# Curved Space-Time: Understanding How Nothing Has Shape

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• bcl
In summary: We say that the circle is curved because we can put it next to a straight line and see that the two diverge. The circle has extrinsic curvature. In summary, spacetime has geometrical properties and can be thought of as the geometry of physics. It is difficult to visualize in four dimensions but can be understood through analogies in lower dimensions. Extrinsic curvature involves comparing a curved space to a straight space, while intrinsic curvature can be measured within the space itself. Theories suggest that spacetime is not nothing, but its exact makeup is still unknown. Some propose it may be a form of energy or dark energy, and clues may be found through observing the appearance and disappearance of subatomic
bcl
TL;DR Summary
Mass creates curvature in space, but how can nothing have form?
So I understand that mass creates curvature in space-time. But what I struggle with is how nothing has a shape. I picture the space around say a planet, but how does the space (nothing) actual have form? Does anyone have a good intuitive explanation for that?

PeroK
The obvious implication is that spacetime isn’t nothing. Spacetime has geometrical properties, so at least in part you can think of it as the geometry of physics.

I suspect the relevant distinction is between intrinsic and extrinsic curvature.

At a guess, the picture you have, @bcl, is that something is "curved" if you can put it up against something straight and see that it does not line up. That is an example of extrinsic curvature. When we say that space-time is curved, we are talking about intrinsic curvature.

It is difficult to visualize four dimensional geometry. So we are pretty much forced to train our intuitions based on analogies with one, two or three dimensions.

Extrinsic curvature involves one space that is "embedded" in a higher dimensional space. The simplest example would be a circle (a one-dimensional space) drawn on a piece of paper (a two-dimensional space). We say that the circle is curved because we can put it next to a straight line and see that the two diverge. The circle has extrinsic curvature.

Intrinsic curvature is different from that. Unfortunately, one needs at least a two dimensional space to encounter intrinsic curvature. So let us consider a piece of paper (a two-dimensional space) that is rolled up into a tube within our ordinary three-dimensional space. We put a flat sheet of paper up next to the rolled up piece. The two diverge. Again, that is extrinsic curvature.

Suppose that we had drawn a triangle on the flat piece of paper before rolling it up. The angles on the three corners of that triangle added up to 180 degrees before it was rolled up. They still add up to 180 degrees after it has been rolled up into a tube. A hypothetical ant could walk all over the surface of that paper and (assuming it did not walk off one edge and back onto the other) never realize that the paper was anything but flat. Similarly, the ant could measure the radius, circumference and area of a circle and get r, ##\pi r## and ##\pi r^2##.

By contrast, consider the surface of a globe. You can draw a triangle on the surface of a globe (using great circle arcs for the edges since those are the closest things to straight lines on the surface) and find that the angles at the corners of a triangle add up to more than 180 degrees. That is intrinsic curvature. You can measure intrinsic curvature without ever having anything flat to compare against. All you have to do is carry a protractor to the three corners of a triangle and add up the readings. Or you could draw some circles and notice that the circumference and area are no longer ##\pi r## and ##\pi r^2##

Intrinsic curvature is the sort of curvature that can be measured from within a space. It does not require a comparison against some external standard of flatness. An ant or a careful surveyor walking on the surface of a globe can measure the curvature of the globe without having to walk all the way around.

In three and four dimensions, curvature gets rather trickier than in just two dimensions, but the same general principles apply. Intrinsic curvature can still be measured from within the curved space.

phinds
Dale said:
The obvious implication is that spacetime isn’t nothing. Spacetime has geometrical properties, so at least in part you can think of it as the geometry of physics.

If spacetime isn't nothing, are there theories as to what it could be? It doesn't seem to be matter or anti-matter. It is some kind of energy or dark energy? If we create a deep vacuum we can watch sub atomic particles suddenly appear and disappear. Is that a clue at to what space can be, since apparently something is coming out of what we perceive as "emptiness".

jbriggs444 said:
I suspect the relevant distinction is between intrinsic and extrinsic curvature.

At a guess, the picture you have, @bcl, is that something is "curved" if you can put it up against something straight and see that it does not line up. That is an example of extrinsic curvature. When we say that space-time is curved, we are talking about intrinsic curvature.

It is difficult to visualize four dimensional geometry. So we are pretty much forced to train our intuitions based on analogies with one, two or three dimensions.

Extrinsic curvature involves one space that is "embedded" in a higher dimensional space. The simplest example would be a circle (a one-dimensional space) drawn on a piece of paper (a two-dimensional space). We say that the circle is curved because we can put it next to a straight line and see that the two diverge. The circle has extrinsic curvature.

Intrinsic curvature is different from that. Unfortunately, one needs at least a two dimensional space to encounter intrinsic curvature. So let us consider a piece of paper (a two-dimensional space) that is rolled up into a tube within our ordinary three-dimensional space. We put a flat sheet of paper up next to the rolled up piece. The two diverge. Again, that is extrinsic curvature.

Suppose that we had drawn a triangle on the flat piece of paper before rolling it up. The angles on the three corners of that triangle added up to 180 degrees before it was rolled up. They still add up to 180 degrees after it has been rolled up into a tube. A hypothetical ant could walk all over the surface of that paper and (assuming it did not walk off one edge and back onto the other) never realize that the paper was anything but flat. Similarly, the ant could measure the radius, circumference and area of a circle and get r, ##\pi r## and ##\pi r^2##.

By contrast, consider the surface of a globe. You can draw a triangle on the surface of a globe (using great circle arcs for the edges since those are the closest things to straight lines on the surface) and find that the angles at the corners of a triangle add up to more than 180 degrees. That is intrinsic curvature. You can measure intrinsic curvature without ever having anything flat to compare against. All you have to do is carry a protractor to the three corners of a triangle and add up the readings. Or you could draw some circles and notice that the circumference and area are no longer ##\pi r## and ##\pi r^2##

Intrinsic curvature is the sort of curvature that can be measured from within a space. It does not require a comparison against some external standard of flatness. An ant or a careful surveyor walking on the surface of a globe can measure the curvature of the globe without having to walk all the way around.

In three and four dimensions, curvature gets rather trickier than in just two dimensions, but the same general principles apply. Intrinsic curvature can still be measured from within the curved space.

jbrigg2444, Thanks, that's a great explanation of intrinsic and extrinsic curvature. The conceptual difficulty I have is that these explanations require matter to assign the geometry to. But the space I envision around, say a ball in deep space, is not matter, but still has shape, how? Even if I just envision a universe with noting in it, we would say that universe is flat. But even describing it as flat does not make sense to me because we are still assigning a shape to "nothing".

kent davidge said:
@Dale , @jbriggs444 the OP wants to know "what the spacetime is made of"
Not necessarily. If it was made of something, it would be easier to envision it having some kind of form. Do physicists agree that it is made of something then? If not, I struggle with how we can assign shape (or curvature or flatness) to nothing.

bcl said:
Not necessarily. If it was made of something, it would be easier to envision it having some kind of form. Do physicists agree that it is made of something then? If not, I struggle with how we can assign shape (or curvature or flatness) to nothing.
It's not made of matter/energy. Space has a broader definition in mathematics. Spacetime is one of such spaces. A distribution of matter/atoms sometimes can be modeled as constituting a space, for example a ball, a rubber sheet or a cup of tea.

All these type of spaces share in common several features, curvature being one of them.

kent davidge said:
It's not made of matter/energy. Space has a broader definition in mathematics. Spacetime is one of such spaces. A distribution of matter / atoms sometimes can be modeled as constituting a space, for example a ball, a rubber sheet or a cup of tea.
I agree that it has a broader definition in mathematics. But its not just limited to a mathematical description. Its curvature has direct impacts on us, on reality (e.g., gravity).

bcl said:
If spacetime isn't nothing, are there theories as to what it could be?

It's spacetime. Why does it have to be anything else?

bcl said:
It doesn't seem to be matter or anti-matter. It is some kind of energy or dark energy?

No, it's spacetime. Again, why does it have to be anything else?

Do you ask what matter or energy are made of? If not, why not? Why do they somehow get a pass and not have to "be" something else, while spacetime doesn't?

PeterDonis said:
It's spacetime. Why does it have to be anything else?
No, it's spacetime. Again, why does it have to be anything else?

Do you ask what matter or energy are made of? If not, why not? Why do they somehow get a pass and not have to "be" something else, while spacetime doesn't?

That's a good point. I will have to contemplate the universe for a bit on that one.

bcl said:
But its not just limited to a mathematical description. Its curvature has direct impacts on us, on reality (e.g., gravity)
That doesn't mean its real in the sense you are thinking. Its only one description of this phenomenum called gravity. Consider that in Newtons theory (i.e. another description of the same phenomenum gravity) there's no curvature at all. You could say that "force of gravity" is real because it has an effect on us as we learn from Newtons theory. But in Relativity there's no force, what exists is spacetime curvature.

So you see that what exists or not, what is real or not, depends somewhat on the theory you are considering.

bcl said:
That's a good point. I will have to contemplate the universe for a bit on that one.
Okay, I actually do ask what matter and energy are. Matter can be described by its mass, which is E/c^2 (so mass is just a form energy). Energy is the ability to do work. I can picture an energy field and the work it does on some mass moving through it. But, your point is well taken. Perhaps spacetime can be thought of as something in and of itself, something aside from matter or energy.

bcl said:
Okay, I actually do ask what matter and energy are.

Good, at least you're being consistent. And now I can give you an easy answer to all those questions: we don't know.

More precisely: if we take "matter" to mean "stuff that's made of atoms" (or electrons and quarks, at a more fundamental level--i.e., the fermions of the Standard Model of particle physics) and "energy" to mean "stuff that's made of light" (or more generally radiation, i.e., the bosons of the Standard Model of particle physics), then we don't know what matter, energy, or spacetime are made of, in the sense that they are the most fundamental things we have in our theoretical models; they aren't "built" out of anything more fundamental. Everything else is "built" out of them.

bcl said:
Perhaps spacetime can be thought of as something in and of itself, something aside from matter or energy.

That's the best we can do with our current knowledge, yes. See above.

bcl said:
Summary: Mass creates curvature in space, but how can nothing have form?

So I understand that mass creates curvature in space-time. But what I struggle with is how nothing has a shape. I picture the space around say a planet, but how does the space (nothing) actual have form? Does anyone have a good intuitive explanation for that?

Let's consider the geometry of space for a moment, rather than the geometry of the space time.

Specifically, let's consider the geometry of a plane, and the geometry of the sphere (the two dimensional surface of a ball or globe).

The plane is flat, the sphere is not-flat.

Euclidean geometry, which you are probably familiar with to some extent, describes the geometry of the plane, but it does not describe the geometry of a sphere.

You probably can't visualize a curved 4 dimensional geometry using only intuition, but that doesn't mean that it doesn't exist or that it can't be studied. I would recommend, however, dealing with a lower-dimension case, such as the surface of the sphere I mentioned previously, first.

I found "Curving", by E.F. Taylor, a chapter in "Exploring black holes", to be a useful introduction.

This particular chapter is available online from the autor's website at <<link>>.

I'll give a brief quote here, though it'll be better formatted, easier to read, and have diagrams if you visit the link.

I suppose I'd mostly recommend understanding "curved space" first, then understanding space-time and why it's unified, then finally tackling the idea of curved space-time.

It wouldn't be bad at all to interchange the first two points - i.e. you could try to understand space-time first, then understand curvature, but it sounds like you're more interested in curvature at the moment, so you might want to study that first.

Because understanding general curvature in N dimensions is rather difficult, I will repeat my recommendation to consider the simplest possible 2 dimensional case, the sphere, first, before attempting anything more general. This could be accomplished by studying spherical trignometry.

Taylor said:
Nothing is more distressing on first contact with the idea of curved space-time than the fear that every simple means of measurement has lost its power in this unfamiliar context. One thinks of oneself as confronted with the task of measuring the shape of a gigantic and fantastically sculptured iceberg as one stands with a meterstick in a tossing rowboat on the surface of a heaving ocean.Reproduce a shape using nails and string.Were it the rowboat itself whose shape were to be measured, the proce-dure would be simple enough (Figure 1). Draw it up on shore, turn it upside down, and lightly drive in nails at strategic points here and there on the surface. The measurement of distances from nail to nail would record and reveal the shape of the surface. Using only the table of these distances between each nail and other nearby nails, someone else can reconstruct the shape of the rowboat. The precision of reproduction can be made arbitrarily great by making the number of nails arbitrarily large. It takes more daring to think of driving into the towering iceberg a large number of pitons, the spikes used for rope climbing on ice. Yet here too the geometry of the iceberg is described—and its shape made reproducible—by measuring the distance between each piton and its neighbors.The event is a nail driven into spacetime.But with all the daring in the world, how is one to drive a nail into space-time to mark a point? Happily, Nature provides its own way to localize a point in spacetime, as Einstein was the first to emphasize. Characterize the point by what happens there: firecracker, spark, or collision! Give a point in spacetime the name event.
http://www.eftaylor.com/pub/chapter2.pdf

kent davidge said:
That doesn't mean its real in the sense you are thinking. Its only one description of this phenomenum called gravity. Consider that in Newtons theory (i.e. another description of the same phenomenum gravity) there's no curvature at all. You could say that "force of gravity" is real because it has an effect on us as we learn from Newtons theory. But in Relativity there's no force, what exists is spacetime curvature.

So you see that what exists or not, what is real or not, depends somewhat on the theory you are considering.
But Newtons theory is wrong, it's useful but not perfect. For example, it can't predict the orbit or Mercury. Newtons theory does not try to explain what gravity is. It just makes an approximation (usually a good one) of gravity. General relativity explains gravity more accurately and also "why" gravity is.

PeterDonis said:
Good, at least you're being consistent. And now I can give you an easy answer to all those questions: we don't know.

More precisely: if we take "matter" to mean "stuff that's made of atoms" (or electrons and quarks, at a more fundamental level--i.e., the fermions of the Standard Model of particle physics) and "energy" to mean "stuff that's made of light" (or more generally radiation, i.e., the bosons of the Standard Model of particle physics), then we don't know what matter, energy, or spacetime are made of, in the sense that they are the most fundamental things we have in our theoretical models; they aren't "built" out of anything more fundamental. Everything else is "built" out of them.
That's the best we can do with our current knowledge, yes. See above.
Ooh, that's a very good point and very enlightening. At the most fundamental level, we don't know what matter or energy are, so why should we know what spacetime is? At the same time though, I'm thinking "that sucks, we actually don't know what anything is."

bcl said:
At the most fundamental level, we don't know what matter or energy are, so why should we know what spacetime is?

You're assuming that all these things have to be "made of" something else. That might not be the case. It might be that the Standard Model fields and spacetime simply are the most fundamental things there are.

Think about it: suppose I told you that matter and energy and spacetime were all made of gronks. How would that help you? Note that the current position of string theorists is basically this, except they say "strings" instead of "gronks". Either way it just pushes the question back a step: what are strings/gronks made of? You can always keep on asking such questions. But that doesn't mean they'll always have answers.

1977ub
bcl said:
If spacetime isn't nothing, are there theories as to what it could be? ... It is some kind of energy or dark energy?
According to relativity it is the geometry of physics. Why should it be anything more or less than that?

bcl said:
But even describing it as flat does not make sense to me because we are still assigning a shape to "nothing"
We already dispensed with that. Please stop repeating it, it is tedious.

PeterDonis said:
You're assuming that all these things have to be "made of" something else. That might not be the case. It might be that the Standard Model fields and spacetime simply are the most fundamental things there are.

Think about it: suppose I told you that matter and energy and spacetime were all made of gronks. How would that help you? Note that the current position of string theorists is basically this, except they say "strings" instead of "gronks". Either way it just pushes the question back a step: what are strings/gronks made of? You can always keep on asking such questions. But that doesn't mean they'll always have answers.
Your right. Gronks would not help either. What helps me from this discussion is that spacetime is something fundamental. We don't necessarily know what it is, but it's something, and not nothing (like someone might think of the vacuum of space as). Therefore, I don't have to reconcile nothing having some shape.

Dale said:
According to relativity it is the geometry of physics. Why should it be anything more or less than that?

We already dispensed with that. Please stop repeating it, it is tedious.
Sorry I'm being tedious. I did not realize that you had to read and respond to all these.

Of course, there is no place in the universe that has nothing. For one thing, electromagnetic fields are everywhere. What do those fields consider to be "straight"? What does gravity consider to be "straight"?
But even without those fields or any matter currently in a part of space, what path would an unaccelerated particle take? All those things give a meaning to "curvature" in any part of space. The mathematics of "curvature" has meaning even if nothing is at a particular location because it tells what would happen if something did go through that location.

bcl said:
What helps me from this discussion is that spacetime is something fundamental. We don't necessarily know what it is
It is the geometry of physics.

Dale said:
It is the geometry of physics.
I don't disagree with this. I just have a hard time with what it means, it's kind of abstract.

bcl said:
I just have a hard time with what it means, it's kind of abstract.

If you want a physical meaning of spacetime curvature, that's easy: it's tidal gravity. In a flat spacetime, there would be no tidal gravity.

bcl said:
I don't disagree with this. I just have a hard time with what it means, it's kind of abstract.
You have a non-abstract physical table. That physical table has a flat surface and four legs that are perpendicular to the surface, parallel to each other, and equal in length.

Physics involves geometry. That is spacetime. Yes, we do use mathematical abstractions to represent it in our theories and analyses, but what the abstract math represents is every bit as concrete as the table. Geometry is clearly part of the physical world.

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bcl said:
I don't disagree with this. I just have a hard time with what it means, it's kind of abstract.

I suspect I fell victim to "Too long, didn't read", aka TL/DR.

So, the chapter title of the Taylor reference I quoted earlier (http://www.eftaylor.com/pub/chapter2.pdf) gives the shortest meaningful answer.

"Distance determines geometry".

There _are_ some fine points that these three words do not cover, but - that's the basic idea.

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bcl said:
But Newtons theory is wrong, it's useful but not perfect. For example, it can't predict the orbit or Mercury. Newtons theory does not try to explain what gravity is. It just makes an approximation (usually a good one) of gravity. General relativity explains gravity more accurately and also "why" gravity is.
This is not accurate. Everything we do in physics is to make models of how the world works. GR in no way makes a claim to know ”why” gravity exists. All it does is to describe gravity through the use of a 4-dimensional curved geometry. Just like Newton’s theory describes gravity as a force. There is no way we can ever know ”why” because, just like an insisting child you can keep asking that question ad infinitum. You can only know that your description is accurate to within experimental uncrrtainties.

bcl said:
Newtons theory is wrong, it's useful but not perfect.

We know that now, but for a long time, as far as everyone knew, it could have been perfect, since no experimental evidence was known that contradicted it. Right now, no experimental evidence is known that contradicts General Relativity; but that doesn't mean there never will be, any more than the fact that there was no known evidence against Newtonian physics in, say, 1800 meant there never would be.

bcl said:
Summary: Mass creates curvature in space, but how can nothing have form?

So I understand that mass creates curvature in space-time. But what I struggle with is how nothing has a shape. I picture the space around say a planet, but how does the space (nothing) actual have form? Does anyone have a good intuitive explanation for that?

In a way, saying that spacetime has a "shape" is the intuitive explanation.

What we know, from experiment, is that matter moves in response to other matter. How do we explain this? There are two things to explain. For example:

1) How does a satellite know that the planet is there?

2) How does the satellite know how to move?

Physics cannot give you the ultimate answer to these questions. All it can do is give you a mathematical model (plus perhaps an intitive interpretation of the model) that successfully predicts how objects move. To some people this is disappointing: "it sucks" as you say in a later post. But, physics - no matter how deep your theories go - is always going to be based on some postulates, which have to be taken as "the laws of physics".

The mathematical basis for GR is (these are the laws of physics):

1) Einstein Field Equations - determines a "metric" for spacetime, based on the distribution of matter.

2) Lagrangian principle - determines the equations of motion for a particle given a spacetime metric.

The "intuitive" explanation is that the metric defines a geometry, hence spacetime has a "shape", and particles follow that shape (in a sense).

A deeper theory might "explain" this further, but it would still be based in its own, perhaps more fundamental, laws of physics.

Finally, in modern physics you cannot entirely escape abstract mathematics. GR has a 4D "curved" spacetime; but, Quantum Mechanics is much worse, to the point where there are multiple, very different interpretations of the mathematics.

kent davidge

## 1. What is curved space-time?

Curved space-time is a concept in physics that describes the curvature of the fabric of space and time due to the presence of massive objects. It is a fundamental aspect of Einstein's theory of general relativity and helps explain the force of gravity.

## 2. How does curved space-time affect our perception of space?

Curved space-time affects our perception of space by causing objects to follow curved paths instead of straight lines. This is because space and time are intertwined, and the presence of mass or energy can cause the fabric of space-time to bend, altering the path of objects traveling through it.

## 3. Can we observe curved space-time?

Yes, we can indirectly observe the effects of curved space-time through phenomena such as gravitational lensing, which occurs when the light from distant objects is bent by the curvature of space-time around massive objects. This can create distorted or magnified images of the objects behind them.

## 4. How does curved space-time affect the concept of "nothing" having shape?

Curved space-time challenges the traditional idea of "nothing" having shape because even in seemingly empty space, the fabric of space-time is still present and can be curved by the presence of mass or energy. This means that even in areas with no visible objects, there is still a shape to the space-time fabric.

## 5. Are there any practical applications of understanding curved space-time?

Yes, understanding curved space-time is essential for many modern technologies, including GPS systems, which rely on precise calculations of space-time curvature to accurately determine location. It also helps us understand the behavior of celestial bodies, such as planets and stars, and can aid in the development of future space travel technologies.

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