What Does Curved Space Really Look Like? A Visual Guide to General Relativity

[Denton]

It is just something i can't get my head around, how to picture how space is curved around large masses. Is space like uniform jelly with it being denser around the planets, or is it like a void where curvature flattens out the farther away from the gravity source. Id like to hear your interpretation of the look of curved space.

-Denton.

 

[losang]

I have come to two conclusions. Either spacetime is a substance and "stretches" in the presence of mass or there is no spacetime in the substantial sense.

Spacetime and mass are linked in that matter is formed from this spacetime substance. When you condense enough of this substance in a small volume you get massive objects. This condensing has the effect of stretching the neighboring spacetime and you get curvature. The tension in the substance decreases with distance from the mass so the curvature does as well.

Another possibility is that spacetime is not a "substance". It is merely that the equations of motion change in the presence of mass such that inertial motion follows curved paths. Why is spacetime flat in the absense of matter? What we mean by spacetime may just be the geometry of the equations of motion.

 

[Naty1]

The bowling ball sunk on a rubber sheet analogy works for me.

It's no more difficult than trying to imagine what time dilation or length contraction "looks like". And once you get into 11 spacetime dimensions such "curving" of space is impossible for mere mortals to visualize...nor can you "see" electromagnetic radiation except in the very narrow visible light spectrum...alas, we are rather imperfect...

 

[Dale]

I always use geometry on a sphere to understand curved spacetime. On a sphere the sum of the interior angles of a triangle are greater than 180º, straight lines can intersect in two points, and straight lines that are parallel in one spot are not parallel in other spots. These are essentially the kinds of features that are important for understanding curved spacetime in GR.

 

[starkind]

The idea of a bowling ball on a rubber sheet doesn't work for me any more. The sheet has two important dimensions and we are really looking at a three dimensional space. That is, gravity acts on three dimensions, not two. So the sheet idea is too limited. Nowadays instead of curvature I try to think of density.

 

[Legion81]

I usually think of space as a 3D grid of cubes having units of (1,1,1). Near a large mass it is like sticking a (2,2,2) cube into the small one. So each component has to bend for the ends to line up (make a D shape). The extra length of each component can't fit in and have to switch out with parts that are in the cube already (spills over into time dimension), making time slower. Thats how it "looks" in my head.

 

[losang]

None of you have really addressed his question. He's asking about the ontology of spacetime. All the replys are suggestions on the technology of spacetime; assuming spacetime is like this here is how i visualize curved space. He's not asking this. He wants to know what is being curved. Read my post and you'll see (its the 2nd one). It's original as far as i know.

 

[Rafi]

Just don’t try to. There is a limit to what our brain is able to imagine. Space-time curvature is out of its scope. Richard Dawkins wrote in "The Blind Watchmaker":

"Just as our eyes can see only that narrow band of electromagnetic frequencies that natural selection equipped our ancestors to see, so our brains are built to cope with narrow bands of sizes and times."

We can imagine only three dimensions. We need one dimension in which to curve our space, so the maximum dimensions of curved space that we can imagine is two (curved surface). But surface curvature is atrophied. It has no Ricci or Weyl curvature. We can use 2D models like the rubber sheet to help us understand things, but we can really handle time-space curvature only with mathematical tools.

You can read a little more about it in my article here:

http://rafimoor.com/english/GRE2.htm#The_Geometric_Meaning_of_Curvature_Tensors

 

[Ookke]

I think 4-dimensional timespace should be thought as a purely mathematical construction and any imagining of it is actually harmful. It only leads to an interpretation that time is a spatial dimension, which it is not, not even in some sense.

 

[Fra]

Even though there are some visual ideas of ants walking on manifolds embedded in higher spaces, rubbersheet analogies that may help to understand differential geometry I think there is also a chance that it gives false visions. In particular at the next level, when you are incorporating QM. Also once you understand curved spacetime, the next headache is to understand the dynamics of this curvature.

I try to seek other abstractions, you can "picture other abstractions" for yourself that aren't necessarily visual 3D things. These litterally mechanistic visual pictures also easily gives a realist view of everything. This may be fine in classical relativity, but since even a studen of that might eventually want to contemplate QM, maybe it's not a bad idea to look already now for better abstractions.

I think a very interesting and more QM-compatible way of thinking is to consider information spaces and curved information spaces. This could IMO partially be intuitive if you stop thinking of mechanical pictures and start thinking in terms of decision making and communication. A geodesic I think of as the path of minimum speculation, a path of NO speculation typicall doens't exist. But once an observer actually starts to "walk" this path he is fed with new information, and the previously predicted geodesic may now have deformed. Perhaps one can think of the curvature of informationspace as existing in a "map" that the observer encodes.

What would curved space look like to a real inside observer cruising this space? Then the picture of an ant walking a sphere doesn't fit very well as that's an external view.

/Fredrik

 

[DrGreg]

Although it's difficult to visualise curved spacetime, you can visualise curved space (ignoring time) near a (spherically symmetric non-rotating uncharged) black hole.

Imagine a 2D slice of space (ignoring time and one space dimension) through the middle of a black hole. Now think of it, not as a flat horizontal plane but shaped like the end of a vertically-upwards trumpet, with parabolic vertical cross-section. The surface is nearly horizontal at a large distance, but as you approach the hole it curves downwards until it is vertical at the event horizon. At this point the surface stops; there is a hole in the middle because this model breaks down at the event horizon.

Distance measured within the curved surface represents the distance that would actually be measured by a "stationary" observer at each point in the surface. Horizontally-measured radius is the r coordinate that appears in standard equations for such a black hole ("Schwarzschild coordinates"). The closer you get to the event horizon, the larger the discrepancy between the horizontal r distance and the observer's curved distance.

Note, however, there is no distortion in the "tangential" direction -- the circumference of any circle around the centre is still $2 \pi r$.

This isn't just a handwaving approximation, it is mathematically exact: see this thread, in particular post #15 and the second half of post #35, which include a link to a diagram.

It should be stressed the above model deals only with space-curvature. When you add in the time dimension it gets more complicated and it's beyond this model.

 

[losang]

Is anyone going to answer his question or respond to mine (see my second post for an explanation of what he's asking). The problem with this site is everyone wants to be right but few want to think.

visualizing curved space is easy and I'm sure the original poster knows how to visualize a curved line. So obviously his question is on the ontology of ST, not obvious explanations of the difference between a curved and straight line.

this is a perfect example of the problems with physics today. There are no more thinkers, just technologists. If you look back to the early part of the 20th century and compare these physicists with today its is obvious. The is in large part due to the successes of the standard model through the last half of the 20th century. Unfortunately, no major changes are going to occur until physicists regain that creativity that einstein and feynman had.

Has anyone ever heard of the second explanation for curved ST in my first post? Any comments appreciated.

 

[Desh627]

I think of a made bed, in which the sheets have been folded very taut. When you place a mass on the bed, it will cause the sheets to curve around it. It's a bit off from what really happens, but it does the job.

 

[DrGreg]

Is anyone going to answer his question or respond to mine (see my second post for an explanation of what he's asking). The problem with this site is everyone wants to be right but few want to think.

visualizing curved space is easy and I'm sure the original poster knows how to visualize a curved line. So obviously his question is on the ontology of ST, not obvious explanations of the difference between a curved and straight line.

this is a perfect example of the problems with physics today. There are no more thinkers, just technologists. If you look back to the early part of the 20th century and compare these physicists with today its is obvious. The is in large part due to the successes of the standard model through the last half of the 20th century. Unfortunately, no major changes are going to occur until physicists regain that creativity that einstein and feynman had.

Has anyone ever heard of the second explanation for curved ST in my first post? Any comments appreciated.

The original question asked "how to picture how space is curved" and about "the look of curved space". That question seems to have been answered and Denton is welcome to ask for further clarification.

What hasn't been explicitly answered is whether spacetime is a substance "like jelly". The answer is no.

The use of the word "curved" is an analogy, it doesn't necessarily mean anyone will observe a literal curve in 3D space. The curvature of spacetime refers to the non-Euclidean geometry of spacetime. If you consider an expanding circle around a mass and measure the rate of change of circumference with respect to radius, you find the answer is not $2\pi$ as Euclidean geometry would indicate. Why? That's just the way our Universe works.

 

[losang]

Dr. Greg,

good post. Nice to see someone is finally addressing the question. Thank you.

What you are saying seems to be similar to my second interpretation where "curved spacetime" means the laws of motion become non-euclidean, i.e. geodesic motion is no longer motion in a straight line. There is no substance that is actually curving and like you said this is just the way the universe is. I can to this interpreatation just by thinking but don't have any physical justification for one interpreatation over the other.

Could you please clarify why ST is not like "jelly" and it is just our universe is non-eucledian. What evidence is there for this interpreatation.

 

[DrGreg]

Dr. Greg,

good post. Nice to see someone is finally addressing the question. Thank you.

What you are saying seems to be similar to my second interpretation where "curved spacetime" means the laws of motion become non-euclidean, i.e. geodesic motion is no longer motion in a straight line. There is no substance that is actually curving and like you said this is just the way the universe is. I can to this interpreatation just by thinking but don't have any physical justification for one interpreatation over the other.

Could you please clarify why ST is not like "jelly" and it is just our universe is non-eucledian. What evidence is there for this interpreatation.

Well, it depends just what the properties of this supposed "jelly" are. What we do know is that if there were a jelly we have no way of detecting motion relative to it -- otherwise your jelly would be just another name for "aether" which relativity has established there is no evidence in favour of it. If you can't detect motion relative to it, then what is it? The simplest answer is "nothing, it doesn't exist". The onus is on anyone claiming it does exist to describe an experiment to demonstrate its existence.

 

[losang]

This also has me thinking of the explanation that ST is really the gravitational field. It is not a background which gravity and other fields/matter are placed upon. Its just fields on fields as was explained by Carlo Rovelli in his gravity book. Seen in this way the equations incorporating ST are just the fields/matter moving or changing with respect to the gravitational field, there is not stage for the actors. This is a really counter intuitive way to see things but Rovelli's arguments based on GR seem convincing.

 

[starkind]

Jelly is just a model and of course you cannot spread spacetime jelly on a sandwich. The model works in some ways but not in others. In what way does the spacetime jelly idea work? Some phenomena behave like a free object moving through the middle of a jelly. The jelly can have density gradients which can cause some moving objects to be deflected in a smooth curve which depends on local variations in the jelly. The jelly model fails because it provides for a background which fills space, and no such background can be demonstrated in free space.

The variations in density of the model represent the fact that local gradients exist in spacetime which affect the observed paths of objects. Just because we can compare it to jelly does not mean it exists as a material substance or aether. In any case the jelly model is closer to describing reality than is the bedsheet model.

So what are the local gradients for gravity? One clue is that clocks run slower near gravitational masses, and at the horizons they stop entirely. We observe a curved path because the differential crosses the body of the object, causing one part of the object to move differently from another part. If one side runs faster than we might expect and the other runs slow, we will observe a curved path, or an angular momentum.

If you don't like jelly you can use the equivalence principle to convert the space parts of the equation to time parts. Then fixed background space points are not required. Or, if that is too complicated, use the jelly model and just keep in mind that it is not an aether. Or I suppose you could just stay in bed.

 

[stevebd1]

I remember seeing a programme once on tv that showed the old stretched rubber sheet idea but this one had a slight difference. The rubber sheet had a grid on it made up of 100 mm approx. squares and at the centre was a plunger on a rod and clamp representing the source of gravity. The thing that was slightly different was that there was a camera directly over-head looking down on the rubber sheet. As the plunger was pushed into the rubber sheet, the rubber stretched, the grid distorted and the plunger was held in place with a clamp. From eye-level, you could see the distortion in the grid and rubber sheet but from the camera that looked directly down onto the sheet, there appeared to be no distortion, the grid still looked like a grid of 100 mm squares, the only sign that there was something different was when marbles were thrown into the mix. I thought this was a half decent representation of the 'curvature' around a object of mass as the space itself doesn't look any different, it's only when objects pass through it that it's apparent something is different.

 

[DrGreg]

Apologies to those who already know all this, but it's perhaps worth saying more about what "curvature" is.

The main technical reason why we refer to the "curvature" of spacetime, rather than the "density of jelly" or suchlike, is that the mathematical equations that describe the geometry of spacetime are very similar indeed to the equations that describe the geometry of a curved 2-dimensional surface in 3-dimensional Euclidean space. Remember that a particle moving in space is equivalent to a line or curve in spacetime (i.e. a graph of space-position against time), so particle dynamics equate to 4D geometry. If you have difficulty picturing 4D spacetime, just ignore 1 or 2 of the space dimensions -- many problems can be analysed in only 1 or 2 space dimensions.

In many ways, the geometry of flat, 2D Euclidean space is summed up by Pythagoras' Theorem r² = x² + y², which easily extends to 3D as r² = x² + y² + z². In flat (gravity-free) spacetime, this becomes r² = x² + y² + z² - c²t². In curved spacetime, we find that the same equation is approximately true for small values near the observer, but if you try to extend your coordinates to a larger region of spacetime, the equation changes and takes different forms in different places and times. Knowing what this equation is (the "spacetime metric") tells you all you need know about the geometry, the "curvature" of spacetime, and the dynamic behaviour of particles.

This behaviour is almost exactly the same as if you try to apply the 2D Pythagoras Theorem to triangles drawn on a curved 2D surface instead of flat paper. Although we can draw a very accurate map of a small part of the Earth's surface on a flat piece of paper, flat maps that depict large parts of the globe are always distorted. This distortion can be formulated in a way that is mathematically very similar to the gravitational dilation and contraction effects of general relativity.

How can you tell if the spacetime around you is curved or flat? Well, if you look over small distances and times you can't. If you are a free-falling observer, other free-falling particles that are near you are apparently stationary or move at a constant velocity, whether there is gravity or not. (This is known as the "equivalence principle".) But over longer distances and longer times you will notice that isn't really true and other free particles are slowly accelerating relative to you. That tells you that spacetime is curved. If, on flat paper, you draw a distance v. time graph of the particles, they follow curved lines. If the lines were straight you would be in flat spacetime. This is entirely analogous to the fact that two straight lines drawn on a curved surface cannot be truly parallel everywhere: they will inevitably curve relative to each other even though each appears straight in isolation. Just consider two nearby lines of longitude which are parallel at the equator but which meet at the poles.

 

[A.T.]

This is the best visualization I know:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html

More links on GR visualization:
http://www.adamtoons.de/physics/gravitation.swf
http://fy.chalmers.se/~rico/Theses/tesx.pdf (chapter 2)