# I Curved Space-Time

#### bcl

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?

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#### Dale

Mentor
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.

#### jbriggs444

Homework Helper
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.

#### bcl

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".

#### kent davidge

@Dale , @jbriggs444 the OP wants to know "what the spacetime is made of"

#### bcl

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".

#### bcl

@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.

#### kent davidge

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 modelled 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.

#### bcl

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 modelled 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).

#### PeterDonis

Mentor
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?

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?

#### bcl

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.

#### kent davidge

But its not just limited to a mathematical description. Its curvature has direct impacts on us, on reality (e.g., gravity)
That doesnt 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) theres 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 theres 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

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.

#### PeterDonis

Mentor
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.

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.

#### pervect

Staff Emeritus
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

#### bcl

That doesnt 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) theres 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 theres 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.

#### bcl

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."

#### PeterDonis

Mentor
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.

#### Dale

Mentor
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?

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.

#### bcl

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.

#### bcl

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.

#### FactChecker

Gold Member
2018 Award
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.

#### Dale

Mentor
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.

#### bcl

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.

#### PeterDonis

Mentor
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.

"Curved Space-Time"

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