Gravity – Gravité

Table of Contents

Notes on General Relativity (GR) and Gravity – includes notes on the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus International Winter School on Gravity and Light

Notes (LaTeX format) on General Relativity (GR) and Gravity – includes notes on the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus International Winter School on Gravity and Light; link to github, will be the most up-to-date and permanent

Notes (LaTeX format) on General Relativity (GR) and Gravity – includes notes on the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus International Winter School on Gravity and Light; link to github, will be the most up-to-date and permanent

Notes (pdf format) on General Relativity (GR) and Gravity – includes notes on the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus International Winter School on Gravity and Light; link to github, will be the most up-to-date and permanent

Lecture # Lecture name Lecture link Tutorial # Tutorial name Tutorial video link Tutorial sheet (pdf)
3 Lecture 3: Multilinear Algebra (International Winter School on Gravity and Light 2015) https://youtu.be/mbv3T15nWq0 3 Tutorial 3: Multilinear Algebra (International Winter School on Gravity and Light 2015) https://youtu.be/5oeWX3NUhMA tensors_neu.pdf

NOT Updated: NOTES ON GENERAL RELATIVITY (GR) AND GRAVITY in wide, “grande” format; includes notes on the Central Lectures given by Dr. Frederic P. Schuller for the WE Heraeus International Winter School on Gravity and Light from wordpress NOT UPDATED

gravite github repository

github.io page for Gravite

github repository for Gravite

Euclidean space as a Manifold – R^2,R^3,R^n

Using SageManifolds, Euclidean space, R^2, R^3, and R^n, is implemented as a manifold.

Rn.sage – Euclidean spaces as manifolds using sagemanifolds

Features

  • R^2,R^3,R^n as a manifold, with a chart atlas
sage: load(‘‘Rn.sage’’)  
sage: R2eg = R2() 
sage: R3eg = R3() 
sage: R4 = Rn(4) 
sage: R2eg.M.atlas()
[Chart (R2, (x, y)), Chart (U, (x, y)), Chart (U, (r, ph))]
sage: R3eg.M.atlas()
[Chart (R3, (x, y, z)),
 Chart (U, (x, y, z)),
 Chart (U, (rh, th, ph)),
 Chart (U, (r, phi, zc))]
sage: R4.M.atlas()
[Chart (R4, (x1, x2, x3, x4)),
 Chart (U, (x1, x2, x3, x4)),
 Chart (U, (rh, th1, th2, ph)),
 Chart (U, (r, the1, phi, z))]
  • (carefully) define a spherical coordinate and cylindrical coordinate chart on Euclidean spaces, e.g.
sage: R2eg.transit_sph_to_cart.display() 
x = r*cos(ph)
y = r*sin(ph)
sage: R3eg.transit_sph_to_cart.display() 
x = rh*cos(ph)*sin(th)
y = rh*sin(ph)*sin(th)
z = rh*cos(th)
sage: R3eg.transit_cyl_to_cart.display() 
x = r*cos(phi)
y = r*sin(phi)
z = zc
sage: R4.transit_sph_to_cart.display() 
x1 = rh*cos(ph)*sin(th1)*sin(th2)
x2 = rh*sin(ph)*sin(th1)*sin(th2)
x3 = rh*cos(th2)*sin(th1)
x4 = rh*cos(th1)
  • calculate the Jacobian!
sage: to_orthonormal2 , e2, Jacobians2 = R2eg.make_orthon_frames(R2eg.sph_ch) 
sage: Jacobians2[0].inverse()[:,R2eg.sph_ch]
[ cos(ph) -r*sin(ph)]
[ sin(ph) r*cos(ph)]
sage: to_orthonormal3sph, e3sph, Jacobians3sph = R3eg.make_orthon_frames(R3eg.sph_ch) 
sage: to_orthonormal3cyl, e3cyl, Jacobians3cyl = R3eg.make_orthon_frames(R3eg.cyl_ch) 
sage: Jacobians3sph[0].inverse()[:,R3eg.sph_ch]
[ cos(ph)*sin(th) rh*cos(ph)*cos(th) -rh*sin(ph)*sin(th)]
[ sin(ph)*sin(th) rh*cos(th)*sin(ph) rh*cos(ph)*sin(th) ]
[ cos(th)         -rh*sin(th)        0                  ]
sage: Jacobians3cyl[0].inverse()[:,R3eg.cyl_ch]
[ cos(phi) -r*sin(phi) 0]
[ sin(phi) r*cos(phi) 0] 
[ 0 0 1]
  • equip the Euclidean space manifold with a metric g and calculate the metric automatically:
sage: R2eg.equip_metric() 
sage: R3eg.equip_metric() 
sage: R4.equip_metric()

sage: R2eg.g.display(R2eg.sph_ch.frame(),R2eg.sph_ch)
g = dr*dr + r^2 dph*dph
sage: R3eg.g.display(R3eg.sph_ch.frame(),R3eg.sph_ch)
g = drh*drh + rh^2 dth*dth + rh^2*sin(th)^2 dph*dph
sage: R3eg.g.display(R3eg.cyl_ch.frame(),R3eg.cyl_ch)
g = dr*dr + r^2 dphi*dphi + dzc*dzc
sage: R4.g.display(R4.sph_ch.frame(),R4.sph_ch)
g = drh*drh + rh^2 dth1*dth1 + rh^2*sin(th1)^2 dth2*dth2 + rh^2*sin(th1)^2*sin(th2)^2 dph*dph
sage: R4.g.display(R4.cyl_ch.frame(),R4.cyl_ch)
g = dr*dr + r^2 dthe1*dthe1 + r^2*sin(the1)^2 dphi*dphi + dz*dz
  • Calculate the so-called orthonormal non-coordinate basis vectors in terms of the (local) coordinate basis vectors, showing clearly and distinctively the difference between the two (concepts)
sage: e2[1].display( R2eg.sph_ch.frame(), R2eg.sph_ch)
e_1 = d/dr
sage: e2[2].display( R2eg.sph_ch.frame(), R2eg.sph_ch)
e_2 = 1/r d/dph
sage: for i in range(1,3+1):                                                         
    e3sph[i].display( R3eg.sph_ch.frame(), R3eg.sph_ch )
....:     
e_1 = d/drh
e_2 = 1/rh d/dth
e_3 = 1/(rh*sin(th)) d/dph
sage: for i in range(1,3+1):
    e3cyl[i].display( R3eg.cyl_ch.frame(), R3eg.cyl_ch )
....:     
e_1 = d/dr
e_2 = 1/r d/dphi
e_3 = d/dzc

Rn_examples_01Rn_examples_02Rn_examples_03

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