General relativity introduced to us "space-time curvature", and also told us that space can be warped, deformed or curved when mass is acting upon it. Mass has atoms and particles inside but what space-time has? Is it a material thing and has particles like mass? What exactly is the "spatial points" that can be curved?

  • $\begingroup$ Here is another very similar question. physics.stackexchange.com/q/714427/37364 $\endgroup$
    – mmesser314
    Jun 26 at 3:21
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    $\begingroup$ Is the 2-sphere curved? Is it made of particles? $\endgroup$
    – WillO
    Jun 26 at 4:20
  • $\begingroup$ @WillO Yes. Do you think it is not made by atoms? Do you have a new theory about that? $\endgroup$
    – user339151
    Jun 26 at 4:44
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    $\begingroup$ I am entirely confident that the 2-sphere is not made of atoms. Do you think that the number 7 is made of atoms? $\endgroup$
    – WillO
    Jun 26 at 23:02
  • $\begingroup$ I think a theory of quantum gravity might answer this question. Or it could be left as a mathematical idea that has no physical meaning as being made out of something. $\endgroup$
    – Tachyon
    Jun 27 at 2:24

5 Answers 5


In theoretical physics there is always a 'choose your battles' judgement call to make.

My favorite example in the history of physics is Newton proposing the inverse square law of gravity. Many of his contemporaries objected to it on the grounds that no explanation was available as how how this universal gravity was supposed to act over the vast distance of space.

(Descartes had proposed a model that required only a concept of contact force. Descartes has proposed that all planets are surrounded by vortices, and the orbital motion of each planet is subject to a pushing effect from the surrounding vortices, causing the planets to circumnavigate the Sun.)

Imagine a version of history where Newton would have decided against publishing the idea of the inverse square law of gravity, because he could not explain it.

What Newton did was that he insisted: this inverse square law of gravity works so well, we have to accept it as is.

That is what I mean with: 'choose your battles'.

In the case of GR: In order to formulate the theory at all the concept of space-time curvature must be granted.

It's a giant step, because it means a shift to attributing physical properties to spacetime.

John Wheeler coined the expression:
Matter/energy is telling spacetime how to curve
Curved spacetime is telling matter how to move.

So the idea is: spacetime is an entity that is not only acting upon matter (telling it how to move), but it is reciprocal: spacetime is being acted upon. This recognition of reciprocity is a form of unification.

Matter has the property that it has discernable parts that can be tracked through time. A vivid example of that is the traces in a cloud chamber, and in a bubble chamber.

We have of course that spacetime does not have discernable parts that can be tracked trough time. And yet, in order to formualate GR at all physical properties must be attributed to it.

That does not necessarily mean that these physical properties are exhaustively understood. It means that there is sufficient understanding to narrow down to a quantative theory that describes the physics taking place.

As a long time stackexchange contributor I notice a recurring pattern: in many questions the words 'what exactly is' are used.

I surmise you have an expectation that since there is a quantative theory of spacetime curvature it must be the case that physicists have an exhaustive understanding of it.

Well, such an expectation of exhaustive understanding is totally unrealistic.

I submit there are always 'choose your battles' judgement calls at play.

In order to make progress at all the physicist is allowing educated guesses all the time. Sometimes such a guess turns out to fail, but then sometimes an assumption turn out to have great, great mileage.


What exactly is the "spatial points" that can be curved?

As far as we know, spacetime is not made of “things”. When we talk about curvature we are not talking about a material “thing” that is curved.

Curved space just means that the physical geometry is not Euclidean. For example, if you draw a triangle of three physically straight lines then in a curved space you will find that the angles don’t add up to 180 degrees.

  • $\begingroup$ I notice that you use the turn of phrase 'it just means that'. For comparison: when a object is dropped and it hits the ground, where does the kinetic energy go? As we know, the object will be slightly warmer. The kinetic energy has not disappeared; it just means that the kinetic energy has shifted to thermal motion of the constituent molecules. The expression physical geometry of spacetime is not Euclidean is at the same level of in-need-of-explanation as curved space. The 'it just means that' suggests explanation is provided, but the two are actually at the same level of abstraction. $\endgroup$
    – Cleonis
    Jun 26 at 4:33
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    $\begingroup$ @Cleonis yes, that is why I immediately followed with a concrete example. Is the example insufficient? $\endgroup$
    – Dale
    Jun 26 at 10:17

Supposing $M >> m$, we can describe the movement of $m$ using Newton's law of gravity, and its second law:$$\mathbf F = m\frac{ \mathbf {d^2r}}{dt^2}= -GMm\frac{\mathbf {\hat r}}{r^2} \implies \frac{ \mathbf {d^2r}}{dt^2} + GM\frac{\mathbf {\hat r}}{r^2} = 0 \implies \frac{ \mathbf {d^2r}}{dt^2} + \nabla \Phi = 0$$ Where the second term was expressed as the gradient of the scalar function $\Phi = -\frac{GM}{r}$

It is a vectorial equation, and breaking it into coordinates: $$\frac{d^2X^i}{dt^2} + \frac{\partial \Phi}{\partial X^i} = 0$$

It starts to look like a geodesic equation:$$\frac{d^2X^k}{d\lambda^2} + \Gamma^k_{ij}\frac{\partial X^i}{\partial \lambda}\frac{\partial X^j}{\partial \lambda} = 0$$ It can be shown that taking $\frac{\partial \Phi}{\partial X^i}$ as the connections of a $3+1$ spacetime, it is really a geodesic equation, and the curvature (Riemann) tensor derived from the connections is not zero.

So, even the Newton's law of gravity can be interpreted as a geodesic in a curved spacetime. The word 'curvature' must be understood as mathematical concept, that is related to our intuition of a curved surface in some cases, but that is not limited to that intuition.


General relativity can be viewed another way than space-time curvature: e=mc2 therefore gravitation energy has mass that follows Newton. Of course, this is a feedback situation: when gravitational potential (energy) increases, so does energy=mass; when it 'runs away' we get a black hole.

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    – Community Bot
    Jun 26 at 18:57

Spacetime is a thing, it has reality. It allows the placing of something somewhere, in some orientation and at sometime and hence allows motion. It should not be confused with the void which because it is nothing, it has no properties, and certainly not the property of putting something somewhere. Spacetime is the medium of all motion.

Historically speaking, from Newtons time, absolute space and absolute time was the medium of mechanical motion and from Maxwell's time, the luminiferous ether was seen as medium for the motion of electromagnetism. After Einstein's SR, Newtons absolute space and time was welded into a unity and the lumineferous ether, as is commonly suggested in textbooks, was rejected. But this is not what Einstein himself thought. He said in a lecture at the University of Lieden, 1920:

Certainly from the standpoint of the special theory of relativity, the ether hypothesis appears at first as an empty hypothesis ... but on the other hand, there is a weighty argument to be adduced in favour of the ether hypothesis. To deny the ether is ultimately to deny the ether has no mechanical properties. The fundamental facts of mechanics do not harmonise with this view.

In fact, after Einstein's GR, one can identify the gravitational and lumineferous ether with spacetime itself. Whilst the weak and strong force suggest that spacetime has additional internal structure. Spacetime, then, can be identified with force at the fundamental level.

Having described it as something, what else can we describe it as?

Well, nothing we have said so far supposes quantification. Time can pass, motion can occur, but we need not be able to quantify any of this. The principle of spacetime relativity, as enunciated by Liebniz, says in the presence of matter we can measure length, duration and angles. In brief, it has a metric. An empty universe need not have a metric, and in fact according to GR, pretty much doesn't.

So far, the reasoning has been classical. Quantum theory suggests that spacetime has a discrete character of some kind that is yet to be wholly elucidated. So in a sense, we should have something like atoms of spacetime. Loop Quantum Gravity, for example, demonstrates that there is a quanta of length, area and volume that is discrete.


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