# Light rays curved path in General Relativity

I don't understand GR at all, and I have a question that baffles me a lot. This picture is very common:

My question is, if a light ray passes through the central star (maybe the sun here) more closely than the path above, will it fall into or toward the 'pit' or 'bottom' of this surface? Some pictures on the Internet seem to show that phenomenon. But why is it? I mean, there is no force to drag it toward that direction.

For example, like the small ball in the picture below. It moves exactly along 'the shape of the surface'(which I don't understand at all what it stands for).

What exactly is this surface? Why should it be under the star (like in all pictures about General Relativity), instead of maybe above the star, like a hat on a person's head, or other direction (since a star or a ball is symmetric)?

I'm sorry if my question is stupid.

Edit: I found a video just now. https://www.youtube.com/watch?v=tzQC3uYL67U

Is the light path at 6'10'' right or wrong? I am totally lost.

In the picture above, i can understand the green line and the red line , but i don't understand the white line. Why would a light go a path such as the white line ?

• The picture is misleading. There is no bottom of anything “underneath” the star. (What direction would “underneath” be?) “Spacetime fabric” is a pop-sci notion that causes more confusion than enlightenment. Sep 14 '20 at 20:42
• As @G.Smith pointed out, the picture is not very good representation of actual theory. Watch this video, it should clarify a lot: youtube.com/watch?v=wrwgIjBUYVc&ab_channel=ScienceClicEN Sep 15 '20 at 5:10
• Many thanks for your reply! I have reedit my post. Is the light path in the video right or wrong? Thanks again! Sep 15 '20 at 22:10

Welcome to Physics SE!

First of all, if you don't understand GR at all, I suggest you look up something online: there are a lot of "GR 101" videos, for example Eugene Khutoryansky or PBS SpaceTime, that provide a very visual understanding.

Secondly, as the comment by G. Smith pointed out, that kind of image can be very misleading if you don't know the subject well. The reason for this is that, in GR, there is no 2D fabric to be bent: the whole 4D spacetime is. Even if this is not a perfect representation as it obviously only shows only a 3D projected image, you may get a better understanding of what's going on in GR taking a look at pictures like this GIF.

But why is it? I mean, there is no force to drag it toward that direction.

That's exactly what other scientists thought when Einstein showed them his results! I'm going to try to keep things simple: the whole idea behind GR is that there is no "force of gravity": just take any problem with a gravitational force, and delete it. Instead, everything with mass or energy bends spacetime in its vicinity. Then why do thing fall towards each other? Because things always try to go on a "simple path", and while on flat spacetime the simplest path is always a straight line, in curved spacetime the simplest path (or geodesic) is usually a curved line. Therefore an object (any object, be it a rock or a light ray) thrown in the vicinity of another object (a star, in your example) will follow its geodesic, giving to us the illusion that there is a force pulling it towards the star.
To answer your question, specifically: the light ray would just hit the star, as a rock would do. This is one of the most incredible consequences of GR: light is affected by spacetime-bending (or "gravity") exactly like other massive objects.

• @Kovalevskaya it is neither right or wrong: it's a representation. It's right in the sense that it represents the correct thing (the fact that the shortest path isn't a straight line), it's wrong in the sense that there isn't an actual flat sheet, so there is no "path on the sheet". The path is in 3D space. The sheet is a REPRESENTATION of 3D space, in the same way a feynman diagram is a REPRESENTATION of a scattering amplitude, or Google Maps is a REPRESENTATION of the streets in your town. Sep 16 '20 at 0:37
• @MauroGiliberti The path is in 4D spacetime. For slow moving objects, it is time component that is actually dominant, not spatial curvature. For light rays I would guess the time component and spatial components become comparably important, especially for schwarzschild where it holds $g_{tt}=-1/g_{rr}$. This means, that no spatial representation can make full justice to curvature of light path. Sep 16 '20 at 9:42
• @Kovalevskaya that picture being without context, I can't really understand it, but the white path looks straight-up wrong. I can't think of anything in any non-exotic metric that would follow that path. Also, comments aren't for extended discussion: if your initial question is answered, you should mark it as such, and if you have more questions you should ask them separately. Sep 16 '20 at 10:11
• @Umaxo I'm aware of that, but it seemed to me that OP doesn't have a firm grasp on the 4D spacetime that GR uses, and is more interested in a visual explanation. As you correctly said, it's impossible to make "full justice" of 4D curvature: in my opinion, an approximate 3D picture is better than the 2D sheet and while it's not perfect, it's better than nothing. Sep 16 '20 at 10:20
• @MauroGiliberti Personally, I find 2D and 3D picture of the same value, unless of course you take 2D picture literally as some physical surface embedded in 3D universe as OP did. I recently stumbled on this video, check it out youtube.com/watch?v=wrwgIjBUYVc&ab_channel=ScienceClicEN . Seems to me the best explanation of GR I have ever seen on youtube Sep 16 '20 at 10:53

Only the intrinsic geometry of spacetime matters. An actual geodesic on a surface like the one in your second image will bend toward the center because of the roughly conical shape of the surface. While it may be hard to visualize, you could imagine trying to cover the surface with long strips of paper, papier-mâché style. They'll stick the best, with a minimum of folding/tearing, if they curve inwards.

If you flip the surface upside down (or sideways) then it's still the same shape and the geodesics are still the same.

While in principle it makes no difference how these diagrams are oriented, obviously any good tutorial on general relativity would show them as hills (or sideways) so that readers don't fall into the trap of thinking that spacetime curvature has something to do with those gravity-well exhibits in science museums. In practice, virtually all popularizations, and even most textbooks, show them pointing down. I can only assume that the authors want to mislead their readers, or perhaps don't understand GR themselves.

The rubber-sheet or gravity-well picture is a pretty accurate model of Newtonian gravity, if you take the height of the surface to be the Newtonian gravitational potential, and make various idealizing assumptions. In that case, a hill (higher potential) does lead to a repulsive gravitational force.

Aside from being oriented in the worst possible way, both of the images in your question have other problems. In the first image, the bending of the surface doesn't match any solution to general relativity. Also, light is shown as bending in a part of the space that's evidently flat, which makes no sense. In the second image, while the surface is an accurate embedding diagram (of a constant-$$t$$, constant-$$θ$$ slice of the Schwarzschild interior+exterior solution), the curved line shown on the surface isn't a geodesic of the surface. Also, if the red sphere is meant to represent the massive body at the center, then it's far too large; the interior (non-vacuum) part of the space is just the small hemispherical "cap" at the bottom.

Also, of course, both of these spheres should be painted onto the surface, not floating above it in the physically meaningless background embedding space.

The other problem with these diagrams, and all embedding diagrams of this sort, is that they only show a spacelike slice through the full spacetime. If tachyons existed, they could follow geodesics lying on these surfaces. But real worldlines, whether geodesics or not, can't leave the light cone, which means they pass through time "at least as much as" they pass through space, and their worldlines don't lie on these surfaces.

The second diagram appears to show an elliptical orbit, which is not a geodesic of the surface shown in the image. It is a projection of a geodesic of the full spacetime onto the surface, but without any way of seeing the full shape of the spacetime, you can't see that it's a geodesic.

The Fabric of Spacetime is and is not. Einstein proved that space and time are essentially the same thing. The 'ball' represents gravity; if you put a ball on a fabric (that wasn't too tightly mounted) there would be a dimple in the fabric at that spot. So if another ball rolled by the dimple (anything, including light passing near a gravity source) it would have a tendency to change direction inward toward the ball creating the dimple. This second dimple is the force of gravity and though it doesn't create a dimple, it does act as a kind of magnet; with an invisible force acting on everything near it (near being relative). The passing ball or light is not dragged down into anything; similar to a magnet, the passing ball is pulled in the shortest possible distance toward the star or gravitational source; think about the magnet analogy. Those renderings show the passing light ball going 'down' is to help demonstrate why it's changing direction (because the fabric of spacetime is warped in such a way that it's easier to confuse people to try and explain it). This should send you away with a headache, say that passing ball represents a single photon of light. Getting out of it's originating star may have taken hundreds of thousands or more years as it bounced off of one particle after the next on it's random way toward space where it might spend billions of years travelling to your eye so you can see it. Because the speed of light is constant, from that photon's point of view, no time at all has passed.