Notes on Relativity and Cosmology - Physics Department, UCSB

1.3. NEWTONIAN ASSUMPTIONS ABOUT SPACE AND TIME 31
Now, are these well-defined? After some thought, you will probably say ‘I think
so.’ But, how can we be sure that they are well-defined? There are no certain
statements without rigorous mathematical proof. So, since we have agreed to
think deeply about simple things (and to check all of the subtleties!!!), let us
try to prove these statements.
1.3 Newtonian Assumptions about Space and
Time
Of course, there is also no such thing as a proof from nothing. This is the usual
vicious cycle. Certainty requires a rigorous proof, but proofs proceed only from
axioms (a.k.a. postulates or assumptions). So, where do we begin?
We could simply assume that the above definitions are well-defined, taking
these as our axioms. However, it is useful to take even more basic statements
as the fundamental assumptions and then prove that position and time in the
above sense are well-defined. We take the fundamental Newtonian Assumptions
about space and time to be:
T) All (ideal) clocks measure the same time interval between any two events
through which they pass.
S) Given any two events at the same time, all (ideal) measuring rods measure
the same distance between those events.
What do we mean by the phrase ‘at the same time’ used in (S)? This, after all
requires another definition, and we must also check that this concept is welldefined.
The point is that the same clock will not be present at two different
events which occur at the same time. So, we must allow ourselves to define
two events as occurring at the same time if any two synchronized clocks pass
through these events and, when they do so, the two clocks read the same value.
To show that this is well-defined, we must prove that the definition of whether
event A occurs ‘at the same time’ as event B does not depend on exactly which
clocks (or which of our friends) pass through events.
Corollary to T: The time of an event (in some reference frame) is welldefined.
Proof: A reference frame is defined by some one clock α. The time
of event A in that reference frame is defined as the reading at A on any clock
β which passes through A and which has been synchronized with α. Let us
assume that these clocks were synchronized by bringing β together with α at
event B and setting β to agree with α there. We now want to suppose that we
have some other clock (γ) which was synchronized with α at some other event
C. We also want to suppose that γ is present at A. The question is, do β and
γ read the same time at event A?