Download Proof that E = mc2

Proof that E = mc2
sirdanielofchen
August 2016
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Introduction
We prove that E = mc2 , using dimensional analysis.
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Proof
Theorem 1.1: Mass-Energy Equivalence (Einstein). Suppose we have a
physical system with energy E, and mass m, and let c = 3.0 × 108 m/s be the
speed of light. Then the energy and mass are equivalent, in the sense that
E = mc2
Proof: Suppose that in such a physical system, the parameters E, m, c are
sufficiently and necessarily related. We form the following dimensional matrix:


E m c
 1
1
0 M 

D=
 2
0
1
L 
−2 0 −1 T
Each column represents the fundamental dimensions of the quantity specified
at the top of the column. For example, mass (m) has units MASS (M) and this
is indicated with a ”1” corresponding to M as indicated in the first row. Each
row represents a fundamental dimension (M = MASS, L = LENGTH, T =
TIME).
The rank of the matrix is 2, since we can easily see that the second and
third rows are linearly dependent, and the first row is independent from the
rest. So rank(D) = 2, and since we have 3 related quantities, we may apply
the Buckingham Pi Theorem to get a physical relation relating these quantities
involving 3 − 2 = 1 dimensionless and independent quantities (pi blocks) based
on these quantities.
Then it follows that the choice of Π is unique (why?) and is
Π=
E
mc2
,
1
so the Pi Theorem tells us that K =
Then we easily get
E
mc2 ,
where K ∈ R is a fixed constant.
E = Kmc2
and so a single well-done experiment can tell us that K = 1, so that we
conclude
E = mc2
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Questions and Comments
The given proof is highly suspect, since it uses no physics at all, and one questions its validity and rigor. In fact, the proof is rigorous mathematically speaking (except for the fact that point 1 illuminates that we avoided proving part of
the theorem... so maybe it isn’t so rigorous after all), but it is not particularly
satisfying for a number of reasons:
1. It assumes the quantities E, m, c are already related, and that they are sufficiently related (so that no other variables are involved). This is actually
probably the majority of the work put into developing this theorem. Critical experiments and experimental observations needed are furthermore
highly non-trivial. Thus, the soundness of the argument is weak.
2. It makes use of some experimental data (K = 1).
3. It uses dimensional analysis, which no one likes.
K bye.
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