On Nov 6, 2004, at 10:02 AM, Saul-Paul & Mary-Minn Sirag wrote:
[Saul-Paul Sirag] The issue of coordinate dependence of tensors is very tricky. Perhaps the following statements (by Arthur Eddington, 1920) will clarify the issue.
1. "A tensor does not express explicitly the measure of an intrinsic quality of the world, for some kind of mesh-system is essential to the idea of measurement of a property, except in certain very special cases where the property is expressed by a single number termed an invariant, e.b. the interval, or the total curvature."
2."But to state that a tensor vanishes, or that it is equal to another tensor in the same region, is a statement of intrinsic property, quite independent of the mesh-system chosen."
Yes, a good start. Let's continue.
A tensor (or spinor or twistor) transformation is a multi-linear homogeneous transformation on "objects".
A spinor is a complex double-valued representation of some continuous groups like SU(2) that covers O(1,3) of special relativity.
The objects are N-dimensional hyper-matrix arrays of elements of an algebraic field.
Let X be the basic linear transformation.
For example, using summation convention on repeated pairs of identical upper and lower indices, ^ denotes upper index
Au --> Au' = Xu'^uAu
is a simple linear transformation.
S ---> S' = S is a scalar invariant.
For example S = AuA^u
The simplest multi-linear transformation is
Fuv ---> Fu'v' = Xu'^uXv'^vFuv
X in general is a homomorphic image or representation of some GROUP (maybe semi-group and maybe even up to a CATEGORY?)
Therefore we really have X(G) where G is some group.
All laws of physics must be covariant under the physical symmetry groups G.
This means that the laws of physics must be tensor equations in this general sense.
What applies to the laws of physics DOES NOT APPLY to the fundamental VACUUM SOLUTIONS of those laws! Or to any solution.
There is a lot of confusion on this issue.
In the SPECIAL CASE of Einstein's 1916 theory of gravity
G = GCT AKA DIFF(4), i.e. generally NONLINEAR LOCAL coordinate transformations AT A FIXED POINT EVENT P.
X for GCT is the Jacobian matrix of the reversible 1-1
x^u(P) <-----> x^u'(P)
X^u'u(P) = x^u'(x^u(P)),u
,u is ordinary partial derivative
R. Kiehn generalizes this to one-way semi-group mappings that are not 1-1.
This point event P is more than a mathematical point p in a manifold. This is where the confusions of active Diff(4) vs passive Diff(4) comes in.
Active DIFF(4) takes p to p' ON THE MANIFOLD.
*GCT is at FIXED PHYSICAL P that has no definite p.
P depends on a relationship between gravity guv and electromagnetic Fuv since the LIGHT CONE is the essential INVARIANT of 1916 GR.
Without light there is no gravity!
And GOD said: Let there be light and the INCOHERENT (renormalized) micro-quantum ZPF Dirac Sea (with virtual photons) came into Being and Becoming.
But the incoherent Dirac Sea is unstable without gravity and inertia. The inflationary phase transition morphs the incoherent micro-quantum ZPF Dirac Sea into the coherent MACRO-quantum Higgs Ocean with gravity WEP equivalent to inertia. Gravity is emergent without quantum foam and the ZPF is absorbed almost completely solving the cosmological constant problem, but leaving exotic vacua "dark energy" (& dark matter) residues. The Casimir force is not a ZPF force at all, but is a complex electrostatic force between mutually induced electric dipoles in neutral atoms and molecules in the Casimir plates (boundary conditions). The entropy of the post-inflationary Higgs Ocean << entropy of pre-inflationary Dirac Sea and this explains why the Arrow of Time points in the direction of the expansion of 3D space of the universe.