Space Time Is Curved

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What do they mean when they say space time is curved? I keep hearing a lot of talk about Space Time being the 4th dimension, and no matter how many trampoline examples I see, I just don’t get it.

In: Physics

They mean what they say, that it’s curved.

You can see this around objects with high gravity fields. Best examples are so called gravitational lenses. This is where something like an entire galaxy bends the space around it to such an amount that you can see what is behind it. If the space wasn’t curved then the galaxy would block the light. By the space around the galaxy being curved the light from the objects behind it travel in a straight line, because that is what light does, but it’s travelling in a straight line through curved space. Sometimes this means you can see the same object multiple times.

If you’ve seen Interstellar you would have seen it with the black hole.

Example http://assets.cdn.astronomynow.com/wp-content/uploads/2015/02/02153356/Interstellar_black_hole_940x400.jpg

That’s a flat disc but the high gravity has bent the space around it so you can see behind it.

A line is 1 dimensional, and can only be (edit) measured in one direction. For it to be curved, it requires two dimensions. If it goes North/South, any East/West movement is in another “dimension,” or direction of movement.

A plane figure (square, circle, whatever) is 2 dimensional, and requires three to curve it. Imagine a circle, and it slowly deforms into a bowl shape. It’s moving in another dimension, or direction, to curve it.

Now…a sphere or cube is 3 dimensional. The curve is in the 4th dimension. It’s not a dimension that we can see, because our whole lives have been experienced in 3 dimensions. It can be understood as a continuation of the above ideas…curving an object in one dimension higher than it exists in.

This goes beyond what you are asking, but also explains the more common description of gravity being understood as an objects reaction to the curvature of space/time:
[Visualizing General Relativity](https://youtu.be/wrwgIjBUYVc)

In maths lessons you’ve probably worked on paper with grids; lines equally spaced, making a bunch of different squares of the same size.

You can use this easily to make a set of axes (usually x-y) and a grid. This is how we normally think of (2D) space; we could describe points by a grid – any point we can identify by how far it is in the x-direction (or how many lines along), and how far in the y-direction. The two directions are at right angles so any motion in one is independent of the other – we can move freely left-right and/or up-down.

Now imagine instead of that graph being drawn on paper it was drawn on something stretchy, like the elastic you’d get from a balloon if you cut it open and flattened it into a giant square (but still kept flat on a table). Now you could twist or squish the elastic, which would twist or curve the lines. You could get the lines to bunch up in some places, and be more spread out in others, you could get them to curve – although you couldn’t get them to cross over each other – you’d still have the same number of squares overall, some just wouldn’t be square any more, and they’d be of different sizes. You might be travelling left (along an y-direction line), but due to curving find yourself travelling up a bit instead, or vice versa.

It turns out that real space is stretchy/squishy, and mass (or energy) squishes it around. It bunches up space (like the elastic sheet), so what should be straight lines start being a bit curved. Also you might find that two points that should be 5 squares apart (so if you went up 10, along 5 and down 10 you’d get from one to the other) are actually six if you try to cross them directly because the lines are bunched together more there.

And it turns out this doesn’t just affect the three dimensions of space (it’s a bit harder to visualise this in 3D, but don’t worry about that too much), but also the 4th (or 0th) dimension of time. Not only do you find there are places where space is all bunched up, time also gets stretched out as well (so there is more time between two events than there ‘should’ be). And to make it even weirder, space and time aren’t these separate things, but part of a combined space-time thing; so you might find that while you thought you were just travelling in time, your time-direction has been twisted or curved a bit into a space-direction, so rather than just moving through time you seem to move in space as well (when viewed from the outside); i.e. you fall.

The trampoline example is useful for demonstrating orbits, but it’s kind of prone to overthinking (“then what’s pulling the bowling ball down?”). The scenario I like better is if you imagine that you and I are standing 1 meter apart at the equator, and we each have a compass, and we’re both going to walk north at the same speed. If the earth was flat, lines of longitude would be parallel, and every time I would look over at you you would be 1 m away. Because the earth is curved though, longitude lines converge at the north pole. This means very gradually we would find each other getting closer and closer together, even though we’re both following our line north. Now if we didn’t *know* the earth was curved, we would have to conclude that there’s some attractive force between us that’s pulling us together.

This is how spacetime is. Everything (even a beam of light) is following it’s own “straight line” (geodesic) but the curve of those lines is what produces gravity, makes light rays bend, etc.

Think of graph paper, it’s flat. You can draw a triangle on it, just draw 3 dots and connect them with three lines. One of the properties of a triangle is all the angles of the corners add up to 180°. This is always true… On flat spacetime.

Curved spacetime would be instead of flat paper, you draw on a balloon. Do the same thing a draw 3 dots and connect them with straight lines. On a balloon you can draw a triangle with three right angles. This is because while the corners look flat and the lines look straight up close, they are actually curved. And since they are curved, math doesn’t always work as you expect, exactly how depends on how it’s curved.

Spacetime might be curved in the same way, but in three directions, triangles might not always have corners that add up to 180 degrees, parallel lines might not stay parallel, and you might actually be able to look in any direction and see yourself from behind.

The problem is you are not evolved to picture it. All we can do is show you analogies, or we can tell you the experience, but you do not have the grey matter evolved to model it.

Imagine traveling in a perfectly straight line, one where there is no curve, but you end up where you started. The line is straight, the space it is in is curved.

Imagine drawing a triangle, and measuring the three angles, and they end up adding to a number that is slightly different than 180, but the straightest ruler can find nothing but straight sides.

Imagine it being impossible to make two straight parallel lines, ever. If you move along using a spacer to keep them the same distance apart, when you look they clearly curve.

See? Even knowing the consequences you can’t picture it, because your brain is evolved and programmed by this reality, to model a flat spacetime. You just don’t have the tool. All you can do is resolve the consequences, basically use math to describe it.

Imagine your bed made up with a blanket on it.

That is a good example of flat space time.

Now let’s say you are in the middle, under the blanket because it is cold. That would be curved space time and you would be acting as the mass pulling the sheet closer to you.
Both of these are in reference to special relativity.

If you are asking about topology of Space-Time it gets a lot more complicated. Depending on the shape we can infere information that we might not be able to see from where we are.

For reading about the latter, look up Space-Time topogy and particle horizon.

Bonus fact: past the event horizon every path you can take leads into the center of the black hole (singularity).

Okay, start with the concept of spatial dimensions.

Imagine a 3d game of Battleship played on a big wireframe mesh cube, with the coordinates given in terms of upness, rightness and forwardness from the bottom/left/front respectively.

Those three directions are all at right-angles to each other – the lines are all perpendicular, so changing your position in one coordinate doesn’t change your position in any other.

Every object, every atom, every subatomic particle takes up a little chunk of that massive, universe-engulfing grid.

So that grid is what we call ‘space’.

Now we extend that concept: we add *another* dimension to that grid, *at right-angles to all the others*.

This is of course impossible to visualise, but the concept is straightforward enough.

This new dimension is called ‘time’, and its coordinate is given in terms of laterness, from the big bang.

Every object on the grid *also* takes up room in the ‘time’ direction – a dinosaur wouldn’t just be dinosaur-shaped; if you could see in 4d, it would be like a long extruded length of [dinosaur pasta](https://nuts.com/images/rackcdn/ed910ae2d60f0d25bcb8-80550f96b5feb12604f4f720bfefb46d.ssl.cf1.rackcdn.com/1e6c14229d87184e-Y1Shzv_q-large.jpg) that changes shape down its length (or as time went on on, from its own perspective)

That extra-confusing grid is called ‘spacetime’.

Here’s the fun part: you put a big massive object in the grid, and it *bends the actual fucking lattice* that all your objects are stuck to.

Just a little bit, not enough that you’d see it – but it’s there. The whole grid was poorly designed and is made of cheap shitty plastic, so objects don’t *quite* fit properly, and if you wedge a whole bunch of atoms in next to each other, you get some noticeable distortion start to build up.

TFW you order your universe from wish.com…

Anyway, now you’ve got this trippy concept where the strands of the lattice *are no longer completely perpendicular along their length*, so travelling along an X-coordinate strand can actually move you a little bit in the Y direction as well.

It’s like haring down the freeway on your motorbike, when the road starts to take a very slight curve – you keep going in an absolutely straight line, but somehow this means you drift out of your lane, wtf.

And this is how gravity works.

Mass bends the time-strands into spacey directions around it, so when you’re standing still near a planet-sized chunk of mass, just minding your own business going forwards in time, somehow this means you drift out of your position in space, wtf.

(actually, it bends *all* the strands, but we – and planets – are in fact very, *very* long pasta shapes, so the drift from laterness into downness is a hell of a lot more noticeable from our perspective)

Truly empty space-time is flat. A photon traveling through that space-time will travel in a straight line. Anything with mass pulls on space-time so that paths that were once straight now curve toward it. The effect of that curving path on other masses is what we call gravity.

As you add more mass, that curvature increases, further spreading the object’s gravity well. An object enters orbit of a parent body if that thing is falling toward the parent but also has enough angular momentum to “dodge.”

You actually can’t picture it but not for the reason that it’s 4d – that’s actually not a problem. The issue is that curved spacetime is formulated “intrinsically”, meaning from the perspective of people living on the manifold (us). This is done using something called “the metric” which describes the relationship between distances on the surface with respect to a predefined coordinate system. The metric then describes how distance and time measurements vary as you move through space and time along your coordinate grid.

This is like the walking on the globe examples others have given. Something missed is that you can’t prove whether you’re walking along the inside of the sphere or the outside by taking any measurement of the surface. They have the same metric.

The metric tells us everything it is possible to measure about the distances, angles and curvature but it doesn’t allow you to actually make a picture that you could be certain was the true “shape” of spacetime *no matter how many dimensions you could picture*. You can (kind-of) make pictures of several spacetimes that have the same metric as the ones we measure, but it’s not actually that helpful.

Another reason for this is that the metric is strictly local, it actually doesn’t tell you anything about the space as a whole, it just tells you that when you make a measurement at a point, this is what you can measure – and if you then see someone else measure something, how their measurement will be different to your measurement from your perspective. This is subtly different to describing the surface itself.

Tldr: differential geometry is hard and you can only begin to understand curved spacetime by learning a lot of very very dense and quite abstract math (compared to say high school math)

I want to say the trampoline example is bad but I won’t because that’s as close as it gets to see it in real. The below video breaks that misconception (??).

In short, imagine a 3d mesh instead of a single plane like the trampoline. That’ll help you visualize what space time curvature is. We know how it bendd but I guess we still don’t know why!

https://youtu.be/wrwgIjBUYVc