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authorbnewbold <bnewbold@eta.mit.edu>2009-03-31 17:44:02 -0400
committerbnewbold <bnewbold@eta.mit.edu>2009-03-31 17:44:02 -0400
commit8bef4aef6ee30e1c309c636b57a9baf3c912f999 (patch)
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parent0947416bdba01d9b3d3fcebb7d44268d9305866e (diff)
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anomalous precession work
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+
+; based off the development in ch6.2 of Reflections on Relativity at
+; mathpages.com
+;
+
+;========================================================================
+; first prepare a schwartzchild
+; definitions are in calculus/manifold.scm
+
+#| UNUSED see orbital-plane below
+
+(install-coordinates spacetime-rect (up 't 'x 'y 'z))
+(pp (spacetime-rect 'coordinate-prototype))
+; #(t x y z)
+
+(install-coordinates spacetime-spher (up 't 'r 'theta 'phi))
+(pp (spacetime-spher 'coordinate-prototype))
+; #(t r phi theta)
+
+(spacetime-spher 'check-coordinates)
+
+(define dT (1form-field->tensor-field dt))
+
+(define dX (1form-field->tensor-field dx))
+(define dY (1form-field->tensor-field dy))
+(define dZ (1form-field->tensor-field dz))
+
+(define dR (1form-field->tensor-field dr))
+(define dTheta (1form-field->tensor-field dtheta))
+(define dPhi (1form-field->tensor-field dphi))
+
+(define rect-event ((spacetime-rect '->point) (up 't0 'x0 'y0 'z0)))
+(define spher-event ((spacetime-rect '->point) (up 't0 'r0 'theta0 'phi0)))
+|#
+
+; for orbital plane: (tau,r,phi)
+(attach-coordinate-system 'spacetime-polar 'origin R^n-type
+ (lambda (manifold)
+ (define (me m)
+ (case m
+ ((check-coords)
+ (lambda (coords)
+ (and (up? coords)
+ (fix:= (s:dimension coords) 3))))
+ ((coords->point)
+ (lambda (coords)
+ (if (and (up? coords)
+ (fix:= (s:dimension coords) 3))
+ (let ((t (ref coords 0))
+ (r (ref coords 1))
+ (phi (ref coords 2)))
+ (make-manifold-point
+ (up t
+ (* r (cos phi))
+ (* r (sin phi)))
+ manifold
+ me
+ coords))
+ (error "Bad coordinates: spacetime-polar"
+ coords))))
+ ((check-point)
+ (lambda (point)
+ (my-manifold-point? point manifold)))
+ ((point->coords)
+ (lambda (point)
+ (assert ((me 'check-point) point) "Bad point assertion: spacetime-polar")
+ (get-coordinates point me
+ (lambda ()
+ (let ((prep (manifold-point-representation point)))
+ (if (and (up? prep) (fix:= (s:dimension prep) 3))
+ (let ((t (ref prep 0))
+ (x (ref prep 1))
+ (y (ref prep 2)))
+ (let ((r (sqrt (+ (square x) (square y)))))
+ (up t
+ r
+ (atan y x))))
+ (error "Bad point: spacetime-polar" point)))))))
+ (else (error "Bad message: coordinate-transformer" m))))
+ me))
+
+(define orbital-plane (make-manifold R^n-type 3))
+(define orbital-plane-rect (coordinate-system-at 'rectangular 'origin R3))
+(define orbital-plane-polar (coordinate-system-at 'spacetime-polar 'origin R3))
+
+(install-coordinates orbital-plane-rect (up 'tau 'x 'y))
+(install-coordinates orbital-plane-polar (up 'tau 'r 'phi))
+
+(install-coordinates the-real-line (up 'k))
+
+(define dT (1form-field->tensor-field dtau))
+
+(define dX (1form-field->tensor-field dx))
+(define dY (1form-field->tensor-field dy))
+
+(define dR (1form-field->tensor-field dr))
+(define dPhi (1form-field->tensor-field dphi))
+
+(define rect-event ((orbital-plane-rect '->point) (up 'tau0 'x0 'y0)))
+(define polar-event ((orbital-plane-polar '->point) (up 'tau0 'r0 'phi0)))
+
+(define (Schwartzchild-plane-metric M)
+ (+ (* (/ (- r (* 2 M)) r) (square dT))
+ (* -1 (/ r (- r (* 2 M))) (square dR))
+ (* -1 (square r) (square dPhi))))
+
+(define S-p-metric (Schwartzchild-plane-metric 'M))
+
+;(pe (coordinate-system->basis orbital-plane-polar))
+
+(define Gamma
+ (Christoffel S-p-metric orbital-plane-polar))
+
+#|
+
+(show-time
+ (lambda ()
+ (pec ((tensor-field->coefficient-structure Gamma orbital-plane-polar)
+ polar-event))))
+|#
+
+(begin
+ (define orbit-Cs-2
+ (metric->Christoffel-2 S-p-metric
+ (coordinate-system->basis orbital-plane-polar)))
+ (define orbit-Cs-1
+ (metric->Christoffel-1 S-p-metric
+ (coordinate-system->basis orbital-plane-polar)))
+ (se ((Christoffel->symbols orbit-Cs-2)
+ ((orbital-plane-polar '->point) (up 'tau0 'r0 'phi0)))))
+
+(install-coordinates the-real-line (up 'k))
+
+(define test-path
+ (compose (orbital-plane-polar '->point)
+ (up (literal-function 'alpha)
+ (literal-function 'beta)
+ (literal-function 'gamma))
+ (the-real-line '->coords)))
+
+;(se ((orbital-plane-polar '->coords) (test-path ((the-real-line '->point) 'k))))
+
+;((orbital-plane-polar '->point) ((orbital-plane-polar '->coords) (test-path ((the-real-line '->point) 'k))))
+
+(define polar-Cartan-over-path
+ (Christoffel->Cartan-over-map orbit-Cs-1 test-path))
+
+(((((covariant-derivative polar-Cartan-over-path) d/dk)
+ ((differential test-path) d/dk))
+ (orbital-plane-polar '->coords))
+ ((the-real-line '->point) 'k))
+
+
+(((((covariant-derivative polar-Cartan-over-path) d/dk)
+ ((differential test-path) d/dk))
+ (orbital-plane-polar '->coords))
+ ((the-real-line '->point) 'k))
+
+
+(se ((orbital-plane-polar '->coords) (test-path ((the-real-line '->point) 't))))
+
+(define polar-Cartan (Christoffel->Cartan orbit-Cs-1))
+
+
+
+
+(define basis-over-path
+ (basis->basis-over-map test-path
+ (coordinate-system->basis orbital-plane-polar)))
+
+;(basis-over-path (test-path ((the-real-line '->point) 'k)))
+
+(define w
+ (basis-components->vector-field
+ (up (compose (literal-function 'w0)
+ (the-real-line '->coords))
+ (compose (literal-function 'w1)
+ (the-real-line '->coords))
+ (compose (literal-function 'w2)
+ (the-real-line '->coords)))
+ (basis->vector-basis basis-over-path)))
+
+(pe
+ (s:map/r
+ (lambda (omega)
+ ((omega
+ (((covariant-derivative polar-Cartan-over-path)
+ d/dt)
+ w))
+ ((the-real-line '->point) 'k)))
+ (basis->1form-basis basis-over-path)))
+
+(((basis->1form-basis basis-over-path)
+ (((covariant-derivative polar-Cartan-over-path) d/dk)
+ ((differential test-path) d/dk))
+ (orbital-plane-polar '->coords))
+ ((the-real-line '->point) 'k))
+
+
+
+
+
+
+
+;===========================================================
+; for mercury, [TODO: cite these]
+; precesion: ~43.92 arcsec/century
+; ~0.103 arcsec/rev
+; eccentricity k: 0.2060
+; semimajor axis: 57.9 x 10^6 km
+
+
+
+
+