How can “nothing” be warped?

It is said that space is empty, a void, a “nothingness.” Space is the lack of anything.

And, Einstein showed that “gravity” is the warping of spacetime. Objects with mass do not “attract” each other; they tend to move toward each other because their masses warp the spacetime between them.

But, if space is nothing, the absence of anything, then how can it be warped?

• Welcome on Physics SE :) Why do you think space is nothing? – Sanya Dec 25 '16 at 17:31
• Isn't space a "vacuum?" And, isn't a vacuum "nothing?" – shorewood200 Dec 25 '16 at 17:35
• I do not think so - space and vacuum are different in my mind. But let's wait for other opinions. – Sanya Dec 25 '16 at 17:41
• @shorewood200 Spacetime isn't 'nothing', and just as importantly, a vacuum does not imply nothing. In quantum field theory, the vacuum of a theory is not nothingness! – JamalS Dec 25 '16 at 18:02
• In addition space-time is not 'nothing'. Space-time is something (space time has a finite age, and began to exist with the big-bang.) – user34445 Dec 25 '16 at 21:53

[Disclaimer: this is a seriously hand-wavy answer.]

The easiest way to think about this, I find, is to ask the opposite question: why do we think that 'nothing' is flat? In other words, why should Euclid's axioms be true? Other than the fact that, in day-to-day experience they do seem to be true is there any other reason, really?

I don't think there is: I think what makes it surprising that spacetime isn't flat is our normal experience that it is.

So, there's no reason to think it should be flat, but that still leaves the question as to why energy-momentum should cause it to be curved. Well, again, we can think of this another way: rather than thinking that energy-momentum somehow causes spacetime to be curved, I think that a very natural view in GR is to say that energy-momentum is curvature: after all, that's how you know what's there: you compute some function of the curvature, and call it $T_{\mu\nu}$.

It's not quite as simple as that of course: there are lots of cases (Schwarzschild, for instance), where there is curvature but the solution is a static vacuum. But if there is curvature you know that something is going on involving energy-momentum somewhere near at hand, as it were.

So that's not really a serious physics answer, but it's how I think about it, when I do.

• I tend to think of energy-momentum as the same thing as space/spacetime. If the energy density is the same everywhere, your gedanken gridlines are all straight. But if you inject some energy at some location, they bulge outwards, and now you've got curvature. – John Duffield Dec 27 '16 at 13:25

You need to be careful about taking metaphors like the rubber sheet too literally. Spacetime is not a physical thing. It is a mathematical object, and more specifically it is a manifold equipped with a metric. Since it is not a thing spacetime cannot be warped. While it often a convenient metaphor to talk about spacetime being warped by matter, this is just a metaphor and is not the way general relativity describes the physics.

Einstein's equation relates the metric to the mass/energy distribution, but be careful about the physical interpretation of this. Einstein's equation tells us what the metric must be in order to match the mass/energy but there is no sense in which we start with a flat spacetime then warp it by introducing matter.

The relationship between the geometry and mass/energy can be subtle. For example the Schwarzschild and Kerr metrics contain no matter or energy i.e. for these geometries the stress-energy tensor is everywhere zero. The mass associated with these geometries is the ADM mass, which is a quantity calculated from the geometry. So in a sense in these cases it is the geometry of the spacetime that is responsible for the mass rather than vice versa.

• I think this is confusing: Schwarzschild & Kerr contain no matter or energy, but they contain singularities. If we assume singularities are unphysical then they presumably really do contain matter & energy in some sense. Drawing a conclusion from a solution which is singular strikes me as dangerous, useful though those solutions are. – tfb Dec 25 '16 at 19:12

But, if space is nothing, the absence of anything, then how can it be warped?

This is what Alfred Whitehead over a century ago called the "fallacy of misplaced concreteness." Or, as Alfred Korzybski put it decades later, "the map is not the territory." We use maps all the time, for all kinds of things. Those flat pieces of paper represent the Earth, or parts of it, but they are not the Earth.

Galileo, Kepler, Newton, and others laid out a somewhat dated map of the universe. (Their map replaced a much older map.) This Newtonian map viewed space as being three dimensional Euclidean space and viewed time as the independent variable that described motion in that Euclidean space. A newer map, laid out by Einstein and others, views space and time as not quite as separable as that older map. This newer map locally looks like that older map. ("Locally" has a strict mathematical meaning.) To extend that local look, one needs to view space and time as being "curved."

Is this curvature real? Asking that assumes that the map is the territory. Even worse, it assumes the Newtonian view is the correct map. The relativistic map does a better job describing the universe than does the Newtonian map. But it's still just a map. The map is not the territory.

The Pythagorean theorem can be extended beyond 2 dimensions, so that the squared distance between two points is $ds^2:=\sum dx_i^2$, where Cartesian coordinates $x_i$ differ by $dx_i$ between the two points. If you rotate the axes this value is unchanged; a length is a length. Using a technique called the calculus of variations, you can prove the shortest path between two points is a straight line, which may not sound very impressive but bear with me.

In special relativity the equivalent "invariant" quantity also depends on time differences, so we have to think in terms of spacetime instead of space. For close together spacetime events, the line element $ds^2:=-c^2dt^2+\sum dx_i^2$ with $c$ the speed of light in a vacuum, is invariant. However, neither the usual spatial length nor time periods is in general invariant. That's why relative motion can cause length contraction or time dilation. The shortest path or "geodesic" is now motion at fixed velocity, which according to Newton's first law is what happens when no force acts on you. So in relativity we obtain a geodesic path in the absence of forces.

In general relativity $ds^2$ is of the more complicated form $\sum_{ab} g_{ab} dx^a dx^b$ with $x^0:=ct$ for some matrix $g_{ab}$ that depends on the matter distribution. For example, near one big mass we have this approximation. As before the motion in the absence of forces is a geodesic, which has a $g$-dependent shape. Gravity is not considered a force in general relativity, so gravitational orbits are effectively just a generalisation of moving in a straight line at constant speed. Is space "something" or "nothing"? It doesn't matter: the matter distribution determines the shape of geodesics.

How can “nothing” be warped?

It can't be. There's nothing to warp. However...

It is said that space is empty, a void, a “nothingness.” Space is the lack of anything.

Who said that? Einstein didn't. Einstein said space was neither homogeneous nor isotropic where a gravitational field was. He said this:

"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials guv), has, I think, finally disposed of the view that space is physically empty."

He said space isn't nothing. It's a thing, not a nothing, something that is conditioned by energy in the guise of a massive star, this effect diminishing with distance.

And, Einstein showed that “gravity” is the warping of spacetime.

He didn't actually say that. Try finding where he said that. You won't be able to.

Objects with mass do not “attract” each other; they tend to move toward each other because their masses warp the spacetime between them.

That's the popscience version I'm afraid.

But, if space is nothing, the absence of anything, then how can it be warped?

It isn't warped, it's rendered homogeneous. And it isn't nothing, it's something. See this Einstein article from 1929 where he talks about a field as a state of space:

"The two types of field are causally linked in this theory, but still not fused to an identity. It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structure of the physical continuum is not completely described by the Riemannian metric".

A gravitational field is one state of space, an electromagnetic field is another. Also note the Robert B. Laughlin quote here:

"It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed... About the time relativity was becoming accepted, studies of radioactivity began showing that the empty vacuum of space had spectroscopic structure similar to that of ordinary quantum solids and fluids... Subsequent studies with large particle accelerators have now led us to understand that space is more like a piece of window glass than ideal Newtonian emptiness..."