# Plotting space-time curvatures as curved 2D-surfaces

• TheMan112
In summary, the figure attached shows a plot of the volume factor versus spatial position using the Alcubierre metric. I'll work through it with you in Maple starting tomorrow.
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.

#### Attachments

• Alcubierre.png
27.8 KB · Views: 705
Look here under Matlab:
http://fy.chalmers.se/~rico/

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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.

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

TheMan112 said:
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.

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.

George Jones said:
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

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

Explanation to follow.

Code:
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]);

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Thanks, looking forward to it. ;)

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.

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## 1. What is space-time curvature?

Space-time curvature is a phenomenon in which the presence of mass or energy causes the fabric of space and time to bend, resulting in the distortion of the path of objects moving through it.

## 2. How is space-time curvature measured?

Space-time curvature can be measured using the mathematical concept of curvature, which is a function that describes how much a space-time surface deviates from being flat. This is typically done using tools such as the Riemann curvature tensor or the Einstein field equations.

## 3. Why are space-time curvatures often represented using curved 2D surfaces?

Curved 2D surfaces, such as spheres or cones, are often used to represent space-time curvature because they provide a visual representation of how space and time can be distorted by the presence of mass or energy. This allows for a better understanding of concepts such as gravitational lensing and the curvature of space-time around massive objects.

## 4. Can space-time curvature be observed in everyday life?

Yes, space-time curvature can be observed in everyday life through the effects of gravity. For example, the curvature of space-time around the Earth is what keeps objects on the surface from flying off into space. Other observable effects of space-time curvature include the bending of light around massive objects and the slowing down of time in strong gravitational fields.

## 5. How does plotting space-time curvatures aid in understanding the universe?

Plotting space-time curvatures allows scientists to visualize and study the effects of gravity on the fabric of space and time. This not only helps us understand the behavior of massive objects in the universe, but also provides insights into the fundamental laws of physics and the structure of the universe itself.

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