GR just forbids particles from moving faster than light. That's a
physical constraint. It's not built in the definition of the vector
space which is happy to include tachyons.
Brent
On 8/17/2024 12:52 PM, Alan Grayson wrote:
According to Wikipedia and other reliable sources, the Metric Tensor,
denoted as g (without its two subscripts), is a bilinear function of
vectors *u *and*v,* in the *_vector_ _space_* resident in the tangent
space, say at point P, of an underlying manifold, which maps to the
real numbers. There’s also the concept of the Metric Tensor *Field*/,/
which presumably has a *unique* real value at every point in the
underlying manifold. But there exists an uncountable infinite set of
pairs (*u*,*v*), at which g can be evaluated at each point P where the
tangent space contacts the manifold. So, which *specific pair* must we
choose to define the Metric Tensor or its Field? Or suppose we know
the Energy-Momentum Tensor in some region of Spacetime (the assumed
manifold), how do we calculate the presumed unique value of the Metric
Tensor in this region if it is defined on an uncountable infinite set
of pairs (*u*,*v*) at each point P of the manifold?
A related problem is the *construction of the tangent space in GR*,
say at point P on the manifold. If the manifold is Spacetime, we can
imagine the paths of all test particles moving at speeds *less than
c*, the speed of light**(in order *not* to violate one of the
postulates of GR), which are tangent to the manifold at P, where these
vectors presumably *define* the tangent space at P. But for these
vectors set to form a *_vector_ _space_*, the satisfaction of the
linear additive property of a vector space under the field of real
numbers will yield vectors representing test particles traveling at
speeds *greater than c* on the tangent space, and presumably at such
speeds on the underlying manifold. The only way out of this problem is
to assume the test particles on the manifold include those moving at
speeds greater than c. But then we’re violating one of the basic
premises of GR.
TY, Alan Grayson
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