Hmmm... I just purchased a book Faster Than the Speed of Light
by Joao Magueijo. It is an autobiographical sketch of one physicist's exasperated struggle to get his idea heard within the physics community: namely, that the speed of light was at one time much
higher than it is now.

I've always been puzzled by the c^2 part of this equation. It seems to me that c (speed of light) squared just gives you a really big constant (in essence, a big fudge factor), and whenever I have seen this equation rassled into a practical context, it was accompanied by other big fudge factors no matter what units were being used. It seems to me one could just use E=M and pick a fudge factor based on that, as easily as use the classic rendition. Am I out to lunch here? Any physicists lurking?

Paula, the same thought about c-squared has occurred to me. Hmm. Meanwhile ...



"You'd think we'd know by now just how little we know." You'd think so, but we don't. And we never will, I'm afraid.

Trust me, physics is mostly fudge.

As an (ex-)physicist, I suppose I should comment. Bruce, no one in any field wants to devote time to a total reconstruction of their science. It's as difficult for a physicist to back out of standard relativity & quantum theory and start theorizing anew as it would be to learn biology by one's self -- it would feel like a new science. I'm sure other scientists were interested in Magueijo's ideas but were unwilling to put years into them.



Paula -- a mass times a velocity squared has the dimensions of energy . . . thus the kinetic energy of a moving body is 1/2 m v^2. So for your "fudge factor" you're not free to pick any constant, you have to pick a velocity squared. The remarkable thing is that the universe appears to have only one characteristic velocity -- the speed of light -- whereas logically it could have had several. God somehow managed to formulate the laws of nature in an elegant and simple form, and still create intelligent life.

From the NYT article:



'The most recent buzz in V.S.L. circles is about something called "doubly special relativity." '



No doubt proposed by physicist Dean Wormer....

The main reason relativity hangs on is that it still remains a useful and predictable means of describing the universe. When that ceases to be, a new theory will be devised.



The same thing happened with geocentrism. The Ptolemaic system lasted as long as it did because it could predict the movement of the starts. It wasn't until Kepler came along and introduced elliptical orbits, thus making sense of Copernicus' system, that a heliocentric model could be taken seriously.



As PJ noted, it takes a lot of work to do something like that--Kepler had to throw away years of work, in fact, before he got anywhere. But sooner or later, if the theories are wrong, then the facts will start contradicting them, and someone will have to work out another hypothesis.

Paula:



Nope, no fudge factor; now, if you give M and c in pounds and mph respectively, your answer will come out in terms of the energy required to accelerate 1 pound at a rate of 1 mph/hour for 1 mile, which isn't very useful, so we generally throw in conversion factors to get the energy in sensible units.



(I'd need pencil and paper to confirm I got the units right there, but you get the idea).

Derek, it isn't only relativity that's "hanging on". Humble, old Newtonian physics is hanging in there just fine, even in the 21st century. Ask any engineer (engineers being people who have to get things done and who have no time for prissy speculation like relativity).

Mike, pounds are a weight measure, not a mass measure. (That pompous triviality aside...) the energy released by the conversion of an equivalent mass (one that on Earth would weigh one lb) would far exceed the amount of energy needed to accelerate one lb 1 mph/hr so I am not sure what your point is. My point is why bring c^2 into it if it does not have any more intrinsic value than a conversion factor which is essentially arbitrary except insofar as it conforms to the known data (the point again being c^2 without extra conversion factors doesn't -hey, I am going in circles, aren't I?)

Yeah, yeah, I know pounds aren't mass, but who the heck wants to work with Slugs?



Anyhow, you forgot the c^2; converting one 'pound' to energy releases exactly 34596000000 times the energy required to accelerate 1 pound by 1 mph/hour for 1 mile, where 34596000000 is the speed of light in mph^2.



The only 'constant' in Newton's K = .5 m v^2 is one half - this is no different.



The only reason conversion factors are required is that people like to measure energy in odd units like tons of TNT, rather than the power units implicit in their chosen units of time, mass, and distance. Contrary to what you might intuit, having units for time, mass, distance, and energy is redundant, which is where the conversion factor comes from, unless you use sensible energy units.

This is from my textbook titled _A First Course in General Relativity:_



"The student is assumed already to have studied: special relativity, including the Lorentz transformation and relativistic mechanics; Euclidean vector calculus; ordinary and simple partial differential equations; thermodynamics and hydrostatics; Newtonian gravity (simple stellar structure would be useful but not essential); and enough elementary quantum mechanics to know what a photon is."



I humbly submit that most people can't spell the above, let alone understand them, and that's the minimum you would need to make intelligent speculation as to the validity of the theory.



It's not accurately expressible as anything other than very high-level mathematics. Anything else is just hand-waving.



Noel Erinjeri

Noel, always nice to hear from a fellow Heinlein fan. I take it from your post that you're either reading a text on General Relativity, or have written one. Given your expertise in the field, please explain, how, practically, it would have made any difference to start with E=M and apply conversion factors (as opposed to starting with E=Mc^2 and applying conversion factors). Thanks.

Paula -



E = mC^2 was not a post-hoc rationalization, if that is your concern (I may not be reading your question correctly), but a prediction
of GR. As PJ mentions, the units must make sense and a velocity-squared is necessary. I have found a simple derivation of E=MC^2 using

quantum-mechanical and classical physics:











"Effective" mass of the photon is m (yes, photons have no mass but work with me here).





The photon has a velocity c, momentum p, energy E, a wavelength l, and a frequency n.





Therefore,





p=mc (from classical physics)





c=nl (definition of frequency and wavelength)





E=hn (from Planck's energy-frequency law)





l=h/p (from DeBroglie wavelength-momentum law)





Therefore, c/n = h/p = h/(mc)





Thus, c/(E/h) = h/(mc), or E=mc^2.







This ain't the way ol' Uncle Albert did it, to be sure. It fits well with empirical data because GR is a good theory (although imperfect if the NYT article is to be believed), and predicted the relationship rather than being fitted to experimental data - the crutch upon which most fluid flow problems eventually rely . Does this answer your question?

Paula:



If you use E=Mk, your constant will not be dimensionless; it will have units of distance^2/time^2. People using 'constants' really aren't using equations of the form e=mc^2k, they're just transforming by identities and possibly being sloppy with dimensional analysis.



If you're consistant with units, the conversion factors will all disappear.

Orrin's original point -- a very odd one from

an announced antirationalist -- was that

any scientific theory would have to be

discarded.



This is not correct, but if it were, on what

grounds?



Some theories are much better supported

than others -- "more robust" is the

term. The theory of the conservation of

angular momentum is not under attack

by even the wildest speculative minds.



Relativity certainly is, as in quantum theory.

In fact, almost all of theoretical physics

today is devoted to attacking one or the

other.

I couldn't come up with a derivation of E = m c^2 from memory, but I found this:



2.3 The Inertia of Energy




If you follow the math you'll find E=mc^2 about a third of the way down the page. So you can accept that the equation comes from Einstein's Special Relativity -- not General Relativity -- and isn't merely an empirical finding with a fudge factor.



You *can* say "E = m" ... if you use a system of units in which the speed of light is defined as 1. :-)





Google couldn't find it on the web, but there's a Sidney Harris cartoon showing how Einstein worked out the equation:



E = m a^2 ... "No"

E = m b^2 ... "No"

E = m c^2 ... "Yes!!!"

Holy smokes, Paula, I'm about 50 IQ points shy of being able to write a GR textbook. I took one class in it as an undergraduate. My "expertise" in the field is pretty much non-existent. But I also have an ego the size of Montana, and you've stroked it pretty well, so I'll try to answer your question.



Actually, I started trying to answer it, and then I saw the link Bill posted, which is a lot better than I could ever do.



Noel Erinjeri

Bruce, thanks for your response. It is indeed a substantive one. I remain with the complaint though that it seems to me that you introduce as a premise the notion that the energy (or technically the momentum) of a photon is a function of its velocity (by dint of which you are forced to posit its mass), and you end by appearing to demonstrate that the energy of a photon is a function of its velocity. It all seems pretty circular. Sorry, I don't expect a response I can understand since I'm already out of my depth. Thanks for your post.

Paula,



It's pretty simple.



Today we express all demensions in terms of three - length, mass, and time.



For example, acceleration is expressed in m/s^2. A Joule (unit of energy) is Newtons*m. A Newton (unit of force) = kg*m/s^2, so a Joule is kg*m^2/s^2.



So the fascinating thing about E=mc^2 is that E kg*m^2/s^2 = Mass kg * (c m/s)^2 with no fudge factor of any kind required at all.



This doesn't depend on some clever choice about the definition of a Joule. In fact, we just define Joules in terms of meters, seconds, and kilograms. You would get exactly the same result if you worked in feet, days, and slugs. So the choice of c here is not arbitrary at all - if we chose any other velocity we would need a special fudge factor to make the math work. With c our fudge factor is 1.



So yes, it is pretty amazing, isn't it?

Mike, you demonstrate my point. One can achieve any arbitrary effect, not impacting an actual variable, according as one defines one's units; it should be obvious by your very logic that if c is treated as a constant then its presence in the equation is optional.

Paula, I don't think you get it.



They key point here is that this doesn't come from how you define your units. For example, if I replaced Meters with Feet and used the equation E kg*ft^2/s^2 = Mass kg * (c ft/s)^2 it would still work.



It's not a matter of how you define units. I could replace Kg with earth masses (e) and use E e*ft^2/s^2 = Mass e * (c ft/s)^2 it would still work.



Then I could replace seconds (s) with hours (h) and it would still work. E kg*ft^2/h^2 = Mass kg * (c ft/h)^2.



Another way to look at it is that E / (mc^2) is a dimensionless constant - all the units cancel out. And it is equal to 1.



E kg*m^2/s^2

————————————--

Mass kg * (c m/s)^2



E kg*m^2/s^2

= ————————————--

Mass * c^2 kg*m^2/s^2



E (Note: All units have

= ——————--- canceled out - this is

Mass * c^2 dimensionless)



= 1









=

Damned HTML formatting! I think you can still read it.



Anyway, if you really think you can choose units to make E=mc^2 not true and, say, to replace it with something different (say E = M (c/2)^2 ) can you show us how?

Dimensional analysis can only take you so far. You can prove that both sides of the equation have the same units, but that isn't enough to prove that, say

E = 2 pi M c^2 ,

isn't the correct equation.