Eric Gisse wrote:
On May 10, 9:01 pm, Tom Roberts wrote:
Exercise for the reader: why is the HOMOGENEOUS Lorentz
transform used, and not the full Poincaré group (aka the
inhomogeneous Lorentz group, which includes translations)?

a) Electromagnetism is Lorentz, not Poincare, invariant?

No. Maxwell's equations are invariant under a 15-parameter group called
the conformal group, which is larger than the Poincare' group. The other
5 parameters are changes of scale, so if one keeps the same units only
the 10-parameter Poincare' group applies.


b) The inhomogeneous Lorentz group contains parity inversions and only
continuous operations are allowed?

No. Maxwell's equations are invariant under time reversal and parity
inversion.


c) Because the Faraday tensor does not transform as a tensor under the
Poincare group?

No. It does. Indeed, it transforms as a tensor under any coordinate
change whatsoever.


Can one use the full homogeneous Lorentz group, or only
the component of it which includes the identity?

Are you wearing your continuous transformation only hat?

I have no such hat.


Hint: remember that when one expresses a tensor F in terms of its
components {F_ij}, the tensor itself is:
F = F_ij e^i e^j
where the {e^i} are the basis vectors of the coordinate system. Ask
yourself how a translation affects the basis vectors.

Hint2: remember that the components of a tensor {F_ij} transform:
F_i'j' = dx^i/dx^i' dx^j/dx^j' F_ij
and compute dx^i/dx^i' for a translation.


Tom Roberts