Undergrad Favorite Visualization of General Relativity?

[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