I Favorite Visualization of General Relativity?

[Devin Powell]

I am collecting ways to visualize the curvature of spacetime -- and movement of objects affected by gravity -- as per Einstein's GR. Alternatives to the bowling ball / trampoline image so often used in the popular press. Images that show the similarities / differences between Newton & Einstein. That explain how GR addressed Mercury's orbit. Etc.

Do you have a favorite graphic? Thanks!

 

[m4r35n357]

Do you have a favorite graphic? Thanks!

Here is a site with downloadable video that shows travel through a simulated wormhole. I am taking the word "visualize" literally here!

 

[jambaugh]

I'll suggest an alternative of the bowling ball/trampoline image that removes some of the improper analogy. Freeze the trampoline's shape and turn it upside down. Visualize a line of ants walking along the surface and then one's intuition that their path still must bend toward the center to take the shortest path makes clearer the geodesic aspect and removes the artifact of the "baseball rolling down hill since the trampoline has been depressed." which is not a GR effect in the model. An alternative to the ants is a taut string or rubber band held to two points on that surface thereby following geodesic. Something like this:

 

[Devin Powell]

I'll suggest an alternative of the bowling ball/trampoline image that removes some of the improper analogy. Freeze the trampoline's shape and turn it upside down. Visualize a line of ants walking along the surface and then one's intuition that their path still must bend toward the center to take the shortest path makes clearer the geodesic aspect and removes the artifact of the "baseball rolling down hill since the trampoline has been depressed." which is not a GR effect in the model. An alternative to the ants is a taut string or rubber band held to two points on that surface thereby following geodesic. Something like this:
View attachment 235009

Much better than the trampoline! One frustration I have with visualizations like this, though, is that they leave out time. Which seems to leave out the equivalence principle.

 

[PeroK]

Visualize a line of ants walking along the surface and then one's intuition that their path still must bend toward the center to take the shortest path

I'd like to see a graphic that shows that the ants actually take the longest path!

 

[Dale]

One of our members, @A.T. , has some excellent graphics

 

[A.T.]

Alternatives to the bowling ball / trampoline image so often used in the popular press. Images that show the similarities / differences between Newton & Einstein. That explain how GR addressed Mercury's orbit. Etc.

Here are some good illustrations:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime
http://www.relativitet.se/Webtheses/tes.pdf

And here is an animation based on them:

 

[martinbn]

This might be of interest. https://arxiv.org/abs/0708.2483

Visualizing curved spacetime

Rickard Jonsson

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.​

 

[Devin Powell]

Here is a site with downloadable video that shows travel through a simulated wormhole. I am taking the word "visualize" literally here!

Here are some good illustrations:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime
http://www.relativitet.se/Webtheses/tes.pdf

And here a an animation based on them:

Thanks! I own a copy of Lewis Carroll Epstein's book -- so great!

The rolling up of the curved piece of paper (and its subsequent effect on the line of the falling object) makes a lot of sense. How does this compare to the more trumpet-shaped diagram? Is that one more accurate? In pictures:

This one has straight sides when rolled (like a paper cup):

But this one curves like a trumpet (more accurate? ... but harder to roll out flat?):

 

[DrGreg]

Copied from my old post: https://www.physicsforums.com/threa...in-a-gravitational-field.673304/#post-4281670

This is my own non-animated way of looking at it:

• A. Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
  
• B1. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.
  
  B2. Take the flat piece of paper depicted in B1, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B2 shows exactly the same thing as B1, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
  
• C. Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B2. This is the equivalence principle in action: if you zoomed in very close to B2 and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.

 

[pervect]

I am collecting ways to visualize the curvature of spacetime -- and movement of objects affected by gravity -- as per Einstein's GR. Alternatives to the bowling ball / trampoline image so often used in the popular press. Images that show the similarities / differences between Newton & Einstein. That explain how GR addressed Mercury's orbit. Etc.

Do you have a favorite graphic? Thanks!

Well, the first thing that one needs is a graphic representation of space-time. This is well known to be a space-time diagram. This is not terribly hard to imagine, one can imagine time being represented by a time-line, hopefully a familiar exercise from history, with a different time-line of events that happen strictly locally, at each point in space.

This space-time diagram is then the graphic representation of space-time, where every point on the space-time diagram represents an "event" in space-time, an event that occurs at a particular place, and a particular time.

Most illustrations will use diagrams with 1 spatial dimension and 1 time dimension, one can with more work add more spatial dimensions

Then all that curved space-time means is that one draws this space-time diagram on a curved surface. A lot more could be said about curvature, but for introductory purposes I think it's sufficient to say that a plane is flat, and a sphere is not flat (but curved).

While space-time diagrams are not particularly difficult to imagine, and are mentioned in almost all textbooks, I feel a certain amount of resistance when I suggest to posters in the forums that they actually draw one.

Some simple examples to draw would be to start with the space-time diagram of a motionless particle. Then a space-time diagram for a beam of light. Then combine the two, with three motionless particles and two beams of light, to create a light-clock.

The diagrams are much easier to draw on a flat sheet of paper of course, but one could imagine drawing them on a globe. More exotic surfaces can be useful, but a globe (or a section of one) is an easy one to use.

There are several such diagrams in this post of drawing space-time diagrams on various curved surfaces. But before one can draw or interpret such a space-time diagram on a curved surface and gain a useful understanding, one needs to be able to draw a space-time diagram on a flat sheet of paper.

 

[Dale]

While space-time diagrams are not particularly difficult to imagine, and are mentioned in almost all textbooks, I feel a certain amount of resistance when I suggest to posters in the forums that they actually draw one.

Me too, which is weird. Same with free body diagrams, but not with circuit diagrams.

 

[pervect]

One more comment about drawing space-time diagrams on curved surfaces. On a flat plane, a straight line is the shortest distance between two points. On a sphere, a great circle is the shortest distance between two points. For the purposes of illustrating general relativity, one can regard drawing a great circle on a sphere or section of a sphere to be the equivalent of drawing a straight line on a space-time diagram drawn on a flat plane.

On a more general curved surface, one needs to be able to figure out what curve, lying totally within the surface, represents the shortest distance between two points. But I don't wish to go into any of the math required for this. It's much simpler to recommend drawing the diagrams on spheres and using great circles.

This is actually slightly over-simplified, the notion of a "straight line" as being "the curve of shortest distance" eventually becomes too limiting. But it works well enough for a B level introduction.

 

[George Jones]

I am collecting ways to visualize the curvature of spacetime -- and movement of objects affected by gravity -- as per Einstein's GR.

When I think about general relativity, I usually don't use any visualizations.

 

[Daverz]

One of my favorite usages of a visualization in GR is finding the space contribution to the orbital precession by fitting a cone tangent to the Flamm parabaloid and then "cutting and flattening" the cone to find the angle deficit. This is in Rindler's book.

 

[A.T.]

One of my favorite usages of a visualization in GR is finding the space contribution to the orbital precession by fitting a cone tangent to the Flamm parabaloid and then "cutting and flattening" the cone to find the angle deficit. This is in Rindler's book.

This is from Epstein's book:

http://demoweb.physics.ucla.edu/content/10-curved-spacetime

 

[1977ub]

Most people do tend to confuse "curved space-time" with "curved space" (and to not even know about the latter)
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html

 

[George Jones]

Most people do tend to confuse "curved space-time" with "curved space" (and to not even know about the latter)

This is particularly true for standard Friedmann-Lemaitre-Robertson-Walker cosmologies.

 

[PeroK]

Most people do tend to confuse "curved space-time" with "curved space" (and to not even know about the latter)
https://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_massive/index.html

Curved time, to me at least, is considerably more difficult to visualise than curved space.

 

[1977ub]

Curved time, to me at least, is considerably more difficult to visualise than curved space.

Would that not require multiple time axes?

 

[Nugatory]

Would that not require multiple time axes?

Not necessarily. The video above by @A.T. video does quite well with one time axis and one space axis, and the curvature is quite literal - the axes are not straight lines.

 

[1977ub]

Not necessarily. The video above by @A.T. video does quite well with one time axis and one space axis, and the curvature is quite literal - the axes are not straight lines.

The "curvature" of time is not independent of the curvature of space, though. So it is a curvature of space-time.

 

[Orodruin]

The "curvature" of time is not independent of the curvature of space, though. So it is a curvature of space-time.

It actually isn't, it is still flat spacetime (it is drawn on a flat surface), but in curvilinear coordinates that happen to coincide with our regular notions of time and space.

 

[1977ub]

It actually isn't, it is still flat spacetime (it is drawn on a flat surface), but in curvilinear coordinates that happen to coincide with our regular notions of time and space.

Nothing non-euclidean happens until space is added. Time is shown to curve back on itself or anything.

 

[Orodruin]

Nothing non-euclidean happens until space is added. Time is shown to curve back on itself or anything.

You are wrong, it is flat spacetime. It is a coordinate patch that is small enough for curvature to be negligible so I repeat that the illustration does not show curved spacetime. Furthermore, there is nothing Euclidean about spacetime. Spacetime follows Lorentzian geometry, which is fundamentally different from Riemannian geometry.

 

[1977ub]

You are wrong, it is flat spacetime. It is a coordinate patch that is small enough for curvature to be negligible so I repeat that the illustration does not show curved spacetime. Furthermore, there is nothing Euclidean about spacetime. Spacetime follows Lorentzian geometry, which is fundamentally different from Riemannian geometry.

I misunderstood you. I thought we were both referring to the "curvature of time".

 

[Orodruin]

I misunderstood you. I thought we were both referring to the "curvature of time".

There is no such thing. Curvature (or more specifically, intrinsic curvature, which is the kind of curvature we discuss in relation to GR) requires at least two dimensions.

 

[1977ub]

There is no such thing. Curvature (or more specifically, intrinsic curvature, which is the kind of curvature we discuss in relation to GR) requires at least two dimensions.

see #19

 

[Orodruin]

see #19

It is not an issue of visualisation. It is about curvature of a one-dimensional manifold not making any sense.

 

[Dale]

It is about curvature of a one-dimensional manifold not making any sense.

To expand on this, one of the measures of curvature is the sum of the interior angles of a triangle, you cannot form a triangle in a 1D manifold. Another indication of curvature is the difference in a vector which is parallel transported from one point to another through different paths, but in 1D there is only 1 path. Similarly with all things associated with intrinsic curvature.

 

[Orodruin]

To expand on this, one of the measures of curvature is the sum of the interior angles of a triangle, you cannot form a triangle in a 1D manifold. Another indication of curvature is the difference in a vector which is parallel transported from one point to another through different paths, but in 1D there is only 1 path. Similarly with all things associated with intrinsic curvature.

To expand on the expansion and start counting degrees of freedom, the curvature tensor has $n^2(n^2-1)/12$ independent components in $n$ dimensions. For $n=1$ this evaluates to zero.

 

[Dale]

To expand on the expansion

Accelerated expansion?

 

[Devin Powell]

Question about this: Would B2 be equivalent to Newton? What exactly does the extra curvature in C (trumpet shape instead of coffee cup) signify? Thanks!

Copied from my old post: https://www.physicsforums.com/threa...in-a-gravitational-field.673304/#post-4281670

This is my own non-animated way of looking at it:

View attachment 235117

• A. Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
  
• B1. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.
  
  B2. Take the flat piece of paper depicted in B1, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B2 shows exactly the same thing as B1, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
  
• C. Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B2. This is the equivalence principle in action: if you zoomed in very close to B2 and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.

 

[Orodruin]

Question about this: Would B2 be equivalent to Newton? What exactly does the extra curvature in C (trumpet shape instead of coffee cup) signify? Thanks!

$B_1$ and $B_2$ are completely equivalent and correspond to a non-inertial frame on flat (i.e., Minkowski) spacetime. In $C$ there is some actual spacetime curvature.

 

[Devin Powell]

If there is no spacetime curvature in B1 and B2, wouldn't that mean there is no gravity? And if there's no gravity, how can an object in B1 and B2 still fall downward, as per Lewis Carroll Epstein's diagram of a falling object?

$B_1$ and $B_2$ are completely equivalent and correspond to a non-inertial frame on flat (i.e., Minkowski) spacetime. In $C$ there is some actual spacetime curvature.

$B_1$ and $B_2$ are completely equivalent and correspond to a non-inertial frame on flat (i.e., Minkowski) spacetime. In $C$ there is some actual spacetime curvature.

 

[Orodruin]

If there is no spacetime curvature in B1 and B2, wouldn't that mean there is no gravity?

Gravity and acceleration are locally indistinguishable. It would be more appropriate to say that there are not tidal forces.

 

[A.T.]

If there is no spacetime curvature in B1 and B2, wouldn't that mean there is no gravity?

No, it means that there is no tidal effect. B1 and B2 represent an uniform gravitational field. C represents an non-uniform gravitational field.

 

[PAllen]

Note the waves in the following, must be gravitational, due to Einstein :

https://www.google.com/search?safe=.....1...33i299.3ec-iYgiDeU#imgrc=0ZG6HU3kReBiiM:

 

[Orodruin]

Note the waves in the following, must be gravitational, due to Einstein:

https://www.google.com/search?safe=.....1...33i299.3ec-iYgiDeU#imgrc=0ZG6HU3kReBiiM:

For once I get to point out that those are Einstein's gravity waves, not gravitational waves, instead of the other way around ...

 

[jambaugh]

Much better than the trampoline! One frustration I have with visualizations like this, though, is that they leave out time. Which seems to leave out the equivalence principle.

Yes but you cannot properly embed the indefinite geometry of space-time into an intuitive picture which exists in our definite geometry of curved surfaces and such. There is a trick I worked up in Special relativity where you switch proper time and coordinate time to get an Euclidean metric structure. Problem is that points on the picture can represent events occurring at distinct times. It is useful for demonstrating the resolution of the Twin's paradox.

So you start with a standard coordinate graph but you label the axes x and \tau for coordinate spatial position and proper time. Draw a curve representing an accelerating observers world line, say, starting at the origin but keeping it monotone-non-decreasing in the tau direction. (No fair letting your observer's proper time run backward.)

Now the coordinate time can be calculated as the arclength since: (in c=1 units) dt^2 = dx^2 + d\tau^2.

But you must be careful with assumptions in this system. Two observers are at the same space-time event and thus can causally communicate immediately with each other if they are at the same x coordinate and at the same distance along their arc-length.

What's nice about this picture is that there's no "well sorta" the quantitative effects are exactly represented and not qualitatively analogized. The problem is that the graph points are not single space-time event points so you can't then go and invoke GR by curving the surface.

 

[A.T.]

There is a trick I worked up in Special relativity where you switch proper time and coordinate time to get an Euclidean metric structure. .

This is also the approach Epstein uses in his book mentioned above. The space-proper-time diagrams for both twins (traveler has constant acceleration) would look like this:

Here is a comparison of Minkowski and Epstein diagrams for the 3 inertial frames of the twins (traveler has constant speed with instantaneous turn around):
http://www.adamtoons.de/physics/twins.html