Schwarzschild to the limit

We use (dr'/dt_o). See [5]

(%i1)

assume ( G > 0 , M > 0 , r0 > 3 · G · M ) ;

[G > 0, M > 0, r0 > 3 \cdot G \cdot M]

(%i2)

assume ( rm > 0 , rm < r0 ) ;

[rm > 0, r0 > rm]

(%i3)

assume ( rp > 0 , rp > r0 ) ;

[rp > 0, rp > r0]

(%i4)

frm : 1 − 2 · G · M / rm ;

1 - \frac{2 \cdot G \cdot M}{rm}

(%i5)

frp : 1 − 2 · G · M / rp ;

1 - \frac{2 \cdot G \cdot M}{rp}

(%i6)

fr0 : 1 − 2 · G · M / r0 ;

1 - \frac{2 \cdot G \cdot M}{r0}

The derivative of the test particle is negative (free falls towards the center)

(%i7)

drm_dt : − sqrt ( 2 · G · M · ( 1 / rm − 1 / r0 ) ) ; /* Eq.[25] */

- (\sqrt{2} \cdot \sqrt{G} \cdot \sqrt{M} \cdot \sqrt{\frac{1}{rm} - \frac{1}{r0}})

(%i8)

dt_drm : 1 / drm_dt ;

- (\frac{1}{\sqrt{2} \cdot \sqrt{G} \cdot \sqrt{M} \cdot \sqrt{\frac{1}{rm} - \frac{1}{r0}}})

(%i9)

δtm : integrate ( dt_drm , rm , r0 , rm ) ;

\frac{\sqrt{{r0}^{2} \cdot rm - r0 \cdot {rm}^{2}} + {r0}^{\frac{3}{2}} \cdot atan (\frac{\sqrt{r0 - rm}}{\sqrt{rm}})}{\sqrt{2} \cdot \sqrt{G} \cdot \sqrt{M}}

(%i10)

δrm : drm_dt · δtm ;

- (\sqrt{\frac{1}{rm} - \frac{1}{r0}} \cdot (\sqrt{{r0}^{2} \cdot rm - r0 \cdot {rm}^{2}} + {r0}^{\frac{3}{2}} \cdot atan (\frac{\sqrt{r0 - rm}}{\sqrt{rm}})))

(%i11)

drp_dt : ( 1 / rp ) · sqrt ( 2 · G · M · ( r0 ^ 2 / rp − rp ^ 2 / r0 ) + rp ^ 2 − r0 ^ 2 ) ; /* Eq.[18] */

\frac{\sqrt{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{rp} - \frac{{rp}^{2}}{r0}) + {rp}^{2} - {r0}^{2}}}{rp}

(%i12)

assume ( ε > 0 , χ12 > 0 ) ;

[ε > 0, χ12 > 0]

χ12 is the ratio (drp/drm)².
Afterwards, we'll unify rp=r0+ε*γ, rm=r0-ε for ε->0+

(%i13)

χ12_a : drp_dt ^ 2 / drm_dt ^ 2 ; /* Eq.[18]²:Eq.[25]² */

\frac{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{rp} - \frac{{rp}^{2}}{r0}) + {rp}^{2} - {r0}^{2}}{2 \cdot G \cdot M \cdot (\frac{1}{rm} - \frac{1}{r0}) \cdot {rp}^{2}}

Initial χ1 guess... Weak Equivalence Principle sugests 1

(%i14)

χ1 : 1 ;

1

χ1²=limit χ12_a(γ) ε->0+ is recursive!
We hope that there's a "fixed point" in the recursive function

(%i15)

χ12_b : subst ( rp = ( r0 + ε · abs ( χ1 ) ) , subst ( rm = ( r0 − ε ) , χ12_a ) ) ;

\frac{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{ε + r0} - \frac{{(ε + r0)}^{2}}{r0}) + {(ε + r0)}^{2} - {r0}^{2}}{2 \cdot G \cdot M \cdot (\frac{1}{r0 - ε} - \frac{1}{r0}) \cdot {(ε + r0)}^{2}}

Find the limit when ε->0+

(%i16)

χ12 : factor ( limit ( χ12_b , ε , 0 , plus ) ) ;

\frac{r0 - 3 \cdot G \cdot M}{G \cdot M}

Remember the result χ12 is the ratio squared.
Iterating, you can see that χ1 converges finally to the value %o16 shown above.
To prove it, we let χ1=(value of χ1²), instead of its squareroot.

(%i17)

χ1 : − χ12 ;

- (\frac{r0 - 3 \cdot G \cdot M}{G \cdot M})

(%i18)

χ12_b : subst ( rp = ( r0 + ε · abs ( χ1 ) ) , subst ( rm = ( r0 − ε ) , χ12_a ) ) ;

\frac{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0} - \frac{{(\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0)}^{2}}{r0}) + {(\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0)}^{2} - {r0}^{2}}{2 \cdot G \cdot M \cdot (\frac{1}{r0 - ε} - \frac{1}{r0}) \cdot {(\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0)}^{2}}

(%i19)

χ12 : factor ( limit ( χ12_b , ε , 0 , plus ) ) ;

\frac{{(r0 - 3 \cdot G \cdot M)}^{2}}{G^{2} \cdot M^{2}}

So, we feed a value and the squareroot of the result is the same value.
It's the solution.
We know γ is negative for r0>3GM.

(%i20)

χ1 : − sqrt ( χ12 ) ;

- (\frac{r0 - 3 \cdot G \cdot M}{G \cdot M})

(%i21)

δrp : χ1 · δrm ;

\frac{(r0 - 3 \cdot G \cdot M) \cdot \sqrt{\frac{1}{rm} - \frac{1}{r0}} \cdot (\sqrt{{r0}^{2} \cdot rm - r0 \cdot {rm}^{2}} + {r0}^{\frac{3}{2}} \cdot atan (\frac{\sqrt{r0 - rm}}{\sqrt{rm}}))}{G \cdot M}

(%i22)

dΘ_dt : ( r0 / rp ^ 2 ) · sqrt ( frp ) ; /* Eq.[31] */

\frac{r0 \cdot \sqrt{1 - \frac{2 \cdot G \cdot M}{rp}}}{{rp}^{2}}

(%i23)

drp_dt : ( 1 / rp ) · sqrt ( 2 · G · M · ( r0 ^ 2 / rp − rp ^ 2 / r0 ) + rp ^ 2 − r0 ^ 2 ) ;

\frac{\sqrt{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{rp} - \frac{{rp}^{2}}{r0}) + {rp}^{2} - {r0}^{2}}}{rp}

(%i24)

dΘ_drp : dΘ_dt / drp_dt ; /* Eq.[31]:Eq.[18] */

\frac{r0 \cdot \sqrt{1 - \frac{2 \cdot G \cdot M}{rp}}}{rp \cdot \sqrt{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{rp} - \frac{{rp}^{2}}{r0}) + {rp}^{2} - {r0}^{2}}}

(%i25)

δΘ : dΘ_drp · δrp ;

\frac{r0 \cdot (r0 - 3 \cdot G \cdot M) \cdot \sqrt{\frac{1}{rm} - \frac{1}{r0}} \cdot (\sqrt{{r0}^{2} \cdot rm - r0 \cdot {rm}^{2}} + {r0}^{\frac{3}{2}} \cdot atan (\frac{\sqrt{r0 - rm}}{\sqrt{rm}})) \cdot \sqrt{1 - \frac{2 \cdot G \cdot M}{rp}}}{G \cdot M \cdot rp \cdot \sqrt{2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{rp} - \frac{{rp}^{2}}{r0}) + {rp}^{2} - {r0}^{2}}}

(%i26)

χ2a : r0 · δΘ ^ 2 / δrm ;

- (\frac{{r0}^{3} \cdot {(r0 - 3 \cdot G \cdot M)}^{2} \cdot \sqrt{\frac{1}{rm} - \frac{1}{r0}} \cdot (\sqrt{{r0}^{2} \cdot rm - r0 \cdot {rm}^{2}} + {r0}^{\frac{3}{2}} \cdot atan (\frac{\sqrt{r0 - rm}}{\sqrt{rm}})) \cdot (1 - \frac{2 \cdot G \cdot M}{rp})}{G^{2} \cdot M^{2} \cdot {rp}^{2} \cdot (2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{rp} - \frac{{rp}^{2}}{r0}) + {rp}^{2} - {r0}^{2})})

(%i27)

χ2b : subst ( rp = ( r0 + ε · abs ( χ1 ) ) , subst ( rm = ( r0 − ε ) , χ2a ) ) ;

- (\frac{{r0}^{3} \cdot {(r0 - 3 \cdot G \cdot M)}^{2} \cdot (\sqrt{{r0}^{2} \cdot (r0 - ε) - r0 \cdot {(r0 - ε)}^{2}} + {r0}^{\frac{3}{2}} \cdot atan (\frac{\sqrt{ε}}{\sqrt{r0 - ε}})) \cdot \sqrt{\frac{1}{r0 - ε} - \frac{1}{r0}} \cdot (1 - \frac{2 \cdot G \cdot M}{\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0})}{G^{2} \cdot M^{2} \cdot {(\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0)}^{2} \cdot (2 \cdot G \cdot M \cdot (\frac{{r0}^{2}}{\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0} - \frac{{(\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0)}^{2}}{r0}) + {(\frac{(r0 - 3 \cdot G \cdot M) \cdot ε}{G \cdot M} + r0)}^{2} - {r0}^{2})})

(%i28)

χ2 : limit ( χ2b , ε , 0 , plus ) ;

2 - \frac{r0}{G \cdot M}

(%i29)

χ0 : ratsimp ( χ2 − χ1 ) ;

- 1

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