# What's being curved, when mass bends the spacetime continuum?

## Main Question or Discussion Point

What's being curved, when mass bends the "spacetime continuum?"

Okay, I've read lots of books about Einstein's spacetime being curved by mass and all, but I've never read a really good hypothesis as to what, exactly, is being curved. You can't curved nothing (sounds like a double negative, but it's not, I guess).

If there is nothing present, then there's nothing to curve. And, if something is being curved, then it's some kind of entity/phenomenon that being acted upon by the mass in some way. What is that?

All of the nice graphics of rubberlike sheets and their grid lines being bent in the locale of a star or whatever are great to look at, and they get the basic idea of mass being responsible for the curvature of spacetime, but those are simply two-dimensional models of something much different that's taking place in three space dimensions and one time dimension.

I'm not really expecting a definitive answer, as such, because, if anyone had one, they'd be in Stockholm, standing before the king of Sweden. So, please, if you respond, don't pontificate. I would just like to hear what others might visualize, when they think of "spacetime" in this context.

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Drakkith
Staff Emeritus

Well, unfortunately this is hard to answer. Spacetime geometry, which is what General Relativity describes gravity as being caused by, is described by a mathematical tool known as a Metric. This is basically a way to organize numbers together in order to do specific kinds of calculations. When we speak of gravity we say that "the metric itself" is being curved or "it's in the metric". However this seems to avoid the question since a metric is just a way of describing something and working with numbers.

Since the metric describes how things will act in a specific area of spacetime I find it usually accurate enough to say "spacetime itself" is curved. But again, this avoids the question. The only answer I could give you that would be appropriate to me is the following.

The way objects interact in spacetime can be described as if there is an "underlying structure" known as spacetime. It would be this structure that would be curved. It is, in it's basic form, simply advanced geometry.

Whether spacetime is "something" or not is beyond our capability to answer at this time.

Well, thanks for that answer, "Drakkith." Yeah, it's certainly one of those unfathomable things, isn't it? Like you said, calling it by a name, like a "metric," is really just avoiding the whole question. And like I said in my original post, I realize that nobody has the answer, of course, but I guess it's just kinda fun to toss out such impossible questions, to see what others think. I wish that I had the smarts to even go as far as those who can comfortably dwell in the thought realm of such advanced geometry, even though they don't have the answer, either. Oh well, I can take solace in the knowledge that, at least in this one thing, I'm not so far behind, lol!

look at it this way: lets say you have a piece of graph paper.

you draw an equilateral triangle on it.

then press down on the center of the triangle with a pencil. lets imagine the graph paper doesn't exist, but the *x-y coordinate system still does*. this is a 2-D analog of what actually happens.

Hi "chill factor," thanks for the reply, but you just re-stated what I already said about the nice, two-dimensional models of planes containing graphic coordinates that are fun to look at, but you've said nothing about what the spacetime continuum actually is. Nothing there directly addresses the question of what actually constitutes "spacetime." But, like I also said, I don't really expect a definitive answer. If anyone had such an answer, it would make headlines all over the world. Whatever spacetime is, it isn't matter in the normal sense of the word as we know it, or it would be visible, so the oft-used term "fabric" of spacetime is obviously only a metaphor. The metaphor is good for helping us to visualize the interaction between mass and this so-called "spacetime continuum," but it does nothing to explain its nature. Using cartesian coordinates to explain it is only sidestepping the question, because x-y coordinates are just x-y coordinates, part of a mathematical construct; they are not the spacetime continuum, itself.

a useful way to look at it is that spacetime is just a mathematical coordinate system that gives you the right answers.

i'm no astrophysicist but qualitatively, what i've been told is that if you were in a curved space time, you'd have absolutely *no idea* you were in a curved space time as opposed to just accelerating in a flat one without the use of measuring instruments or looking outside. even if you sailed right past the event horizon of a black hole, you'd have *no idea* you passed it until (classical) tidal forces start getting too strong for your own structural integrity. this is called Einstein's equivalence principle. Since the 2 cases are indistinguishable, is it really useful to talk about "curved" spacetime as if it was absolute and actually curved?

"a useful way to look at it is that spacetime is just a mathematical coordinate system that gives you the right answers."

Yes, it's a useful coordinate system, for sure; e.g., it allows you to perform astrophysical calculations, much as they did during the "Boomerang" experiment, when they combined their results with those of the COBE satellite, to determine the geometric structure of the universe. In order to do that, they had to deal with incredible complexity, correcting for such things as space density, differences in cosmic temperature as a function of their telescopes' angular resolutions, the cosmological constant, etc. Really heady stuff. The take-home message was that the universe is a four-dimensional structure, but is flat. But all of that still does not address the question: what is the spacetime continuum? What is flat?

A coordinate system is not the answer. It is only a mathematical construct that came from the human mind. Numbers are not things; they only represent the idea of "numberness"/how many things. They are tools, nothing more. Same with coordinates. Coordinates don't really exist, except in the human mind. Just like there is no such thing as a pure mathematical point, a line, a plane, or a three-dimensional shape, like a cube. Those only exist in the human mind. As soon as you try to draw a line with something like a pencil, you have made something that would be, to a bacterium, a long hill of carbon atoms. You haven't really produced a line, because you can't. Bu then I'm sure that I'm stating what you already know. The point I'm trying to make is that it's the same with the idea of saying that the spacetime continuum is an x-y coordinate system. The x-y coordinate system is only a human construct that is used to represent the continuum...it is not the continuum, itself. An x-y coordinate system is useful, but it doesn't bend.

The whole thing reminds me of how successful, yet unfathomable, quantum electrodynamics is. It's a wildly successful theory that produces incredible precision, as expressed by Feynman, when he compared it to producing the precision of being off by only a hair's width, when measuring the distance from New York to Los Angeles. And, yet, in the next breath, he admits that no one understands how those quantum equations really work. The equations are one thing, but the reality is something else again. Crazy, fascinating stuff.

Drakkith
Staff Emeritus

Let me ask you this. Does anything require that spacetime be something, as opposed to just being geometry? Hope that makes sense.

I've been wondering about this for a while now. In order for space to be warped, space itself has to be made of something or have something affecting it which will incline it NOT to be warped in the absence of mass... Sucks that all we can reall say is, "well it's math, get over it." haha

I honestly know next to nothing about physics, but the way I look at it is this, if Space were made up of - or was being affected by something theoretical like (X) and I mean it was just completely jam packed full of (X), and mass has very little (X) or even none at all, then a rock (which has mass) would have equal pressure from all sides pushing on it at all times by (X). Then, when that rock passed close enough by another rock which was also being squished on all sides by (X), there would be less pressure acting on each rock in the area between the two making them move towards each other.

So if I'm right, your question is: what is this (X) which keeps space pushing all this mass together. No idea.

I kinda like those analogies which use the rubber sheets with metal balls on them where the ball kinda sink in showing "warping". Thing is, we know that the connection and elasticity of the rubber is what makes the sheet a sheet and able to support the weight of a ball instead of a bunch of pellets of rubber the ball falls though or breaks apart. If I'm right, what you're asking is: what is this elastic force keeping mass from breaking space or falling through instead of just bending it. No idea. I definitely think (X) is a thing though, probably no one interested in entertaining the idea because it's probably unprovable. Shoot, following this line of reasoning, it's possible black holes are literally where all that mass actually did break through. I'm sure this sounds like complete rubbish to everyone, keep in mind it's all theorizing and speculation :)

Either way, that's the question I'd like to know. If space IS made of something like (X) which keeps it together and nice and taut, I've got lots of theories that would change the way we looked at the big bang.

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Hmm, now that's an interesting way to look at it, alright. I dunno. I guess that, just offhand, I'd have to wonder how you might define geometry, in that case. To me, the term "geometry" can have two meanings: it's either a pure branch of math., in which case it's once again only a human construct (and pure human constructs/concepts can't be curved or bent, as stated)...or it's a term used to describe something (as in saying "the geometry of spacetime"), in which case we're back to talking about a real thing that can be curved or flat--spacetime.

But, in a more practical, less semantical, sense, I would have to ask how a pure geometry could interact with energy, such as light. If mass can warp spacetime, as has been demonstrated so many times (gravitational lensing, for example), then it surely can't just be "geometry." It would have to be the geometry/shape/structure of...something. How could you have a geometry of nothing? Sounds kinda crazy. After all, one definition of physics is that it's the study of the interaction between matter and energy.

In reading the last sentence of my own last post, I just realized that it might have sounded as though I was implying that spacetime might be matter in the ordinary sense. ("...one definition of physics is that it's the study of the interaction of matter and energy") I definitely was not implying such a thing. Whatever the "spacetime continuum" is, it doesn't seem to be matter of the ordinary type. I suppose it could be some kind of exotic matter, the nature of which we've yet to discover, as in the idea of "dark matter." The Michelson-Morley experiment, among other things, seemed to have disproved the existence anything in space that could interact with normal matter.

Drakkith
Staff Emeritus

I think a key point to understand is that eventually all of science boils down to "just math". So what we think of as "real" and what isn't gets kind of muddled up when we get down to it.
Objects are made out of composite particles which are made up of fundamental particles which we describe using a mathematical model that tells us how and why they interact the way they do. It's like asking "what is an electron". I could only tell you it's properties according to the model and how it interacts with other objects, which are all described by a model as well.

If we add gravitons to the mix, do we still get warping space? I've always visualized the bed sheet pictures as a graphical representation of how gravitons work. But I'm not sure if they conflict with GR or not. If gravitons are in play, then maybe nothing has to warp at all?

I don't think gravitons as a concept are useful at all. It is impossible to detect a graviton *even in principle*. There was an arxiv paper on this. If you can't detect it by any means, even in principle, does it really exist?

Geometry does exist independent of matter: the distance between something. How far is it between you and the computer screen? What if you took a step back? What changed? Your distance. But how do you know it changed? Did something get *added* between you and the computer? Maybe if you were in air, but what about in vacuum?

Now lets imagine that the closest distance between 2 things was *no longer a straight line*. Does that count as geometry being wrapped, without any matter being wrapped?

I think a key point to understand is that eventually all of science boils down to "just math". So what we think of as "real" and what isn't gets kind of muddled up when we get down to it.
Objects are made out of composite particles which are made up of fundamental particles which we describe using a mathematical model that tells us how and why they interact the way they do. It's like asking "what is an electron". I could only tell you it's properties according to the model and how it interacts with other objects, which are all described by a model as well.
I disagree. There's some science that's clearly real and not just math. for instance, we have all fields of engineering, materials science, biology, chemistry, geosciences and many fields of physics such as biophysics and optics. I don't think anyone can say that a rock or dog doesn't exist and is just math. On the other hand, alot of physics has gotten to the point where experiments are impossible to verify the theory, even in principle. I mean, there's predictions of particle physics that simply cannot have experiments done on them, there's predictions of astrophysics that can never be observed... at some point you just ask "why?"

Bandersnatch

I don't think anyone can say that a rock or dog doesn't exist and is just math.
And here's where you're wrong. ;)
http://arxiv.org/abs/0709.4024
article said:
I advocate an extreme "shut-up-and-calculate" approach to physics, where our external physical reality is assumed to be purely mathematical. This brief essay motivates this "it's all just equations" assumption and discusses its implications.

HallsofIvy
Homework Helper

"For the purposes of this problem, lets assume a spherical dog"!

Actually, once in on a problem to calculate how long it should take to hard boil an egg, I used Bessel Functions rather than Spherical Harmonics. Fortunately the professor had a sense of humor- he wrote "I see you prefer cylindrical eggs" but gave me partial credit.

Hi.

I understood the original question in two ways.

A) What exactly is space-time/geometry/metric curving with respect to?

B) What exactly is space-time?

Rendering A) has a simple answer. Namely, curvature of space-time is expressed using metric tensor $g_{\mu \nu}$. This metric tensor describes how to bend straight lines of the underlying totally flat infinite dimensional eucledean space in order to become light paths. Yes, there is an underlying archetypal space behind any metric. Namely, one has $g_{\mu \nu}=\partial y^n / \partial x^\mu \; \partial y_n / \partial x^\nu$. There is an underlying function $y$ defining geometry of space-time. We can always imagine any curved space-time as embedded in larger flat space of higher dimension.

So, what is then this underlying archetypal space? This leads us to rendering B)...

Rendering B) has no simple answer at all. That's because of quantum theory. If You want to find answers about anything particular, You have to measure it. So how do we measure empty space we want to learn more about? How can we measure nothing? Well, we can... This nothing when measured has fantastic properties. This nothing is called quantum vacuum. It causes particles to decay into void and to appear from void again. This quantum vacuum has enormous energy density and it fluctuates locally. When we talk about geometry we really talk about particles moving along some lines: lines that we then parametrize with respect to some coordinate system. The thing actually being measured is particle of course. And when measured, quantum vacuum becomes visible. The more energy particle has, the greater the interaction with quantum vacuum. In other words - space-time bends more. So Your answer might be within the realm of quantum field theory.

None of these 2 answer is simple. There is no simple answer to Your question. Actually, there may happen to be no answer at all... You basically asked what is space. Questions such as what is space, mass, time, ... have no answer. There are always some axiomatic underlying structures we understand effortlessly because we are programmed to, genetically or otherwise. No-one ever had to explain to any of us the concept of here and there, or the concept of now and yesterday and tomorrow, or the concept of this object and that object, or the concept of number one... Those are basic questions. If one is to answer such basic questions, one must answer them with respect to another god-given well accepted and respected ideals. So, in order to explain space, one may try to explain it referring to ... to what? See, it's a brick wall.

And yet, we are always so amused by just brain-storming such questions. Analytic thinking just has to stop at those questions. And stopping analytic thinking process is always a blast. But that's another topic for some other forum.

However, one level above these basic objects are measurable objects such as quantum vacuum and particles. So Your answer might be with them.

Cheers.

And here's where you're wrong. ;)
http://arxiv.org/abs/0709.4024
Nope I totally disagree with that approach, and that paper is philosophy, not science. Also, a supercomputer could not calculate the future evolution in the universe at arbitrary resolutions even in principle due to 1.) statistical fluctuations and 2.) chaotic phenomena that amplify those fluctuations. That's where experiment and observation comes in. There will also probably never be a theory of everything that would allow this.

Lol! This thread is quickly spinning off into a metaphysical one, with talk about "what's real," and physicists asking, "...at some point, why?" Sounds like we have the makings of a great wine-and-toga group, here. But then I guess that's about all we can do, really; just ask guys like Heisenberg and Kurt Goedel. Seems like we're doomed to always be a bit uncertain and incomplete, in many ways.

By the way (and as a kind of a picayune aside), a physicist would never ask "why?" about anything, but rather "how," since "why" implies creative design, or, purposiveness by a Designer, which lies outside the realm of modern physics. Not that I, personally, am close-minded to such a thing, but it just wouldn't normally be a part of our society's mainstream physics dialogue. Actually, I ask "why?" all the time, but that's another whole matter. So, although I would have to agree with you on that, "chill_factor," a hard-core physicist probably wouldn't risk his/her scientific reputation and grant money on such a question.

Like "Drakkith" said, in our quest to understand it all, we descend downward, from molecules to atoms to protons to quarks to maybe strings, to who knows what? Ultimately, pure energy in some form, according to Einstein. But, at that point, we'd realize that we've been just kinda stalling for time, as it were, because we'd then be faced with the question "Then what's energy?" And so on, ad infinitum. It's enough to make your head swim, because each questions creates the next.

But back to the topic at hand. I would also have to agree with the statement that geometry exists. Of course it does. It's a whole branch of mathematics that's undeniably real. But so what? What does that have to do with the discussion of spacetime's real nature? Measuring the distance between a person and a computer screen, before & after taking a step back, has absolutely nothing to do with the space in between the person and the screen. The measurement is only a measurement, not the thing being measured. If I measure a fish, that doesn't tell me anything about the fish. Is it a salmon? a cod? Is it rotten? fresh? Measuring the shortest distance between two points on a sphere, such as the earth's surface, requires spherical geometry, geodesics, of course. But, once again, so what? Calculating that geodesic tells me absolutely nothing about the nature of the thing being measured, the earth. I don't see how your example relates to the question about the true nature of spacetime. It seems like we just keep saying the same things over & over again. Pass the wine.

Hi.

Ha ha, yes, we all do agree and yet we all have something to add!

We all do realize the question has no physical meaning: no interpretation within physics. On the other hand, it is amusing to ponder over it for a while.

Cheers.

pervect
Staff Emeritus

To and try and get things back on track, it is necessary and sufficient to be able to measure distances in order to define curvature.

So if you don't have any philosophical angst about measuring distances, you shouldn't have any philosophical angst about measuring curvature.

Philosophically it does require that one be able to "mark" events somehow in the space and/or space-time that one is measuring distances between, so that it's possible to talk about the "distance between events", even if actual events never happen (i.e. in a vacuum).

So, if you have a set of points, and you can measure the distance between any two points (or any two nearby points, if you prefer), you have enough information to calculate the curvature.

For the curious, the mathematical machinery looks something like this (I've simplified slightly)

$$\left(\nabla_a \nabla_b - \nabla_b \nabla_a\right)$$ defines a map from vectors to a rank 3 tensor defined at some point P. This map is a rank 4 tensor, called the Riemann curvature tensor.

All you need to know to be able to perform these calculations is contained in the (infinite) table of distances between all points in the neighborhood of P. Rather than specify an infinite table, we usually express the distances by a formula, which is known as a line element or metric.

IT's really simple, except perhaps for the details....

I might also add that it's profitable to consider measuringn the Lorentz interval when considering space-time curvature rather than the "normal" distance, but it' perfectly reasonable to use the curvature tensor on purely spatial slices using "ordinary" distances.

Hi.

We all agreed the original question tends to be philosophical... However, there might be a physical twist to it. Please allow me to explain.

If there is no mass present at all within some region of space-time, at that region one has
$G_{\mu \nu}=0$
Here $G_{\mu \nu}$ is Einstein tensor $G_{\mu \nu}=R_{\mu \nu}-1/2 R$, with $R_{\mu \nu}$ being Ricci tensor, and $R$ is scalar curvature. The point is: equation $G_{\mu \nu}=0$ characterizes empty space. There is nothing occupying the region. And we have curvature due to gravity alone. So there is gravitational field present in empty space.

If there is mass=energy present in region, then one has
$G_{\mu \nu}=-8 \pi T_{\mu \nu}$
Here $T_{\mu \nu}$ stands for 4-impulse flow density tensor.

Ok. Now suppose we don't have any particles nor fields in region. $G_{\mu \nu}=0$ should hold then. Let us now change referent potential energy level. Equation $G_{\mu \nu}=0$ now becomes
$G_{\mu \nu}=C$
Here C stands for constant. So, as we can see, trivial redefinition of referent energy level changes curvature. Term $C$ is also known as cosmological constant or vacuum energy density.

This brings us to this question: if there were no gravitational fields present at all and if the region was completely devoid of any energy, and if we then redefined referent energy level of this "empty", our equations defining completely empty space $0=0$ would change to $G_{\mu \nu}=C$. There would be curving suddenly...

So, what's curving when there's nothing there at all? And when presented this way, question does make sense physically. It's quantum vacuum. Quantum vacuum being physically measurable through particle scattering higher corrections and in measuring the Casimir effect, this makes question physically and observationally plausible.

I don't have anything else to say about it at this moment, but ideas will rise with time. Please do comment if You feel inspired by this topic.

Finally, I oversimplified my notation for the sake of simplicity. To be precise, one should have $C \equiv \Lambda g_{\mu \nu}$ and also $R \equiv R g_{\mu \nu}$.

Cheers.

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I think of the curvature of space-time as the way we perceive it straying from a flat space-time which is after all just a set of coordinates we define, and simply the way our universe works. Space-time is just a name, not physical. Curvature is the deviation because of mass.

Hi.

Yes, curvature is a deviation of photon's path from a straight line. Author's question is, as I get it: "photon's path in What?"

Cheers.

Photon's path in the coordinates we choose. After all, it follows different paths according to how we define coordinates and perspectives.