[originally posted 2/17/08. Update added to end.]

This post is in response to a comment on my post Physics Factoid: E = m c squared, which asks "Why is the conversion factor c

^{2}?" This will be slightly more technical than most of my posts, but should make sense to anyone who has had high school physics.There are two parts to the question. First, why is the factor something squared? Second, why is that something c, the speed of light?

The answer to the first question is that the factor has to be some speed squared to get the units right. Most people don't value units enough. You can often solve a problem just knowing the units involved, and you can certainly tell if something is askew if the units aren't right. For example, if you ask someone, "what's the area of your living room?" and they answer, "Thirty feet," you know they misheard the question because an area has to be in square-feet.

The units of energy are mass·speed

^{2}, for example, kg·m^{2}/s^{2}. The kinetic energy of a mass m moving and velocity v is ½ m v^{2}. [application to space junk]So any formula writing energy in terms of mass has to involve something with units of speed squared.

But why the speed of light? Well, the theory of relativity is based on the idea that all observers, no matter how fast they are moving relative to you, must observe the same physical laws. And there is only one special speed which they all measure to be the same—the speed of light. In fact, you can show that the theory doesn't work if you try to add a second special speed. A short way of putting it is that in relativity there is only one constant to work with, the speed of light.

Now I could derive the formula for you, or show you it gives the correct kinetic energy in the low-speed limit, but a simple answer is that there is no other speed it could be!

So in summary, the factor has to be a speed squared to get the units right, and the only speed it could be is the speed of light.

[Answer to Comment added 3/8/08]:

Here is an attempt to answer the comment by 'dubiousraves'. I'm going to use some algebra, so stop reading now if you have math phobia! :)

I assume that you accept that the rest energy of any particle is some constant K times its mass, ie

rest E=K m

So we just need to figure out what K is. Further, its kinetic energy (at speeds much smaller than the speed of light) is ½ m v

^{2}. Now the total energy of a particle is its rest energy plus its kinetic energy. The theory of relativity says it has to be proportional to the factor called gamma, γ= 1/√(1-v^{2}/c^{2}), which is the same factor that accounts for the length contraction and time dilation. The total energy is thustotal E=γ K m.

When v=0, the particle is at rest and γ=1. In that case the total energy equals the rest energy (when something is at rest, it has no kinetic energy). Now when the speed v is a lot less than the speed of light (which is always true in everyday life), then v

^{2}/c^{2}is much smaller than 1. In that limit we can use the formula 1/√(1-e) ≈1+e/2 (this comes from a Taylor series expansion), for any quantity e that is much smaller than 1. So for small v, γ≈1+½ v^{2}/c^{2}. If we plug this into the total energy, we get,total E=γ K m≈(1+½ v

^{2}/c^{2}) K m = Km + (K/c^{2}) ½ m v^{2}.So to get the kinetic energy right, K needs to be c

^{2}. As a check, let's plug in K=c^{2}:rest Energy= K m = m c

^{2},total Energy = γ K m = γ m c

^{2 }≈ (1+½ v^{2}/c^{2}) m c^{2 }= mc^{2}+ ½ m v^{2}.I hope that helps.

## 36 comments:

Hi eyesopen. I've been enjoying your blog.

However, I wonder if you'd mind taking another whack at the "Why units squared?" question. I still don't get it. Why does the speed have to be squared, as opposed to say, cubed?

Just saying it's because those are the right units doesn't answer the question for me.

I think understanding this will help me understand the e=mc2 formula as a whole -- which, as I read the recent Einstein biography -- I've been trying to do. Interestingly, Walter Isaacson, whose explanation of Special Relativity is quite cogent, completely takes a pass on explaining e=mc2. I take from that a sense of how difficult it really is to explain this equation to laypeople.

Personally, I'm going to keep reading about it until I get it.

Thanks, and please do try to break it down more simply.

DR

PS: Go Obama!

Hi. I've posted an addendum to answer your question. I hope it isn't too technical.

-EO

Thanks for posting that, Prof. It's gonna take me a while to digest that, probably with help from other sources, so I'll let you know if I ultimately do.

Meanwhile, fascinating fact I learned from Walter Isaacson's Einstein bio: around 1935, when Einstein had started working at Princeton, the school took a poll of incoming freshman, asking them who the person was they most admired.

No.2 was Einstein.

No.1 was .... wait for it.... Adolf Hitler!

Hello Eyesopen, sorry to be late, I like your blog very much, very interesting.

I found it on a general search for information about c squared.

Actually, I would like to have your comment on the following.

I believe the question could go like this: In our equation, we use C2, but in observing the reality, we don't measure C2, but only C. Physics looks like telling us that what we are measuring is only the square root of what is really happening. The true constant is C2 (call it K, as you did), and write down e=mK which is the right equation. The question which follows is thus why the hell are we measuring only the square root of K instead of K, and what is that constant with strange units m2/s2 representing?

Michel, born 1960.

Hi Michel:

A fundamental quantity can appear in multiple ways. It used to be that the meter was defined by a metal bar. It could be used to define a length (1 m) or an area (1 m squared). In the same way we DO measure c squared when we measure c. You take the measurement for c, plug the value into a calculator and square it. Units all work like this. Oh, and if it were easier to measure some constant squared, then one could just do that measurement and take the square root to get the first power of the quantity.

I hope this helps.

Thank you. I am interested at the moment in c2, because c2 is, as much as I know, the only squared constant. I hope being right at this. Usually, variables are squared, constants are just constants. Like in Phi R squared. Phi, which is the constant, is not squared. That's the reason of my interest.

p.s. i am french speaking, Michel=male, not female.(just for disambiguation) thank you again for your interest.

Hi Michel: Yes I figured it was French male name. Constants appear with all sort of powers and in combination. For example, look up the "fine structure constant" and you will see it is formed by a combination of fundamental constants, e^2/h-bar c.

Hello Sir gud to see you

Im neither a theoretical physicist nor a degree holder in physics.

Im just a student of 17 years who has physics mania

I have my own developed theory which I believe is very promising logically, but as soon as u start applying mathematics which is the heart of any theory , I fall into deep trouble because of ONLY 1 thing and that is the function of c "SQUARED" ........ If e = mass*twice of energy then OMG I've established a relation b/w mass & time but this SQUARE is just a headache ..........

I was wondering if i could find the logic of this on the internet but no hope as every1 is saying its just due to "UNITS"

I believe mathematics is the LANGUAGE OF PHYSICS ......... mathematics SPEAKS , it tells you how a theory is designed ACCURATELY , how our Universe works, It is not there no Justify the Units(although an important priority) but the 1st duty of mathematics behind a theory is to speak accurately the image in the mind of theory's developer as of Einstein in E=mc squared.

So i just want to know what is the LANGUAGE of c squared, if there is some logic or if not, its just a derivational function, Its language has yet to be discovered. then you are welcomed to inform me in both situations !

Thank You !

Hi Di-G: Units are very important. For example,m has units of mass, mc, a mass times a speed, has units of momentum and mc^2 has units of energy. If the units don't agree, it means something's incorrect. For example, height is measured in feet or meters, NOT square-feet or square-meters.

Also, E=mc^2 is very well tested, so if you got something else, I'm afraid it is wrong.

Hi all;

I think the questions asked are quite profound. And the Prof's response quite straightforward and wonderful.

In essence, I interpret the question this way: Einstein stated that the speed of light (massless photon) is an absolute speed limit. Nothing can go faster. (Perhaps the Neutrino experiment at LHC is the first newly discovered exception, {and spacetime expansion}).

So is it not then contradictory to speak in terms of C^2 ? Is it not useless from a practical p/o/v?

The Prof responded, as i understood it; C^2 is the convention used for units of measurement, when relating energy to mass... ie- velocity must be squared. And he gave the example of area of a room or space; we use the convention - square footage in order to describe the area of a room.

The questions are insightful because they recognize that there is a contradiction.

Something else has to be done with E=MC^2 practically speaking...does it not? I'm not clear as to whether the Prof is saying it is ok to use the sq root of E & M and then lower the power of C to 1.

In other words, although E=MC^2 makes sense on a chalkboard at MIT, we still, as yet cannot accelerate particles to twice the speed of light. Therefore, a speed limit - less than C must be absolutely imposed - practically speaking.

C^0.999 is the best we can do. So E^0.4999 = M^0.4999xC^0.999 ... perhaps that violates convention, or some calculus must be used to get an expression that makes sense.

I'm not sure, also being a layman.

I think the real question is somewhat philosophical.

If nothing can go faster than the speed of light, why use math that squares the velocity of light.

Another convention should be found, unless lowering the powers on both sides of the equation (or using square roots & setting C = 1) is permissible.

thanks much,

Mestral Adeona

"Hi Di-G: Units are very important. For example,m has units of mass, mc, a mass times a speed, has units of momentum and mc^2 has units of energy."

But those are not 'units' you describe they are dimensions. Mass is a dimension, kg is a unit of mass!

Bt chosing different untes Einsteins equation can be reduced to E=m.

what ar the units in Einsteins classical version of the relations ship?

If you all have and would use common sense the answer is simple using very easy math. Mass can be converted to energy and energy to mass. Take a writing pen for example. The atoms that comprise the pen and give it mass are held together by nuclear forces. If the atoms comprising it were unstable enough, as in plutonium, you could turn it into a bomb, or in other words, convert the nuclear forces that hold the atoms together into pure energy. If that were to happen common sense should tell you that the energy would be released in three dimensions, not in a two dimensional (flat) plane. To convert a two dimensional plane into a three dimensional plane you square it. Energy travels at the speed of light because light is also pure energy. Therefore the speed of energy (light) squared accounts for all of the energy that would released once multiplied by the mass. Simple! That simplicity is why the equation E = M * C squared is so beautiful!

3rd dimension, I get it. That's why it's sqared. Thanx all

Squared. Sorry for the spelling error, screens getting a little blurry

Sorry May 31, it has nothing to do with the number of dimensions. If we live in two-dimensional flatland the kinetic energy would still be 1/2 m v^2 and the rest energy of the particles, E, would still equal m c^2.

Sorry May 31, it has nothing to do with the number of dimensions. If we live in two-dimensional flatland the kinetic energy would still be 1/2 m v^2 and the rest energy of the particles, E, would still equal m c^2.

"If you can't explain it simply, you don't understand it well enough."

--Albert Einstein

well if you take the wave of an oscilating particle of light then c becomes c^2 cause each unit it seems to go in aparent lengh is actualy squared in real warped lengh

^--- what i said above might be wrong.. but it may be a starting point for the author of this blog to understand the kind of explanation we want to get.. We came here cause we want to understand the fabrics of the universe, not because we want to know mathematical ways to check if units are right.

We all know the math behind this must be flawless, but we want to understand the real meaning behind the numbers. thanks for a great blog promoting science and critical thinking keep up the good job ;)

I think that schollars have an amazing inability of understanding lay people's questions. Everybody here, as much as I could tell, just wanna know how is it possible to have c times c on the equation if the velocity of light can't be exceeded. I've asked this question so many times to teachers and no one grasps it and only talk about mathematical units. Sometimes i think very intelligent people are kinda stupid in a way... Btw, sorry for my poor English, I'm Brazilian.

Re my comment of September 27

Oh dear. I‘ve just realised my ‘explanation’ is wrong! I thought I understood this, but on reflection I now agree with previous contributors who show ‘c squared’ is down to the units involved. You live and learn!

To the Editor – please delete my comment of September 27.....it’s misleading. I now agree the original question has been answered perfectly satisfactorily in the earlier explanations. Thanks.

To the Editor – As requested please can you delete my comment of September 27. I am sure you would want to respect the wishes of a contributor and reader. Thank you.

You can propose that in a black hole Energy stays the same, Mass decreases, and the speed of light increases. How do you increase the speed? Simple add more dimensions and square speed again.

In a black hole I believe that the mass is converted into a force or an infinitely small partial. This particle bounces back and forth between existence and not existing at a very high frequency when exiting the black hole.

I don't know if I'm missing the point here, or if everyone else is. e=mc^2 provides the correct relationship between amounts of energy and mass. But what I want to know is WHY is c^2 involved in the equation. We could probably provide an equation that correctly describes the mathematical relationship between the angle of a particular car accelerator pedal, and the speed at which the car is moving, but the explanation of WHY that particular relationship exists is a whole other issue. Why is it that the event of transferring from mass to energy involves c^2? Does anyone understand this event, other than what it results in?

The answers to the two questions framed are crystal clear authenticated by simple algebra. These answers have enabled me to grasp some enigmatic puzzles and their simple solutions.

From the point of view of someone who barely scraped through algebra and never studied physics ... this whole equation absolutely frustrates me. But something you said ... seems to trigger an idea.

"A fundamental quantity can appear in multiple ways. It used to be that the meter was defined by a metal bar. It could be used to define a length (1 m) or an area (1 m squared). In the same way we DO measure c squared when we measure c. "

Energy is obviously "non-dimensional" or can't be measured linearly. If it's "three dimensional" ... about as close as I can get to defining it .... then what it equals must also be three dimensional.

Therefore entire equation must be "three dimensional". And the only way to do that would be to "square" the c ... as a technicality more than anything else. Since as far as we know the speed of light is fixed.

Or am I totally confused?

Hi, when we consider mass within the context of the E=mc2 why is time not considered as the other c that c is multiplied by. All matter is moving through time as it moves physically in space as I understand it, Does that not have an effect?

Thanks

Fil

OK...we know that e=mc2 works. But, I do not understand how and why Einstein decided to use the speed of light and then squared it to establish the energy component.

Another related question is: having established the speed of light early on, how and why was it then related to the energy product of any mass? The old story about the expanding rocket ship really gets me crazy. Is it the CERN experiments that gave proof to it? Or was Einstein's guess more than a guess. So, it would be helpful to know why Einstein plugged the speed of light into the energy formula and squared it.

Units are everything in physics and this explanation is spot on. If you're having trouble understanding E=MC^2 still, it's not any part of that equation that's confusing you, it's the unit dimensions of energy itself.

If you have an apple that's keeps moving forward two feet every second, you can say it has a speed in distance per time, say 2 ft/sec. Give that handy little equation a second, and it'll always give you back two feet, give it five seconds and it'll give you ten, it's like a mini computer program that keeps track of where your apple is for you.

You can also say that a moving mass has a momentum. A half pound apple moving at two feet per second has 1 lb*ft/sec s of momentum. An eight pound bowling ball crawling forward a foot every eight seconds would have the same momentum (8/8 lb*ft/sec s), and if your apple met your bowling ball head on you could see these were the same as they stopped each other or sent each other back at the same speed.

Energy adds even MORE units but it becomes even more useful. If you have an apple or bowling ball standing still (relative to you), and you want to GIVE it a momentum with a certain amount of stored energy you have, the energy dimensions tell you how much space and how much time it will take to do it (if it can be done at all). Because, as the mass ACCELERATES towards your target momentum (adding more feet/second every second, or ft/sec^2), it's moving further all the time, and because taking more DISTANCE to accelerate to your target momentum (where your distance square comes in: ft^2/sec^2) implies using more energy, the most effective way to use your energy is essentially all at once, not a little over a long time. Therefore: if you have a mass and a speed you want to accelerate it to, the dimensions of energy itself (mass times speed^2) tell you how much minimum energy you need to do it.

So, if we have an apple and we want to convert it to pure energy, i.e. have every quanta and particle in it decohere from the others and begin moving at the speed of light, we are asking what kind of energy it would take to give a half pound apple the momentum 1/2 lb * C. Of course, accelerating it over time wouldn't do, not only would that never reach C, but eventually it would just add more mass to the apple messing up our calculations. Plus, we know taking more space and time is always less energy-efficient. We would have to accelerate our apple to the speed of light INSTANTANEOUSLY. Fortunately, Einstein figured out that instantaneity itself has a speed: the speed of all massless particles, all information spread, causality, and the limit speed of space-time itself. This speed is also C. The maximum energy that a mass can be turned into therefore equals that mass times by C, then times by C again. If we could flip a switch and turn all the mass in an object into massless energy instantaneously, it would HAVE TO deliver exactly this much energy.

hi im a 10 year old boy and about space time i think light years would change because C changes with space time and the length is longer but and the time is slower so that means mc2 really has is c2 in space so space time is like earths speed of light which both of are part of c2 because c2 is made of mass and the speed of light but its different with out an atmosphere. this proves that c2 is slower in space and that the speed of light is not the same in space because there is no atmosphere so that leads to space time so in space E=c2

Zane Chambers offers some interesting points. I would like to expand on his ideas.

This is rather long but necessary for a reasonable understanding.

This is what I would envision happening. We have a particle at rest. Energy is involved in transforming some or all of the mass of the particle into energy (a photon for example).

The current belief is that the total mass of the particle is converted to energy. That is what we see from the equation.

There is a very short period where this conversion takes place. That is the period where external energy is involved in the conversion. Coming out of the other side of the conversion there is a photon of light moving at a constant rate of speed (Light Speed).

This "belief" is confusing in light of the equation because the equation involves the idea of acceleration (m/s^2). And it is hard to

envision acceleration being involved in the conversion of the particle from one state to another.

On the other hand, if we consider that there is no conversion of mass to energy, but rather just a conditional change, the equation starts to make more sense (at least to me).

Envision this: External energy affects the state of a particle. The particle starts to vibrate very rapidly. This vibration results

in the observation of a wave characteristic. The particle has not transitioned into another state. It has only taken on a new

characteristic.

Some of that external energy is also involved in causing the particle to start moving in an additional direction (i.e., from point A to point B (e.g., from an exploding star to Earth where it is observed by you and me)). So we have a particle that is vibrating back and forth and also proceeding in a specific direction at a constant velocity (Light Speed).

Now to move the particle from a stand still to it's maximum velocity (Light Speed, for example) it has to accelerate. It doesn't reach maximum velocity instantly. This acceleration, probably occurs very rapidly and is the only point at which external energy is involved. Once it reaches maximum velocity it stays that way unless, again, influenced by some outside force.

Our equation does involve acceleration (m/s^2). The acceleration is not linear which is why the "seconds" are squared. That part

makes sense. What does not make sense is how the Speed of Light (c) has units of m/s^2. The Speed of Light is said to be constant.

That means it has to be linear and therefore cannot be squared.

As well, Einstein's equation seems to imply a closed system. It does not seem to account for the external energy affecting the particle. A particle cannot start to vibrate nor start moving from one point to another without outside influence.

The equation seems to imply that the particle is capable of converting itself back and forth between a state as a particle (mass)

and energy (joules) at will without outside influence.

In fact, this is likely not the case, because particles do not constantly exhibit wave characteristics. Does light energy emanate from an atom constantly? No it does not (at least not without some external energy influence).

So, my suspicion is that Einstein's equation is incomplete.

The equation has to account for the amount of energy needed to achieve motion given a particular mass.

It should be more like this: E (in) = m (kg) * a (m/s^2)

E (kg * m/s^2)

We also need to include the maximum velocity we wish to achieve (e.g., Light Speed but not necessarily).

Let's assume it is Light Speed.

Starting with basics we have:

v = a * T T = amount of time (in seconds) to achieve maximum velocity (v) given a certain acceleration (a)

Now expanding on this in conjunction with Einstein's equation:

c (m/s) = T (s) * E (kg*m/s^2)/m (kg)

T = time (in seconds) needed to accelerate a particle of given mass to Light Speed

Re-orienting in terms of E (in): E (in) = m * c/T

The units balance and we no longer have a problem with the Speed of Light not being linear.

Comments?

Why, if e = "E"nergy,

m= "M"ass

c= speed of light?????

why not "SL" or "S" or "L" ???

Can someone explain why the speed of light is "C"

"As for c, that is the speed of light in vacuum, and if you ask why c, the answer is that it is the initial letter of celeritas, the Latin word meaning speed."

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