Re: General Relativity experts

Kent Paul Dolan <xanthian@xxxxxxxx> wrote:

[...]
Tensor calculus (one of mathematics'
more ugly step-children) was a _tool_, not the
container of his theory. Your ignorance is now in
obvious display.

At the time Einstein formulated general relativity, tensor
calculus was the fundamental language of differential geometry.
The modern concept of a differentiable manifold, for instance,
was introduced by Whitney, but not until 1936. Tensor calculus
may be a "tool" of general relativity, but only in the sense that
calculus is a "tool" of Newtonian mechanics and linear algebra
is a "tool" of quantum mechanics.

[And at least the one course I took in differential
geometry did not employ tensors _at all_. Partial
differential equations skills more than sufficed.]

It must have been a pretty elementary course, then. The basic,
defining quantities for a Riemannian manifold, for instance, are
the metric tensor and the curvature tensor. The tangent bundle
and frame bundle are defined in terms of tensors (tangent vectors).
If you look at Kobayashi and Nomizu's _Foundations of Differential
Geometry_, tensor algebras are introduced in the second section
of chapter 1; in Warner's _Foundations of Differentiable Manifolds
and Lie Groups_, chapter 2 is "Tensors and Differential Forms";
in Petersen's _Riemannian Geometry_, tensors are introduced in
chapter 2 (and it's assumed that you know the tangent bundle
before you start reading).

I can imagine doing differential geometry of curves and two-
surfaces without tensors by treating them as objects embedded
in a flat three-dimensional space. But this misses the basic point
of differential geometry, the idea of an intrinsic geometry that's
independent of any embedding. It's hard for me to see how you
could do handle this with just "partial differential equations skills"
-- how do you define a geometry with a metric or a connction?

What textbook did you use?

Steve Carlip

.

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