my_screen_name's profile - activity

@eric_g: Thanks for the further update, and I'm glad to learn this will be fixed in the next release.

eric_g: Thanks for the answer. I assume it is correct, but it seems a surprising situation. Since $\psi_3\circ\psi_1^{-1} = (\psi_3\circ\psi_2^{-1})\circ(\psi_2\circ\psi_1^{-1})$, I don't see why the inverse maps should need to be defined. In particular, in the example I'm thinking of (converting standard Schwarzschild coordinates to Regge-Wheeler tortoise coordinates, then Eddington-Finkelstein null coordinates, and then Kruskal), it is impossible to invert the first coordinate transform. Does this mean that, while I can directly define a transition map from the first coordinate system to the third, I can't define it by going from the first to the second and then the second to the third? This seems like an unnecessary restriction.

rburing: I would also like clearer error message.

In sagemanifolds, I would like to change from an initial chart to a second and then, from the second to the third. My attempts fail. Is this my fault or a sagemanifolds problem? (Apologies if I already asked this.)

Here is a minimal (not) working example. I run the following file:

theManifold = Manifold(1, 'M', r'\mathcal{M}')
theOpenSet = theManifold.open_subset('U')

theFirstChart.<x> = theOpenSet.chart(r'x')

g = theManifold.riemannian_metric('g')
g[0,0] =  1

show(g.display())

theSecondChart.<y> = theOpenSet.chart(r'y')
ChangeFirst_to_Second = theFirstChart.transition_map(theSecondChart,[x])

show(g.display(theSecondChart.frame()))

theThirdChart.<z> = theOpenSet.chart(r'z')
ChangeSecond_to_Third = theSecondChart.transition_map(theThirdChart,[y])

show(g.display(theThirdChart.frame()))

The output is

g=dx⊗dx
g=dy⊗dy

and then a few screens of error messages, of which the most clear is "no common chart for the multiplication".

I'm probably doing something obviously wrong, but I'd appreciate help with the following.

I load the following file

theManifold = Manifold(3, 'M', r'\mathcal{M}')
theOpenSet = theManifold.open_subset('theOpenSet')
CartesianChart.<x,y,z> = theOpenSet.chart(r'x y z')
g = theManifold.lorentzian_metric('g')
g[0,0] = 1
g[1,1] = 1
g[2,2] = 1
show(g.display())
sphericalChart.<r,th,ph> = theOpenSet.chart(r'r th:(0,pi):\theta ph:(0,2*pi):\phi')

spherical_to_Cartesian = sphericalChart.transition_map(CartesianChart,[r*sin(th)*sin(ph),r*sin(th)*cos(ph),r*cos(th)])
Cartesian_to_spherical = spherical_to_Cartesian.inverse()

Sage then gives the error message "ValueError: no solution found; use set_inverse() to set the inverse manually".

If I replace the final two lines with

 Cartesian_to_spherical = CartesianChart.transition_map(sphericalChart[sqrt(x^2+y^2+z^2),arctan(z/sqrt(x^2+y^2)),arctan(y/x)])
 spherical_to_Cartesian = Cartesian_to_spherical.inverse()

then Sage gives the following unable to make sense of Maxima expression '[if((-pi/2<parg(_sage_var_xxxx0))and(-pi 2<parg(-_sage_var_xxxx0="" sqrt((4<em="">e^(2I_SAGE_VAR_xxxx1))/(e^(4I*_SA ... [dozens of lines of similar] ... tan(_SAGE_VAR_xxxx1)^2+1)],union())]' in Sage.

I am aware that Sagemanifolds has native commands for using spherical coordinates. I'm attempting to work on a more complicated manifold, and this is is a minimal working example to demonstrate my problems.

I think what you want is

e0 = M.one_form()
e0[0] = sqrt(1-2*m/r)

This gives a unit timelike one-form. Of course, relativity being relativity, there are infinitely many other choices of unit timelike vector. This becomes a more serious issue when working in Kerr.

When I set the defaults based for theOpenSet, I get (1/2) y dx⊗dx. If I use

theManifold.set_default_frame(secondChart.frame()
theOpenSet.set_default_chart(secondChart)

then I do get (1/8) y dy ⊗ dy, as desired. I don't see the logic, but thanks for getting this to work.

The Bel decomposition does not appear in the sagemanifolds reference document, so I assume it is a default command. Nonetheless, it is fairly easy to implement. Assuming that one already has a metric g and a 1-form e0 (perhaps from the 0th element of the coordinate coframe)

theVolumeForm = g.volume_form()
theStarForTwoForms = theVolumeForm.up(g,0).up(g,1)

theRiemann = g.riemann()
theRiemannAllDown = theRiemann.down(g,0)
theStarRiemann = theStarForTwoForms['^{ef}_{ab}']*theRiemannAllDown['_{efcd}']
theStarRiemannStar = theStarForTwoForms['^{ef}_{cd}']*theStarRiemann['_{abef}']

theElectrograviticTensor = (theRiemannAllDown['_{abcd}']*(e0.up(g))['^d'])['_{abc}']*(e0.up(g))['^b']
theMagnetograviticTensor = (theStarRiemann['_{abcd}']*(e0.up(g))['^d'])['_{abc}']*(e0.up(g))['^b']
theTopograviticTensor = (theStarRiemannStar['_{abcd}']*(e0.up(g))['^d'])['_{abc}']*(e0.up(g))['^b']

I have probably mangled some signs in the choice of order of raising and contracting indices on the volume form.

(Some tricks to be aware of: When using up and down, since sagemanifolds stores all contravariant indices first, for a fully contravariant tensor, raising index 1 and then lowering index 1 gives an error since the raised index is automatically moved to index 0. The * operation on tensor only works on a pair of tensors. The latex notation for indices (e.g. ['^{ab}']) can be used to take products and contract tensors, but the latex indexing is lost once the * operation is complete. Thus for a product of three indices, one needs to open a parenthesis, apply the indices to get the correct contraction and product for the first two tensors, close the parenthesis, apply a new set of indices, and then multiply by the third tensor.)

Regarding (2), I couldn't reproduce this. In particular, when I create a manifold (and define some coordinates on it) and then run

p = theManifold.point(name='p')

TpM = theManifold.tangent_space(p)

v = TpM.an_element()

v.dipslay(),

I get the error message

ValueError: no basis could be found for computing the components in the None

I load the following file

theManifold = Manifold(1,'M',r'\mathcal{M}')
theManifold = Manifold(1,'M',r'\mathcal{M}')

theOpenSet = theManifold.open_subset('U')

firstChart.<x> = theOpenSet.chart(r'x')

g=theManifold.riemannian_metric('g')
g[0,0] = x

show("g with first chart")
show(g.display())

secondChart.<y> = theOpenSet.chart('y')

first_to_second = firstChart.transition_map(secondChart,[2*x])
second_to_first = first_to_second.inverse()

show("g with secondChart.frame() and secondChart as arguments to display")
show(g.display(secondChart.frame(),secondChart))

theManifold.set_default_frame(secondChart.frame())
theManifold.set_default_chart(secondChart)

show("g with secondChart.frame() and secondChart as defaults")
show(g.display())

The output is

𝚐 𝚠𝚒𝚝𝚑 𝚏𝚒𝚛𝚜𝚝 𝚌𝚑𝚊𝚛𝚝
g=x dx⊗dx
𝚐 𝚠𝚒𝚝𝚑 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝.𝚏𝚛𝚊𝚖𝚎() 𝚊𝚗𝚍 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝 𝚊𝚜 𝚊𝚛𝚐𝚞𝚖𝚎𝚗𝚝𝚜 𝚝𝚘 𝚍𝚒𝚜𝚙𝚕𝚊𝚢
g=1/8 y dy⊗dy
𝚐 𝚠𝚒𝚝𝚑 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝.𝚏𝚛𝚊𝚖𝚎() 𝚊𝚗𝚍 𝚜𝚎𝚌𝚘𝚗𝚍𝙲𝚑𝚊𝚛𝚝 𝚊𝚜 𝚍𝚎𝚏𝚊𝚞𝚕𝚝𝚜
g=1/4 x dy⊗dy

In particular, when the secondChart and secondChart.frame() have been set as defaults, the metric g is expanded with respect to the second frame, but the components are still expressed in the first coordinates.

Is there a way to set defaults so that g appears as 1/8 y dy ⊗dy?

I upgraded to sagemath 8.6 (the latest version with binaries available for my system), and this now works. Thanks for pointing out to use newer versions.

The output of

show((1-cos(x)^2.simplify_trig())
show((cos(x)^2-1).simplify_trig())

is

sin(x)^2
cos(x)^2-1.

Even worse, the result of

 (sin(x)^2/(1-cos(x)^2)).simplify().simplify_trig()

is

 -(sin(x)^2/(cos(x)^2-1)

when it should be 1. Is there anyway to get sagemath to simplify trigonometric expressions more intelligently?