# E= Mx (C Squared) by why squard?

1. Aug 24, 2010

### Tony Batchelo

we all accept the E=mc2 formula but what is the mathmatical proof for the c2, why for example is it not c3 or c4?

2. Aug 24, 2010

### Pengwuino

For one, the units don't match up.

Do a search for the countless number of threads that discuss this.

3. Aug 24, 2010

### Kevin_Axion

4. Aug 25, 2010

### tom.stoer

In Newtonian Mechanics we have F = ma; why not m², m³, ...?

Asking such questions is silly. No such formula can be understood w/o its context ... and from the context it becomes clear how to proof or at least why to introduce it.

5. Aug 25, 2010

### sophiecentaur

If you start the Lorenz SR formula for mass and relate the mass change to the velocity and the difference in Energy input, you get an expression which, with a little bit of sloppiness / 'unrigorousness', more or less yields E = mc2.

6. Aug 25, 2010

### HallsofIvy

Many of us do not just accept it but know its derivation- it can be found in any introductory text on relativity- and so understand why it is $c^2$. I see Keven Axion has given a link to the derivation.

But, as penquino said, you don't have to know the derivation to see why it cannot be "$mc^3$" or "$mc^4$".

Any form of energy can measured in "Joules" which, itself, reduces to "kilograms meters^2/second^2" and any energy measurement, in any units, must reduce to those bases: mass times distance squared over time squared. Since m is a mass and c a speed which is measured by "distance over time", $mc^2$ gives precisely those units while $mc^3$ or $mc^4$ do not.

(This does not prove "$e= mc^2$", it shows that it is consistent while the others you suggest are not.)

7. Aug 25, 2010

### Theorem.

You could ask the same thing about absolutely any formula. I suggest you look for a derivation. Like HallsofIvy stated, we don't just accept it, we understand its derivation. It has been proven.

8. Aug 26, 2010

### jonlg_uk

The answer lies with the discovery made by a brilliant French aristocrat, Emilie du Chatelet.

At the age of 23, du Chatelet discovered a talent for advanced mathematics which she relished. So much so that she began to formulate ideas of her own; ideas that challenged the great physicists, including Sir Isaac Newton.

Newton stated that the energy (or force) of a moving object could simply be expressed as its mass multiplied by its velocity. But while corresponding with a German scientist called Gottfried Leibniz, du Chatelet learned that Leibniz considered the energy of a moving object is better described if its velocity is squared. But how to test this? Du Chatelet tried an experiment that would prove her point – dropping lead balls into clay.

Newton's formula says that doubling the velocity of a ball would double its energy and so one would expect it to travel twice as far into the clay. But if the velocity is squared, as Leibniz and du Chatelet believed, the force should be four times greater, and the ball should travel four times the distance into the clay.

Du Chatelet conducted her lead ball experiment and sure enough, doubling the velocity of the ball (by dropping it from twice the height) resulted in the ball travelling four times further into the clay. This simple but brilliant experiment proved that when calculating the energy of moving objects, the velocity at which they travel must be squared. The energy of an object is a function of its velocity squared – it is for this reason that the speed of light in Einstein's equation must be squared.

This was a factor that profoundly changed the meaning of Einstein's equation – since c is already a large number, once squared it is vast. Thus, a vast amount of energy (E) can be associated with a very small amount of mass (m) because mass is always multiplied by the speed of light (c) squared – a vast number. Under these laws, even a tiny amount of mass will equate to a huge amount of energy.

9. Aug 26, 2010

### Kevin_Axion

Why are people still responding to this thread? There are many people who provided sufficient answers including a mathematical derivation. This is a formula that is used continually in technology and Theoretical Physics, why ponder and challenge it?

10. Aug 26, 2010

### jonlg_uk

Nobody was challenging it, it has been put through many experimental rigours before. I think my answer will benefit the questioner alot more because it gives some history to the problem. Therefore it makes accepting the magical equation alot easier.

11. Aug 26, 2010

### sophiecentaur

A ponder is fine.
To challenge is a waste of time.

12. Aug 26, 2010

### Kevin_Axion

Yes, pondering is fine, but pondering it and assuming it to be senseless without full disclosure to the Mathematics isn't.

13. Aug 31, 2010

### jerich1000

People are responding to this thread because it is an interesting topic that people are contributing their portion of knowledge to. If anyone doesn't wish to participate then that is fine. Not everyone here is at the same level. I myself prefer to be supportive wherever possible.

14. Sep 1, 2010

### Staff: Mentor

From a previous instance of this question:

15. Sep 1, 2010

### jonlg_uk

I ANSWERED THIS QUESTION GIVING A FULL HISTORY OF WHERE c^2 arrived from....

16. Sep 1, 2010

### Staff: Mentor

But your answer doesn't really make sense. Du Chatelet showed that the object's kinetic energy is proportional to its speed squared. But mc^2 is not a kinetic energy; c is not the speed of the object.

And kinetic energy being proportional to speed squared is only true for low, non-relativistic speeds.

17. Sep 3, 2010

### zincshow

Thank you, very interesting post of something I did not know.

18. Sep 4, 2010

### nickthrop101

there is no answer to this question, it is squared becouse the laws of nature tell us it is, it is squared so that the mathma\tical evidense adds up to it in e=mc2
:)

19. Sep 5, 2010

### Born2bwire

The above comes from some BBC documentary which seems to relate a lot of garbage, at least in the written summary that I have read previously. The old link I had to the offending summary appears to be gone but a quote from an old post reveals it to have read:

A response of mine to this followed:

So, to reiterate, the connection of du Chatelet and Einstein is incorrect. As stated by a previous poster in this thread, the kinetic energy of an object and its relativistic mass-energy equivalence equation is something completely different. In fact, the kinetic energy is an additional term in this equation since the actual equation is
$$E^2 = (mc^2)^2 +(pc)^2$$
So that is incorrect. Also stipulating that Newton incorrectly gave the relatonship of kinetic energy is incorrect as far as I have found. I have looked through the Principia and did not find this and I do not think that Newton even had a kinetic energy equation. He mainly worked with forces and the use of energy is not necessary to derive the results that he did in the Principia.

EDIT: I forgot the most glaring mistake of all. If you double the height of a dropped object it does NOT double the speed of the ball, it only increases it by \sqrt{2}. I really REALLY hated that bloody BBC documentary.

20. Sep 6, 2010

### cjameshuff

Aside from the full derivations (and simplifications from the full equation including momentum) spitting out E=m*c^2 as a simple result from mathematical rules, the alternatives don't make dimensional sense. The du Chatelet example claims Newton used an erroneous equation for kinetic energy of m*v, but m*v doesn't even give units of energy. The example also incorrectly equates energy and force.

As I recall, concepts of energy were not well defined at the time. Newton didn't get an equation for kinetic energy wrong, he just computed things in terms of momentum instead, which is a perfectly valid and often more convenient approach. du Chatelet didn't correct any of Newton's equations, she recognized that the quantity we call kinetic energy is useful but distinct from momentum. The ones who got it wrong were those who equated the vis viva of Leibniz with the momentum of Newton, and assumed that one of the two was incorrect.

As for the original question...energy as expressed in SI base units is m^2*kg/s^2. Force, m*kg/s^2. Velocity of course is m/s. m*c would give units of m*kg/s, which are units of momentum, not energy. m*c^3 would give m^3*kg/s^3...again, not units of energy, or anything else particularly useful. (watt-meters?)

Calculations for specific quantities will differ, but the dimensions must match up.
Potential energy in a uniform gravity field:
U = m*g*h, where m is mass, g is 9.81 m/s^2, and h is height. kg*m/s^2*m = m^2*kg/s^2 = joules of energy
Classical kinetic energy:
kE = 0.5*m*v^2, kg*(m/s)^2 = m^2*kg/s^2 = joules of energy

The similarity of E=m*c^2 to classical kinetic energy is not coincidence, but it's not of any deep meaning either.