Re: For Dr. Enders , spacetime.


Koobee Wublee wrote:
On Dec 30, 1:21 pm, "Ken S. Tucker" <dynam...@xxxxxxxxxxxx> wrote:

An Alternative to the conventional Minkowski
Metric.

For general spacetime, we have

ds^2 = g_ij dq^i dq^j

Where

** i, j = 0, 1, 2, 3

Expanding the above, we have

ds^2 = g_00 dq^0 dq^0 + g_0i dq^0 dq^i
+ g_i0 dq^i dq^0 + g_ij dq^i dq^j

Where

** i, j= 1, 2, 3
** g_0i = g_i0

Because of the symmetry in the metric matrix, the spacetime becomes

ds^2 = g_00 dq^0 dq^0 + 2 g_0i dq^0 dq^i
+ g_ij dq^i dq^j

By dividing both sides by ds^2, we have

g_00 dq^0/ds dq^0/ds + 2 g_0i dq^0/ds dq^i/ds
+ g_ij dq^i/ds dq^j/ds = 1

By dividing both sides by (dq^0)^2 / g_00, we have

Please show your work in the step, that yields,

(1/g_00) ds/dq^0 ds/dq^0 - 2 (g_0i/g_00) dq^i/dq^0
- (g_ij/g_00) dq^i/dq^0 dq^j/dq^0 = 1

These Lagrangians would yield more colorful Euler-Lagrange equations.
Ken

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