Natural Science Forum / Physics / Relativity / December 2006

An Alternative to the conventional Minkowski Metric.
======================================
In flat space you set g_00=g_11=g_22=g_33=1 and
g_0i = -dx^i/dx^0, that works very well when merging
SR with GR.

The thing most researchers avoid are nonorthogonal
metrics like g_0i, but aberration happens when relative
speed separates CS's, so you can't really hide it by
using sqrt(-1), just because g_0i looks complicated.

Unfortunately SR was established algebraically,
and a sort of evolution occurred employing the use
of tensors by Einstein for GR. What should have
happened, in a more ideal historical hindsight,
is to begin with,

  U_u U^u = 1   {u=0,1,2,3}  U^u = dx^u/ds

as a definition in SR. Then expand that to
detail time and space as,

  U_0 U^0 + U_i U^i = 1  ,{i=1,2,3}

The U_i U^i is *absolute velocity* and since one
can always find a CS where motion of something
is zero, is the same as saying motion is relative,
hence,

  U_i U^i = 0 = absolute motion

is the covariant way (for all CS's using tensors)
of writing "motion is relative".

Of course relative motion is retained by U^i
(by choice) and being non-zero generally produces,

  U_i = 0 generally.

Now you can use association to obtain,

  U_i = g_iu U^u = 0

and expand index "u" in time and space {0,i} to,

  0 = g_i0 U^0 + g_ij U^j .

Use a bit of algebra and see,

  g_i0 = - g_ij U^j/U^0 = - g_ij dx^j/dx^0 .

Specifing a flat space-time metric g_ij
simplifies to the Kronecker delta and so,

  g_i0 = - dx^i/cdt  simplified,

and is aberration...a real effect well
established by experiment.

Now let's stick those nonorthogonal critters in

  ds^2 = g_uv dx^u dx^v

by expanding indices "u" and "v" over time and space,

  ds^2 = g_00 dx^0 dx^0 + 2 g_0i dx^0 dx^i

       + g_ij dx^i dx^j

  ds^2 = g_00 dx^0 dx^0 - g_ij dx^i dx^j  (generally).

Sub in a simplified metric g_00, g_11... =1 and
dx^0 = cdt to get the familiar

  ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2

that Minkowski and later Einstein needed for GR.

In relativity, the important thing is to deal
with relative motion and the vanishing of
*absolute* motion at the outset, otherwise a
lot of spooky-kooky aberrations appear, (I won't
hang other's dirty laundry in public), that I
think is best expressed by the tidy,

  U_i = 0  .

Anyway if the metric system you choose finds

 U_i U^i =/= 0

you may have a problem.

Above, we see the only metric compatible with U_i=0 is,

 ds^2 = g_00 dx^0 dx^0 - g_ij dx^i dx^j  (generally).

Note the total absence of g_i0 terms when absolute
motion is excluded and relativity is strictly invoked,
essentially "collapsing" the metric relativistically.

Finally, we need to ask if a non-zero result of the
equation, U_i U^i is truly indicative of absolute
motion as I've assumed?

Regards
Ken S. Tucker
kxsxt