On 26-03-2019 20:29, agrayson2...@gmail.com wrote:
On Tuesday, March 26, 2019 at 11:29:08 AM UTC-6, John Clark wrote:
On Tue, Mar 26, 2019 at 1:14 PM <agrays...@gmail.com> wrote:
_> How do the mathematicians prove it?_
Mathematicians can't prove that a physical theory is correct, all
they can do is show that changing the coordinate system (for example
by rotating the X and Y axis) does not result in different physical
predictions. Only exparament can tell you if the predictions is
right, or at least mostly right.
John K Clark
I'm not asking if GR is correct; rather, whether it is covariant.
Moreover, for SR we can prove covariance, since under the LT, the law
of physics don't change and the SoL is c in any inertial frame. ME are
also invariant under the LT. AG
There are many ways one can do this, the most elegant way is to start
with a Lagrangian of a field theory and then demand that it be invariant
under general coordinate transforms, which requires factors of the
square root of the determinant of the metric tensor to be inserted to
compensate for the Jacobian of a coordinate transform. This seemingly
rather trivial insertion, will yield the field equations of GR as far as
the coupling with the fields desribed by the field theory are concerned.
Compare this with the way you can derive the Maxwell equations from
scratch. You start of a scalar field theory, that is invariant under
global gauge transforms phi ---> exp(i alpha) phi. And then you make the
constant alpha an arbitrary function of space-time, which destroys the
invariance due to derivatives generating additional terms. But you can
compensate for these derivatives by including a gauge potential. This
then yields the coupling of the scalar field to a new field that we can
call the electromagnetic field, and this field will have its own gauge
invariant term proportional to the field strength tensor squared.
So, as Paul Davies puts it, in principle mathematically gifted cave men
who had never done any experiments involving electromagnetism and
gravity could have deduced the Maxwell equations and the Einstein
equations of GR from scratch based only on mathematical elegance.
Saibal
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