# Spacetime and Gravity?

This is not a homework question- it's a theory of mine.

We had this Indian lady come and teach us Einsteinium physics. She talked about spacetime and gravity and free fall etc., but there is one thing I don't get. In Google images, if you type 'spacetime' you get images of a body in the middle of a grid making a dent. Apparently this is a gravity well or something. But how does that work? Where is the proof? I don't see how space can bend and ripple and whatever (that's what she told us). She said it's like gridlines on a map- but space is 3d (geometrically). But in space there is no up or down, left or right. It's relative to the observer. Planets are not all in line with each other. How can this model be correct? And how do we know gravity isn't a force but some spacetime thing (that's what she told us, sorry i don't remember details).

p.s I'm year 9 so no complicated stuff pl0x...

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Nugatory
Mentor
This is not a homework question- it's a theory of mine.
Actually it's not a theory, it's question... and a good one too.

there is one thing I don't get. In Google images, if you type 'spacetime' you get images of a body in the middle of a grid making a dent. Apparently this is a gravity well or something.
You'll see these pictures all over the internet, but they are terribly misleading. If you search the relativity forum here for "rubber sheet model", you'll find some of the bad things we have to say about it. Also, look for a video by user A.T. - it is a much better starting point.

Drakkith
Staff Emeritus
You are correct in that the images of spacetime you usually see are not accurate. The problem is that it is not possible to make a picture of curvature in 3+ dimensions, so we have to simplify it to 2 dimensions in order to visually represent it. However, the theory of General Relativity is not based on pictures, but on math, and the math is very specific about what is happening. This math describes gravity as a curvature of spacetime in a similar way to how we use math to describe the curvature of a 2d surface such as a sphere or saddle (The surface of both objects is 2d, even though the objects are 3d).

pervect
Staff Emeritus
This is not a homework question- it's a theory of mine.

We had this Indian lady come and teach us Einsteinium physics. She talked about spacetime and gravity and free fall etc., but there is one thing I don't get. In Google images, if you type 'spacetime' you get images of a body in the middle of a grid making a dent. Apparently this is a gravity well or something. But how does that work? Where is the proof?
I don't see how space can bend and ripple and whatever (that's what she told us).
The image is just a visual aid, and not a terribly good one at that. Don't expect the visual aid to provide proof. If you look up the proof part in reliable references, though, you will find that General Relativity has made a lot of predictions which have turned out to be true. If you look further, at the idea of how things are "proved" by science, you'll see that theories (like General Relativity) are never really "proven", rather they are tested. These tests are performed by performing experiments, often experiments that are technical and difficult to understand. While the details of the experiments are published and open to all, in practice the average nine year old (and many adults) wont understand all of the details :(. This is unfortunate, but there's not much that can be done about it. In the end, a certain amount of trust in the scientific process is needed. This trust can be occasionally wrong, especially as some who claim to be "scientists" may turn out to be be mistaken, liars, or what is often called crackpots, people with sincere beliefs in crazy things that few other than themselves believe.

Ignoring the issue of the complexity of the tests for now I can say that in the opinion of the scientific community,, GR has passed all the tests that people have thrown at it, and is a well accepted theory. But of course, to evaluate your reaction to this, you might well ask "who is this person who says that?"

She said it's like gridlines on a map- but space is 3d (geometrically). But in space there is no up or down, left or right. It's relative to the observer. Planets are not all in line with each other. How can this model be correct? And how do we know gravity isn't a force but some spacetime thing (that's what she told us, sorry i don't remember details).

p.s I'm year 9 so no complicated stuff pl0x...
If you get the idea from the visual aids that GR says that the actual geometry of space isn't as simple as the Euclidean geometry you are or will be studying, that's about as much as one can expect I think.

Later on, you may learn more details, about what we mean when we talk about "space-time", and how it is different from the purely spatial diagrams that you have seen in the illustrations. And how we go about testing the geometry of space-time, with experiments like deflection of light rays, and radar propagation delay tests, and perihelion shifts, and gravity probe B, and some of the other tests that have been done.

You may not ever fully understand the theory unless you get interested in it and take a lot of rather demanding and advanced math courses.

PeterDonis
Mentor
2019 Award
if you type 'spacetime' you get images of a body in the middle of a grid making a dent.
As Nugatory said, these images are very misleading. One big reason is that they are not images of curved spacetime, they are images of curved space. (The video by A.T. that Nugatory referred to, by contrast, actually shows you images of curved spacetime--though they are still images with a reduced number of dimensions. Nobody knows how to make 4-dimensional images. ;))

I don't see how space can bend and ripple and whatever (that's what she told us).
Again, it's spacetime curvature that's important, not space curvature. (Space can be curved in GR too, but I think it's better to start by trying to understand what spacetime curvature is.)

She said it's like gridlines on a map- but space is 3d
Yes, and spacetime is 4d. But you can still draw gridlines in 3d space, right? It will just be a 3-dimensional grid instead of a 2-dimensional one.

The real question is, if we try to "draw gridlines" in 4d spacetime, how do we draw the lines in the time direction? And the answer is, we draw them by looking at the motion of freely falling objects--objects that feel no force. (You can test for such objects by attaching accelerometers to them; if the accelerometers read zero, the object is freely falling. For example, the International Space Station is a freely falling object. So is a spacecraft like the Voyager probe. So is the Earth in its orbit about the Sun.) The paths of these objects provide the "grid lines" in the time direction in spacetime.

Another important point: any gridlines we draw have to be straight, in the appropriate sense. If the manifold (that's a general term that can be used to refer to spacetime, or ordinary 3d space, or a 2d surface like a sheet of paper) is flat (like a sheet of paper laid flat on a table), then the gridlines just need to be straight in the ordinary sense you're used to. But if the manifold is curved, the gridlines might not look straight to you--for example, "straight" grid lines on a 2-sphere, like the surface of the Earth, are great circles (like the equator, or meridians of longitude, on the Earth). The technical term for these "straight" lines is "geodesics", and we say that the gridlines we draw in any manifold must be geodesics. This requirement is why we have to use freely falling objects to draw the gridlines in the time direction: only the motion of those objects is "straight" in the sense of being geodesic.

Once we have geodesic grid lines, the definition of curvature is easy. Take any pair of nearby geodesics that are parallel at some point. If they don't stay parallel as you move along them, then the manifold is curved. For example, take two nearby meridians of longitude on the Earth. At the equator, they are parallel; but as you move along them towards one of the poles, they don't stay parallel--they converge. This tells you that the surface of the Earth is curved.

The spacetime version of this uses two nearby "gridlines" in the time direction--i.e., the motion of two nearby freely falling objects. "Parallel" in this case just means the objects are at rest relative to each other at some instant. If they don't stay at rest relative to each other, then spacetime is curved. For example, take two nearby objects that, at some instant, are both at rest in free space above the surface of the Earth (we'll suppose someone tossed them upward very precisely so they both came to rest at the same instant at slightly different altitudes). At the instant they're both at rest, the "gridlines" marked out by their motion are parallel. But they don't stay parallel: the object that is closer to Earth will start falling slightly faster, so the distance between the objects will increase--the "gridlines" marked out by their motion diverge. This tells us that spacetime is curved. The shorthand way of referring to this phenomenon is "tidal gravity", and we can then say that the presence of tidal gravity is what tells us that spacetime is curved.

how do we know gravity isn't a force but some spacetime thing
Two reasons. First, objects moving solely under the influence of gravity are freely falling--note that two of the examples I gave above of freely falling objects (the ISS and the Earth) are in orbit around other objects, under the influence of gravity. So objects moving solely under gravity feel no force, and in GR, that means there is no force--a force in GR has to be felt.

Second, the trajectories that different objects follow under gravity are independent of the objects' mass, and indeed of any particular features of the objects (internal structure, etc.). The trajectories depend only on the initial conditions: where the object starts and how fast it starts moving. For example, during the Apollo 15 mission, astronaut Dave Scott dropped a feather and a hammer on the Moon, both from the same height, and they both hit the surface at the same time. No other force has this property, and it makes us think that gravity must be something different from an ordinary force--in fact, that it must be due to some property of spacetime itself.

I hope this wasn't too long, but your questions are good questions and I felt they deserved a fairly detailed answer.

• cr7einstein
A.T.
Apparently this is a gravity well or something.
A gravity well is different from curved space-time, and merely looks similar to curved space (without time) around a mass:
http://en.wikipedia.org/wiki/Gravity_well#Gravity_wells_and_general_relativity

But the curved space (without time) cannot explain the main effects of
gravity. You need to include the time dimension for this:

but space is 3d (geometrically).
Space-time is actually 4D, which even harder to
visualize. That's why we reduce it to just 2D for illustrations. Here is more:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
http://www.relativitet.se/spacetime1.html

Is this all theoretical? Where is the proof? It just doesn't make sense

A.T.
Is this all theoretical?
It's a physical theory, that makes predictions which can be compared with the real world.

Where is the proof?
You cannot prove physical theories. You can only disprove them by observation. So far General Relativity hasn't been disproved.

It just doesn't make sense
A physical theory has to match observation, not "make sense" to you.

Fredrik
Staff Emeritus
Gold Member
It just doesn't make sense
Sure it does. But you have to study general relativity to really see it. And before that, you need to study calculus, multiple-variable calculus, linear algebra, real analysis, topology and differential geometry.

You cannot prove physical theories. You can only disprove them by observation. So far General Relativity hasn't been disproved.
I'll just add that you can prove that the predictions made by one theory are better than the predictions made by another. In particular, you can prove that the predictions of GR are vastly superior to those of Newton's theory of gravity. There's an enormous amount of evidence for that.

Is this all theoretical? Where is the proof? It just doesn't make sense
It's a mathematical model that makes predictions about physics. And the proof is in its usefulness: it is more accurate than Newton's model of gravity.

If you like, you can make sense of it in other ways. For example, Einstein spoke of the "gravitational field": space has physical qualities according to his theory. So, you can say that a heavy mass like the Earth changes the space around it in such a way that objects fall towards it.

There are other predicted effects that Newton's theory did not predict, for example the prediction that clocks near the Earth tick slower than clocks higher up. And that effect was measured. - http://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html (See p.198 of the English translation: "Thus the clock goes more slowly if set up in the neighbourhood of ponderable masses")

PeterDonis
Mentor
2019 Award
Is this all theoretical? Where is the proof?
Was the stuff I described in post #5 "theoretical"? I talked about actual physical happenings, and how GR describes them.

The images are misleading, as everyone above said. 4d spacetime cannot be visualised, so the question of visualising curvature doesn't come up. You should note that though the curvature precisely cannot be measured or quantified experimentally, the theoretical description's ( Einstein field equations) predictions are tested to be accurate. So, in a sense, it actually IS correct, even though you can't draw its mechanism on a sheet of paper. As @pervect pointed out, the math is terrible. You can find out other mathematical details in a good book on differential geometry and tensor calculus ( I suggest Barry Spain).

Now, as for proof, almost all the theoretical predictions( gravitational lensing- the bending of trajectories through curved spacetime, perihelion shift of mercury, etc) have been tested and found to be true. If you really want to go into mathematical rigour, then there's an equation called "Raychaudhuri equation", derived by Indian physicist A.K Raychaudhuri. Now, as @PeterDonis explained, the most fundamental property of general relativity(GR) is that an objects travel curved path in a gravitational field. In fact, in GR, bodies always follow straight lines in four dimensional SPACETIME, but nevertheless appear to move along curved paths in 3 dimensional SPACE. (an analogy: when an airplane flies over a hilly ground, even though it follows a straight path in 3 dimensional space, Its shadow follows a curved path on the uneven 2d ground). These trajectories are called geodesics, and this is the shortest route between 2 points. In FLAT space, it is a straight line. In GR, it is a curve. (NOTE: the straightness in 4d space time and curve in 3d space is compensated by slowing down of time, "stretching" of time- another accurate prediction by GR). Now, the Raychaudhuri equation shows that all neighbouring trajectories ( geodesic congruence in GR) in a gravitational field CONVERGE, i.e. their paths are BENT towards the source of curvature, which is mass/energy. This is exactly what GR says- paths followed by an object in gravitational fields are curved, and these curves are merely the result of the curvature of spacetime itself. So, even though Raychaudhuri equation is derived and based upon GR, in a sense, it kind of "proves" the theory's prediction of bending of trajectories. And, it even considers non gravitational effects such as rotation and shear (even though gravity may be their ultimate reason), so there's nothing vague about it. So, here's your mathematical 'proof' of GR's most basic property. I won't delve into mathematical details, but you sure can if you are really interested (its dizzying). SO, it is mathematically beautiful, experimentally accurate, and testable in principle- a perfect physical theory.

To sum it up, I would like to mention a great intuition towards GR, which Raychaudhuri himself used when teaching(yeah, I quote him a lot.. Being an Indian myself, I know a lot about him... :) )
" There are two ways to describe a car's path through a hilly terrain. If you ask a mathematician, he will ask for various parameters to solve Newton's equation ( 2nd order differential equation), and give the path. This is what Newtonian theory does. If you ask the driver, he'll say that the car's path will be influenced by the road's nature; it will move up, down, side according to how the ground beneath is curved. This is GR. Objects simply follow the the directions curved spacetime tells them to, they move according to the GEOMETRY of spacetime..."

And by the way, I really appreciate your curiosity at such a young age........Wish you good luck!!( I am not much older than you, so I know exactly how you feel..... ;D )

A.T.
The images are misleading, as everyone above said. 4d spacetime cannot be visualised, so the question of visualising curvature doesn't come up.
You don't always need all 4 dimensions. There are scenarios, like a radial fall, that involve just 2 dimensions, and can therefore be visualized. There are also effects, like the spatial contribution to light bending, which can also be visualized using just 2 dimensions.

pervect
Staff Emeritus
Is this all theoretical? Where is the proof? It just doesn't make sense
The proof comes via experiment, not by argument, as others have mentioned. Science has long realized that you don't demonstrate the usefulness, utility and truth of theories by argument, instead you test them. The general paradigm is that successful tests don't "prove" a theory, but unsuccessful tests disprove them.

The predictions of GR differ only slightly from that of theories that preceed it, but this slight difference has been confirmed by a series of ever-more-accurate tests.. If you are looking for proof before you learn the theory, you might want to look at things like the WIkki article http://en.wikipedia.org/w/index.php?title=Tests_of_general_relativity&oldid=628399582.that discusses "tests of general relativity".

There are other reasons you might wish to learn the theory. If you want to go into any field which involves precise timekeeping, you'll need to learn at least a tiny bit of the theory in order to be able to understand the details of how we set up timekeeping systems like atomic time (TAI time) on the surface of the Earth. The effects of GR on timekeeping are small, but at the current accuracy of atomic clocks they need to be taken into account if one wants to realize the accuracy of which the clocks are capable of.

Once you've convinced yourself the theory is worth learning, you need to learn first the framework of the theory, then the details of how you use the framework to make predictions. Popularizations are sometimes oversimplified, but the intent is to teach the framework needed to understand the theory itself.

Because GR is built on top of special relativity (SR), it is also very helpful if you learn that theory first before you learn GR. For instance, one might talk about the fact that space-time-diagrams can't be drawn "to scale" on a flat surface,, and compare this to the way in which flat maps of the Earth's surface can never be drawn precisely to scale, but if you're not familiar with the concept of a space-time diagram (which is first introduced in SR), there's little point to it.

Realistically, I also need to point out that to get to the point where you will be able to make preditctions from GR is going to be a very long road, involving calculus, linear algebra, tensors, and differential geometry, though the later is usually taught within GR itself. You might be able to get away with a bit less if you only need near-Earth applications of the theory to understand the effects that the theory predicts on measurements (such as time) here on the Earth. You'll still need calculus and linear algebra as a minimum, though you might be able to regard the metric as a matrix rather than a tensor.

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