Re: Postulates of General Relativity

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Antti's question about the postulates of GR:

> I have a question on postulates of the General Relativity (GR). I
> happened to read from a high school level book (Kerr, Kerr & Ruth
> 1999, 621) that the postulates of GR are:
>
> 1) Mach's principle -Inertial and gravitational forces are
> indistinguishable.

Actually this is the Equivalence Principle, (Mach's Principle being
something else) and the crucially important aspect about it is that
this indistinguishablility is only *local* in a sufficiently small
neighborhood of any particular point of spacetime such that the
geometric effects of any spacetime curvature are insignificant and
the geometry is sufficiently flat on the small scale that SR is
correct in this limit.  A final aspect about this point is that the
way GR locally boils down to SR is that the locally freely falling
frames of GR are the locally inertial frames of SR.

> 2) Four dimensional space-time is curved as a result of the presence
> of mass.

A better way to say this is to state that the curvature of four
dimensional spacetime is responsible for producing the effects seen
in nature that are caused by gravitation.  In addition this spacetime
curvature itself is caused by multiple quantities other than just
mass.  In GR the sources of gravitation (spacetime curvature) are the
local densitiy of mass/energy, momentum flux, and stress (including
hydrostatic pressure, axial stress, & shear stress) of the matter
and or radiation present.  This is quite different from Newton's
theory where the source of gravitation is just plain mass.  Since
the contribution from the momentum flux and the stress from massive
matter tends to appear with a higher power in 1/c^2 than the
contribution from the mass density, their effects on the curvature of
spacetime can often be ignored (especially in the Newtonian limit).

3) Objects take the shortest path between two points in space-time.

This is not quite correct.  In GR the analogs of straight lines in
the flat spacetime of SR are the geodesics in curved spacetime.
Small (test) objects left to themselves (i.e. experiencing no non-
gravitational forces) follow *timelike* geodesics in spacetime. Light
rays follow null geodesics in spacetime.  But nothing (other than
maybe unobserved tachyons) follows spacelike geodesics.  The timelike
geodesics of spacetime followed by massive test particles are *not*
the shortest path between two fixed points in spacetime.  Rather,
they represent the path of the *longest* elapsed proper time along
the path when these fixed points are timelike separated.  The
spacelike geodesics measure the *shortest* proper distance between
two spacelike separated points, but they are *not* followed by any
observable objects (because doing so would require faster-than-c
motion through spacetime).

> Are these postulates correctly stated? Is it really so that GR
> follows from these three postulates?

Not quite.  Even corrected versions of the above postulates are
insufficient to uniquely determine GR.  There are other non-GR metric
theories of gravitation, e.g. the Brans-Dicke theory which are also
compatible with them.  To uniquely get GR we also need to have some
requirement which uniquely selects GR out from all the other
possibilities.  Part of such a criterion is the requirement of (what
Misner, Thorne & Wheeler call) 'no prior geometry'.  It also needs
to be strong enough to enforce a requirement that the coupling
strength between the sources of gravitation and the resulting
spacetime curvature that results is a global constant of nature
coupling strength.  Whatever extra criterion is used, it needs to be
selective enough so that the only form allowed for the action
functional that produces (via Hamilton's Principle) the laws of
nature, is the Hilbert Action (possibly with an additive constant
term for the Cosmological Constant).

> I have wondered what is the difference between postulates in physics
> and axioms in mathematics. Any opinions?

I don't think they are all that different.  I suppose it might be
argued that in mathematics all of the results of mathematical theory
must follow deductively and rigorously from only its axioms. Whereas
in physics the postulates might be said to act a little more as
inspirational guides in the formulation of the theory, and where many
of the deductive steps follow from things other than directly from
the postulates themselves, e.g. the assumed validity of various kinds
of mathematical manipulations, mathematical consistency requirements,
reasonable smoothness assumptions, coordinate system independence of
the results, etc., etc.

David Bowman

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.