The equation of a planetary orbit in the axes of Pseudo-Euclidean Space-Time has, in [2] been derived, in the form of a first order equation, from a Lagrangian analysis of the metric of the General Theory. To obtain that form here, the simplest process is to obtain the first integral of (C8) from which the desired relationship can be obtained directly. The easiest manner to obtain the first integral of (C8) is firstly, via a re-arrangement of (5.4), thus

(B1)

From (5.3) note that

(B2)

which with (3.19) gives

(B3)

Inserting this into (B1) then gives

(B4)

which incidentally can be shown to be the first integral of (3.17), the equation of planar motion in D₁.

Now, from (2.15)

(B5)

so that substitution for from (B4) yields

(B6)

Also from (C7) and (B4)

(B7)

which when inserted into (B6) yields

(B8)

as the first integral of (C8).

Transformation of (B8) to the axes of D₀ via (4.7), (4.12) and (4.13) gives

(B9)

For simplicity write this as

(B10)

where (5.5) has also been inserted

The equation of the orbit, (expressed in the axes D₀), can now be derived in the conventional manner as follows. Put

(B11)

so that

(B12)

Inserting this and (B11) into (B10) yields

(B13)

Expanding, this finally reduces to the desired expression, thus

(B14)

as derived in [2], pp 198, Eq[58.35].

Finally, in (B10) the simplifying identity

(B15)

was inserted. To shown that this is identical to the same parameter in [2], pp197, Eq(58.26), insert (4.7) and (4.18) thus

(B16)

which from (3.5) and (3.19) becomes

(B17)

and which with (2.15) and (2.19) then gives

(B18)

so that insertion of (4.7) again into this finally gives

(B19)

As derived in [2].

Also from (5.5) it can be seen that the constant h in this paper is identical to the parameter m in [2], pp197, Eq(58.27). These results provide additional proof that a central orbit in D₁ is identical to that in the General Theory.

On to the next Section: Appendix C

Back to the title page: Gravitation

Back to the Home page for this Site: Home