Newsgroups: alt.sci.physics.new-theories
Subject: Re: e=mc^2 missing terms
From: mcirvin@husc8.harvard.edu (Mcirvin)
Date: 19 Aug 92 23:05:25 GMT
Keywords: e=mc^2, einstein, matter, energy

et@teal.csn.org (Eric H. Taylor) writes:

>In talking with another physicist today he wanted to know if I had heard
>about the missing term(s) in Einstein's famous equation.

>He said that while in college (University of Colorado, in the 50's) a
>friend of his, tried deriving the second order differential of e=mc^2.

Second order differential with respect to what?

>When he did so, he found that the results were contradictory.
>He then contacted Einstein himself, and Einstein response was
>"Very astute. The problem is that you are missing a term."

Hmmm... this is a puzzle.  The only likely thing I can think of is
the confusion that can happen when you use the wrong mass
definition for your formulae (see recent sci.physics disputes
for further elaboration).  The modern convention, which Einstein
sometimes (though not always) used, is to call the rest mass
m and to mean rest mass when you say "mass."  With such a
convention E = mc^2 (where E is the total energy) is only
correct in an object's rest frame.  The formula of general
validity is

E^2 = m^2 c^4 + p^2 c^2,

where p is the momentum.  

>The missing term is or includes a -1. 

Well, here's one idea.

The way I wrote it, there's no -1, but it's possible to write it
so that there is one, and the -1 actually means something.
Write it as

m^2 c^4 = E^2 - p^2 c^2

and you get a relative minus sign.  This seems like an idiotic
attempt to shoehorn a -1 into the formula, but it's actually a
very important manipulation.  Its significance can be seen
more easily if you change the units so that c = 1

m^2 = E^2 - p^2

and separate the components of momentum:

 2     2    2    2    2
m  =  E  - p  - p  - p
            x    y    z

This looks a lot like the Pythagorean formula for the length
of an interval in space, except that it's got four terms instead
of three, and three of the four have funny minus signs on
them.  These minus signs are consequences of the Minkowskian
geometry of spacetime, in the context of which the mass is
interpreted as the "length" of a four-dimensional vector
made up of the energy and the momentum.  It's a vector with
a "direction" in spacetime; the momentum gives the spatial
components, and the energy is the time component.  In
Minkowskian geometry, there's a relative minus sign between
the squared time and space components when you calculate
the length of a vector.  

You might say, "Why don't you put the momentum components
back on the left side of the equation?  Then you can interpret
everything in terms of Euclidean geometry in four dimensions."
The answer is that the things we want to be analogous to
rotations in space are *changes of inertial frame* rather
than just 4D rotations.  Just as the length of a vector in
space doesn't change under rotations, the length of a four-
dimensional vector, calculated in this way, doesn't change
under velocity boosts between inertial frames.

Of course, whether that was what Einstein was talking about,
I have no idea.
 
-- 
Matt McIrvin, Cambridge, Massachusetts, USA 
Xref: sparky alt.sci.physics.new-theories:1996 sci.med:16158 bionet.sci-resources:162 sci.research.careers:660 sci.geo.geology:1809