Environmental Engineering Reference
In-Depth Information
We know from Section 11.2 that the Lorentz transformations preserve the scalar
product defined in Eq. (13.15). Thus we recognise that the Minkowski space-time
equivalent of Euclidean rotations involving the time dimension are the Lorentz
transformations and that these correspond physically to a change of inertial frame.
We are free to introduce other vectors in space-time. They may be useful in drawing
up the laws of physics provided that they transform according to the Lorentz trans-
formations and that the scalar product between any two four-vectors is determined
by the metric, i.e. for four-vectors A and B
( A ) T
A
·
B
=
g
( B )
=
A 0 B 0
A 1 B 1
A 2 B 2
A 3 B 3 .
(13.16)
Defined this way, the scalar product is guaranteed to be the same in all frames
and is thus a four-scalar. Armed with Minkowski space-time, four-vectors and
four-scalars we can make progress in physics. In fact our logical development has
brought us all the way to the start of Chapter 12. Still, since we have not made use
of Einstein's postulate that the speed of light is a universal constant, c (which has
by now also appeared in the Lorentz transformation equations) remains nothing
other than the constant that calibrates space-time distances in the time direction.
The causal structure of the theory dictates that c must be a limiting speed, i.e.
we require that particles always follow timelike trajectories through space-time.
Only after we have introduced the energy-momentum four-vector does the more
familiar interpretation of c emerge: for there can exist particles for which m
=
0
=
and E
cp but only if such particles travel at a speed equal to c in all inertial
frames. Thus massless particles may exist in a Minkowski space-time provided
they always travel with speed c . Since light is made of massless photons, we may
go ahead and refer to c as the speed of light. From the space-time view there is
clearly nothing very special about light. Indeed, the four-speed (defined as U · U
)
of any particle (including those with mass) is, from Eq. (12.8), always equal to
c , which means that everything travels through space-time with the same speed.
In our three dimensional world, massless particles appear special since only they
travel with the same speed in all inertial frames.
It is fair to say that one of the main goals of this section of the topic is to present
the reader with a new way of viewing Einstein's statement that the speed of light
is the same in all inertial frames. In particular, we have traced the roots of this
statement all the way back to space-time and causality. Incidentally, Einstein's first
postulate, that the laws of physics are the same in all inertial frames follows auto-
matically once we have specified that we should work in Minkowski space-time,
for we know that moving between inertial frames is just a co-ordinate change in
space-time and the laws of physics should be trivially independent of co-ordinates.
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