Why c2 (speed of light squared)?

[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 !

 

[Vorde]

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 !

And this comes from...where? I don't see how this works at all, most importantly how you get the idea that the energy density is a constant.

 

[Kaushik96]

You know what this relation means ? Mass and Energy are both the same ... So Mass and Energy are directly proportional . To match the SI units of both mass and energy , mass is multiplied with a constant c2 (energy density).

You ask where this energy comes from ? In the sun , two protons are squeezed to form a helium nucleus . In this process some of the mass of protons gets converted to energy . Striking a match also converts some mass (insignificant amount ) into energy. As I said earlier energy density is 9×10^16Joule/kg . In other words 1kg of a substance is 9×10^16Joules of energy.

Same way energy can also be converted into mass . But that's a lot difficult because trillions of joules of energy is needed to make a mass in the order of 10^-9 kg.

In different processes , different substances are converted to energy . The process in the sun I mentioned involves protons . Striking a match involves electrons . And that's how it works . I hope you understand .

 

[Vorde]

Ah, I see what you are saying. But you have a rather circular argument.

How are you possibly going to show that the constant of proportionality of $E \propto M$ is $9 \times 10^-9$ without invoking $E=mc^2$ first - something which falls out of the equations?

 

[Kaushik96]

Practically this is what the relation means . But i don't know how c^2(speed of light squared) is energy density . Give me some time . Ill check what's with this constant !

 

[Kaushik96]

Check out this link ! http://www.drphysics.com/syllabus/energy/energy.html . You may understand

 

[Dale]

But i don't know how c^2(speed of light squared) is energy density .

Technically the term is "specific energy", not "energy density". Energy density is the energy per unit volume (J/m³). Specific energy is the energy per unit mass (J/kg).

Personally, I think you are on a reasonable track here. If you have some proportionality between energy and mass, E=bm, then just by looking at the units you know that b needs to have units of J/kg=m²/s². The only combination of fundamental constants that has those units is c², so it has to be c².

Of course, that begs the question, why is it 1 c² instead of 5 c²? And why is it E=bm instead of E=bm²? To answer either of those you really need a full derivation.

 

[Kaushik96]

Yeah ... that's right ! :)

 

[Vorde]

Agreed, but I think its a matter of insight to say c^2. To say its the energy density in J/kg seems to me just to be saying the obvious because of the equation it is in, when you say c^2 you are saying something more meaningful about the universe.

 

[Kaushik96]

@Dalespam : the reason why "e" is not directly proportional to "m^2" is because as i said earlier energy is mass and mass is energy . They need to be proportional .So "e" must be proportional to "m" and not "m^2" . If it is "m^2" that will mean mass and energy are different .

 

[Vorde]

@Dalespam : the reason why "e" is not directly proportional to "m^2" is because as i said earlier energy is mass and mass is energy . They need to be proportional .So "e" must be proportional to "m" and not "m^2" . If it is "m^2" that will mean mass and energy are different .

But you are starting from the assumption they are directly proportional. As DaleSpam said, you can't get that assumption without a full derivation.

 

[Kaushik96]

Oh ! I see ...The full derivation of this equation is given in this link ! http://www.drphysics.com/syllabus/energy/energy.html

It will answer your questions !

Also this derivation is made with respect to light .
you asked why this specific energy should be constant for substances right ?

This relation only involves light matter and light energy !
If this relation e=mc^2 is true for light it applies to all matter because light matter is made of light ! so c^2(specific energy) will be the same for all light matter .

 

[Vorde]

This relation only involves light matter and light energy !

That is wrong.

E=mc^2 is valid for all things (on the super-quantum scale, you'll have to ask someone else if it applies equally well to quantum stuff). It is valid for you, for me and for a car. That website has an interesting explanation for the average reader, but as has been said a full derivation can be done with SR that is valid for any mass.

 

[Kaushik96]

Hey Hey ! Thats what I also meant ... Light matter is what you see everyday (ie) any object like a pencil , a bike or a car or even you and me , all are made up of light energy ! there are two types of energies (light energy and dark energy ) .

 

[DiracPool]

Oh ! I see ...The full derivation of this equation is given in this link ! http://www.drphysics.com/syllabus/energy/energy.html

Where did the Doc come up with this little nugget for velocity, v=E/(Mc). It looks as though he is using the definition of E=mc^2 to derive the same equation! Am I reading this wrong, or is this just a circular derivation that tells us nothing?

 

[Nugatory]

Where did the Doc come up with this little nugget for velocity, v=E/(Mc). It looks as though he is using the definition of E=mc^2 to derive the same equation! Am I reading this wrong, or is this just a circular derivation that tells us nothing?

It's not circular, it comes from conservation of momentum and the known (through observation) fact that radiation transfers momentum when it strikes a surface.

 

[Vorde]

there are two types of energies (light energy and dark energy ) .

I don't know where you are getting this from, but I'd be very careful.

There is something called dark energy, but I've never heard anyone call 'the rest of the universe' light energy so I'm thinking you might be referring to something more pseudo-scientific.

 

[DiracPool]

It's not circular, it comes from conservation of momentum and the known (through observation) fact that radiation transfers momentum when it strikes a surface.

Hmmm, OK, can you give me link where I can see this equation, v=E/(Mc), used somewhere by someone with a straight face who derived it independently from Einstein's derivation, and for some other purpose? I mean, all we have to do is set the velocity here to c and we have E=mc^2. I think Einstein had conservation of momentum and radiation transfer in mind when he derived it too. What I'm taking issue with is that it seems the DrPhysics is using E=mc^2 to derive E=mc^2. I don't see what novel insight he is bequeathing upon us here. Have I missed something?

 

[Kaushik96]

Initial momentum or the momentum with which the light hits the cylinder will be equal to TOTAL ENERGY OF THE LIGHT / SPEED OF LIGHT (E/c).

Final momentum or the momentum with which the cylinder moves after colliding with light is just THE MASS OF THE CYLINDER * VELOCITY WITH WHICH IT MOVES (Mv)

According to Momentum conservation principle Initial momentum will be equal to Final momentum .So E/c = Mv (OR) v=E/Mc

 

[Vorde]

That's a nice point. You are missing a factor of gamma on the right side though - so it doesn't quite work. As a novice's derivation it works well though.

 

[Kaushik96]

You are missing a factor of gamma on the right side though

I can't quite understand . What do you mean by this ?

 

[Vorde]

In relativistic physics momentum isn't defined as $p=mv$ but rather as $p= \gamma m v$ where $\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}$. So your derivation doesn't quite work. It's been a while since I've had to derive E=mc^2 (I had to do it for an extra credit problem on a test) so I'm not sure whether or not the derivation on that site can be simply expanded to be correct or not, but in it's presented form it does not work.

As I said though I approve of it as a teaching method - it's quite simple - as long as readers are aware that a more thorough proof is required.

 

[Kaushik96]

all we have to do is set the velocity here to c and we have E=mc^2. I think Einstein had conservation of momentum and radiation transfer in mind when he derived it too. What I'm taking issue with is that it seems the DrPhysics is using E=mc^2 to derive E=mc^2. I don't see what novel insight he is bequeathing upon us here. Have I missed something?

You can't do something like that because In E=mc^2 relation 'm' stands for the mass of light ! And in this equation v=E/Mc , 'M' stands for the mass of the cylinder ! Even if you put 'c' in the place of 'v' it won't become E=mc^2

 

[kostas230]

Landau's derivation of the equations of relativistic mechanics in his book Classical Field Theory might help you better understand relativity.

 

[Vorde]

In E=mc^2 relation 'm' stands for the mass of light !

You are fundamentally wrong about this, and you won't get anywhere with relativity unless you do research (wikipedia is a great place to start) and realize that the mass can be anything.

 

[Kaushik96]

You are fundamentally wrong about this, and you won't get anywhere with relativity unless you do research (wikipedia is a great place to start) and realize that the mass can be anything.

In the derivation from the external site 'm' refers to the mass of the light . Thats what i meant