Plotting space-time curvatures as curved 2D-surfaces

[TheMan112]

What would be the easiest way to plot a space-time curvature from a metric as a curved 2D surface. For example the Alcubierre metric would be plotted with the result as in the attached image.

The mathematical programs I have at my disposal right now are Maple, Matlab and Mathematica. I'm much more used to using the two former ones though.

 

[A.T.]

Look here under Matlab:
http://fy.chalmers.se/~rico/

 

[TheMan112]

Although Jonsson's scripts are quite impressive, I was thinking more about doing it from scratch with only built-in functions in the program, it's not as much about generating a graph as it is me learning how to do it.

 

[TheMan112]

Should the plot be based on a Riemann tensor, or can I just plot the metric itself?

 

[George Jones]

Should the plot be based on a Riemann tensor, or can I just plot the metric itself?

The figure you attached is a plot of the volume factor versus spatial position. I'll work through it with you in Maple starting tomorrow.

 

[TheMan112]

Thanks a lot, I can't seem to be able to find many resources online on how to actually plot space-time curvatures. Is it that simple, that it should be self-obvious? It just isn't to me, so that would be very kind of you.

 

[robousy]

The figure you attached is a plot of the volume factor versus spatial position. I'll work through it with you in Maple starting tomorrow.

I'd be interested in seeing this develop George. I've plotted the Alcubierre metric before, but was not really sure why certain things were done.

Looking forward to seeing this!

Richard

 

[George Jones]

The Maple code below reproduces the famous Alcubierre warp drive plot.

Explanation to follow.

f := r -> (tanh(sigma*(r + R)) - tanh(sigma*(r - R)))/(2*tanh(sigma*R));
g := D(f);
sigma := 8; R := 1; vs := 1;
plot3d(vs*x/sqrt(x^2 + y^2)*g(sqrt(x^2 + y^2)), x=-2..2, y=-2..2, style = PATCH, grid=[100,100]);

 

[TheMan112]

Thanks, looking forward to it. ;)

 

[robousy]

George,

I have played with this in the past and found the following code gives a nice 'warp drive' in matlab.

--------------------------



[x, y] = meshgrid([-10:.3:10],[-10:.3:10]);



for i=1:length(x);
for j=1:length(x);
z(i,j)=-1*(tanh (2*sqrt(abs(x(i,j)^2+y(i,j)^2-16)-5)+3)-tanh(2*sqrt(abs(x(i,j)^2+y(i,j)^2-16)-5)-3))...
*tanh(6)*x(i,j);
end

end

mesh(x,y,z)
axis([-10 10 -10 10 -10 10 -10 10])
view([158,26])
colormap(hsv)

------------------------------

Major difference being that I multiply by the (2*tanh(sigma*R)) term instead of dividing.

Interested to hear any thoughts on this...

Also, there is a statement in the original Alcubierre paper, "Since the 3-geometry of the hypersurfaces is flat, the information about the curvature
of spacetime will be contained in the extrinsic curvature tensor [itex]K_{ij}[/tex]

I can't find a huge amount on this tensor in any of my books or online. Do you think you could give an explanation?

Looking forward to hearing your thoughts George, I've never met any physicists who know Alcubierres paper very well.