Re: Einstien's principle of equivalence

Cyberkatru wrote:
Einsteins principle of equivalence, as usually explained, seems strange (to
me) from the point of view of (pseudo) Riemannian geometry.
As explained in some relativity texts, the principle states roughly that
there is no (local) physical difference between effects due to an
accelerating frame and effects due to a gravitational field. However,
choosing an accelerating frame seems like just a certain choice of
curvilinear spacetime coordinates. But isn't gravity supposed to be
curvature due to a metric tensor? But whether a Lorentz manifold has zero
curvature or not certainly doesn't depend on a choice of coordinates. How
can I produce curvature on a flat spacetime just by a choice of chart? I
can't! The whole thing seems especially weird since metric and curvature are
defined as geometric objects with an existence that doesn't even need a
chart at all to make sense.
So I don't see what the equivalence principle could really amount to.
What am I missing?

As usually sated, the Equivalence Principle (EP) states that a
*constant* gravitational field is equivalent to uniform acceleration.
So, effects of gravity can be removed by a coordinate transformation
only as far as the gravitational field can be approximated to be
constant. This is entirely consistent with the well known mathematical
fact that at any point x on a (pseudo-)Riemannian manifold, there
exists a coordinate system in which the Christoffel symbols are zero at
x, since it is the Christoffel symbol components that correspond to the
gravitational force field. Curvature makes an appearance only when
derivatives of the Christoffel symbols are evaluated, which is
completely consistent with the EP.

The principle of general covariance is also a problem for me. Does it amount
to anything more than the statement that the laws of gravitational physics
can be expressed in terms of tensor fields?

Rather more strongly, must be expressible in terms of tensor fields.

It seems to me that, all sorts
of ad hoc laws could be expressed in this way. In fact, if spacetime had
trivial topology then one chart would do and any law written in that chart
could just be expressed in any other chart by brute force anplication of
tensor trnasformation laws. In this case the idea of general covariance
starts to seem empty.
What am I missing?

To see importance of general covariance, you have to look at how it is
used. And the fact is that in practice it is used not as a guiding
principle, but as an algorithm. It is an algorithm for deducing the
form of physical laws in any coordinate system. Suppose that there is a
preferred reference frame or coordinate system where a physical law is
easy to express, say the rest frame of a given system. Knowing that the
same law is expressible in terms of tensors, one makes a guess at the
tensor form and verifies that it reproduces the correct expression in
the preferred coordinate system (if not, one keeps guessing). Once in
tensor form, this law easily lends itself to expression in an arbitrary
coordinate system. With practice, even the guessing stage can be
automated. Thus, for example, knowing Maxwell's equations in an
inertial frame is easily translated to tensor notation and from there
to any other coordinate system.

Hope this helps.

Igor

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