared)?

[neoweb]

Sorry if this sounds dumb, but in the famous equation e = mc2 why is the speed of light squared?

I recently read E=mc2: A Biography of the World's Most Famous Equation by David Bodanis... thoroughly enjoyed it... but the answer to this question eludes me.

Can anyone help in layman's terms please?

 

[Mike2]

Sorry if this sounds dumb, but in the famous equation e = mc2 why is the speed of light squared?

I recently read E=mc2: A Biography of the World's Most Famous Equation by David Bodanis... thoroughly enjoyed it... but the answer to this question eludes me.

Can anyone help in layman's terms please?

I think if anyone were to have time to do a dimensional analysis, he would discover that the units required a velocity squared times a mass to get an energy.

 

[gonzo]

Do you also wonder why kinetic energy is .5mv^2?

 

[neoweb]

No. Do you?

 

[Alkatran]

Look at the units: If c wasn't squared, the units on the left side of the equation wouldn't match the ones on the right.

 

[Nenad]

Sorry if this sounds dumb, but in the famous equation e = mc2 why is the speed of light squared?

I recently read E=mc2: A Biography of the World's Most Famous Equation by David Bodanis... thoroughly enjoyed it... but the answer to this question eludes me.

Can anyone help in layman's terms please?

you can't just ask a question like that. Why is it c^2, why don't we make it c^5. The reason it is c^2 is because of its derrivation. If you look athe way the equation is derrived, the concludion statememnt equates that energy and mc^2 are the same.
And to answer you question about kinetic energy. It is simple.

$$F = \frac{dp}{dt} = ma$$

now we can integrate: (anti-derivative)

$$\frac{dE}{dt} = p = mv$$

$$E = \frac{1}{2}mv^2$$

do you get it, (my notation is a lotlle off, I don't know how to use the integration symbol in laytex)

 

[neoweb]

The reason it is c^2 is because of its derrivation. If you look athe way the equation is derrived, the concludion statememnt equates that energy and mc^2 are the same.

Could someone kindly describe/set out the equation's derivation or point me to a web page?

 

[Nenad]

visit www.google.ca and do a web search on the eqaution, youll ge plenty of results.

 

[geometer]

Could someone kindly describe/set out the equation's derivation or point me to a web page?

The equation falls right out of the Work-Energy Theorem if you use the 4- momentum and proper time.

 

[da_willem]

In Special relativity (SR) an objects inertia increases as it approaches the speed of light, making it more and more difficult to increase the speed. You could assign this extra inertia to the mass of the object by saying the mass (this is actually called 'relativistic mass' , but I'll call it just mass) increases. Working this out it turns out the mass increases with a factor $\gamma (v)$. This is a velocity dependent function increasing to infinity as the speed v approaches c thus making it impossible to acquire a speed large than c. I SR time slows down and length is shortened by the same factor!

So mathematically this means the mass ($m(v)$) in terms of it's rest mass (m) will be:
$$m(v) = \gamma m$$ with $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

For low speeds this can be approximated mathematically by:
$$m(v)=m+\frac{1}{c^2}(\frac{1}{2}m \gamma v^2)$$

But this last term is a particles low speed kinetic energy divided by c^2! So the kinetic energy of a particle contributes to its mass in a way which is consistent with:

$$E=m \gamma c^2$$

Or in terms of relativitsic mass: $$E=m(v) c^2$$. Einsteins famous equation!



If you are willing to adopt the observation that a photon has a momentum $p=\frac{h}{\lambda}$ and energy $E=hf$ with (h Planck's constant) it's much easier:

$$E=hf=\frac{hc}{\lambda}=pc=mc^2$$

With p=mv=mc for a photon.

NOTE: These are only two out of many derivations of the relation, and I'm not sure they can be understood by 'a layman'. I'm also not certain a derivation of the equation is in place. It will always need other assumptions to be derived and in the end it's just an observation, which is the only true 'proof' of the equation!

 

[da_willem]

And about why kinetic energy is (1/2)mv^2? You need Newtons second law that states the motion of an object changes according to: F=ma. And that the energy of an object changes under influence of a force when moving a distance x parallel to the force (work!) by an amount $W= \int F dx$

$$W=\int F dx = \int ma dx= m\int \frac{dv}{dt} dx = m\int \frac{dv}{dt} \frac{dx}{dt} dt = m\int \frac{dv}{dt} v dt = m\int v dv = \frac{1}{2}mv^2$$

[You could also derive it using the general case (not requiring the movement is parallel to the force) but you would need to treat Force, speed, acceleration as a vector and use the dot-product, but this is essentially the same, and makes things a little bit more complicated]

 

[Recher]

What NEOWEB is getting at is though there are all these incredible numbers vis a vis the fine tuning of the universe (Just Six Numbers by Martin Rees a great read) is C2 is on its own in import.

Another way of writing the formula is E/C2 = M

We know what C2 means in terms of maths but the what and why...We have failed to focus...There is much more to this C2 than we have ever thought of

 

[WonderWatcher]

I think recher is the only one who even came close to answering neoweb’s question.

I think the questions: “Why the speed of light and why the heck does it needs to be squared?” are answered like this:
E=mC^2, or m=E/C^2, C^2=E/m, or however you want to write it...what it’s saying is that it is observed that there is a constant proportion between Energy and Mass. Apparently this constant proportion is equivalent to the speed of light multiplied by the speed of light. The speed of light is a constant, regardless of the mass involved, or time dilation, or any of the other weird things that Einstein talked about. It can’t be the speed of sound, or the number of stars in a galaxy, or the number of farts in an elevator, or the speed of a falling turtle, because none of those things survive as a UNIVERSAL constant. Based on what is observed, only the speed of light is a UNIVERSAL constant. Therefore, it’s a really good number to use.
So, the REAL questions is, what the heck is light, and mainly why does it travel at ~186 thousand miles per second all the time?? I believe that if we could answer this question, we could replace C^2 with that answer in some way. Like, the speed of light is ~186k m/s because the Ether makes it so. Therefore, E=m(c^2) could be written E=m(because the Ether makes it so), or more logically (because the Ether makes it so) = E/m.

What do you all think….is this on the right physics-osophical path?

 

[ZirkMan]

What do you all think….is this on the right physics-osophical path?

The layman's view path is probably right. But I think it's wrong to blame the light for the strangeness of the observed phenomena as c is not only the speed of light in vacuo but a speed limit in general for any form of energy capable of carrying information and light just happens not to cross this limit too. The speed limit itself doesn't have to have anything in common with the physical nature of light and probably has deeper roots.

As for the E=mc2 this can be literally interpreted as that the total energy of a system is mathematically equivalent to that many of light squares (c2) as there are units of rest mass we choose. There must be a deeper meaning why this translation is mediated through multiples of areas of the maximum speed limit in two dimensions but I do not think anybody will be able to give you a satisfying answer just now other than mathematical derivation.
It's because we don't have a final theory of space&energy yet. I think the SR&GR is an important step to such a theory and gives some important clues but on this level it still just provides answers of HOW things as energy-space relate to each other rather than why.

 

[AnotherDave]

Ok, let me rephrase the context of the original question.

Grandpa has been a mailman all of his life. But he likes all that "sciencey stuff" on the discovery channel. He knows his smart Grandson is a physics student and he's always wondered about that whole E=MC2 thing.

So Grandpa comes over and says..."hey, maybe you can explain something to me. That crazy guy, Michio Kaku was on Fox News talking about Einstein and how nothing can travel faster than the speed of light. So why do we have that whole C2 thing?"

The smart physics student respects his grandfather and doesn't want to tell him that he's too old or not smart enough to understand. But he also won't understand an explanation that goes something like:

m(v)=m+\frac{1}{c^2}(\frac{1}{2}m \gamma v^2)

What's the best way to explain "why C2" in non-mathematical terms that the retired mailman can understand?

 

[Muphrid]

First, remember that every object travels through space and time at the speed of light. In mathematical language, that says that every object as a four-velocity $u$ such that $|u \cdot u| = c^2$.

From there, remember that any object also has four-momentum $p = mu$. When the object has no three-momentum, this equation reduces to $E/c = mc$. That just follows from the idea that energy and momentum come together to form a single object.

So what do you have?

1) Every object travels through space and time at the speed of light
2) Momentum = mass x velocity
3) Energy and momentum are part of the same object, the four-momentum
4) A little algebra

That's all it is.

 

[AnotherDave]

First, remember that every object travels through space and time at the speed of light. In mathematical language, that says that every object as a four-velocity $u$ such that $|u \cdot u| = c^2$.

From there, remember that any object also has four-momentum $p = mu$. When the object has no three-momentum, this equation reduces to $E/c = mc$. That just follows from the idea that energy and momentum come together to form a single object.

So what do you have?

1) Every object travels through space and time at the speed of light
2) Momentum = mass x velocity
3) Energy and momentum are part of the same object, the four-momentum
4) A little algebra

That's all it is.

First, remember that every object travels through space and time at the speed of light.

I'm quite sure this was a typo. But you said it twice, so I think you mean to say something else. Everything does not travel through space and time at C. Everything travels at a percentage of C...but for oversimplified purposes, only C travels at C.

Grandpa isn't going to understand four-momentum. Grandpa also isn't going to understand "mathematical language" and his poor old eyes will glaze over if you try :-) Although even Grandpa has some math education and we may be able to remind him about balanced equations. Maybe not Algebra, but certainly fractions. What you do to one side you have to do to the other.

He understands the E=M part. He just doesn't get C2. How can you have the square of something that can't be exceeded? Can you see my problem in explaining this? LOL

 

[Muphrid]

I'm quite sure this was a typo. But you said it twice, so I think you mean to say something else. Everything does not travel through space and time at C. Everything travels at a percentage of C...but for oversimplified purposes, only C travels at C.

I meant what I said. Every non-massless object has a four-velocity with magnitude $c$. 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 $c$) is critical in the chain of reasoning here.

 

[AnotherDave]

I meant what I said. Every non-massless object has a four-velocity with magnitude $c$. 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 $c$) is critical in the chain of reasoning here.

Every non-massless object

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]

That crazy guy, Michio Kaku was on Fox News talking about Einstein and how nothing can travel faster than the speed of light. So why do we have that whole C2 thing?"...

What's the best way to explain "why C2" in non-mathematical terms that the retired mailman can understand?

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]

First, remember that every object travels through space and time at the speed of light. In mathematical language, that says that every object as a four-velocity $u$ such that $|u \cdot u| = c^2$.

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]

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?

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 explanations, 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]

C-squared may represent a particle moving through both the space AND time direction at lightspeed

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

Or from another perspective, at c2 speed through both space and time, both space and time would be frozen, eliminating any reference frame

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

 

[StationZero]

I would refer you to the "Minute Physics" video on youtube

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]

m∫dv dt dx dt dt

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]

So Grandpa comes over and says..."hey, maybe you can explain something to me. That crazy guy, Michio Kaku was on Fox News talking about Einstein and how nothing can travel faster than the speed of light. So why do we have that whole C2 thing?"

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]

Can anyone help in layman's terms please?

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]

...why is the speed of light squared?

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/Primer/how_was_energy_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 :-)

 

[Kaushik96]

Actually it is simple ... In E=mc2 , c2 doesn't actually represent speed of light squared ! c2 is the energy density for unit mass in kg . instead of taking the dimensions for c2 as m2/s2 , take it as joule/kg . And this energy density (9×1016Joule/kg is constant for any substance . Simply energy = mass × energy per unit mass !