I have read this sentence in an article:

The theory [of general relativity] holds that gravity is geometry: particles are deflected when they pass near a massive object not because they feel a force, said Einstein, but because space and time around the object are curved.

Do we have a physical evidence that this is in fact true? By evidence I mean stronger explanation than just Occam's Razor, that dictates that bent curvature of space is more plausible/shorter/elegant explanation than let's say gravitational pull being mediated by graviton particle?

EDIT: I understand now that my question does not make the sense I thought it did. Gravity explained by a force between two objects manifests differently than gravity explained by curvature of space and we have measured the difference. Maybe the first part of my question may be of some interest - whether there are other indication of curved space - but I understand now it is the ONLY valid metaphor we have.

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    $\begingroup$ You can't prove anything in physics. We observe, test, and propose axioms (aka laws) which fit known data and correctly predict future events. To that extent, GR is a valid measure of the universe. $\endgroup$ Nov 18, 2015 at 13:44
  • $\begingroup$ Didn't Einstein also state that mass created space-time? I don't think that is well-regarded at this point. $\endgroup$
    – Jiminion
    Nov 18, 2015 at 19:08
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    $\begingroup$ Does this answer your question? General relativity: is curvature of spacetime really required or just a convenient representation? $\endgroup$
    – Alex R
    Jun 28, 2020 at 0:25
  • $\begingroup$ Why not both??? $\endgroup$
    – Prahar
    Jun 28, 2020 at 20:34

6 Answers 6


I found the Weinberg passage, but to quote it I need to do it in an answer (too long). So here it goes.

We have seen in this chapter that the nonvanishing of the tensor $R_{\lambda \mu \nu \kappa}$ is the true expression of the presence of a gravitational field. We also saw in Chapter 1 that Gauss was led to introduce the Gaussian curvature $K = -R/2$ as the true measure of the departure of a two-dimensional geometry from that of Euclid, and that Riemann subsequently introduced the curvature tensor $R_{\lambda \mu \nu \kappa}$ to generalize the concept of curvature to three or more dimensions. It is therefore not surprising that Einstein and his successors have regarded the effects of a gravitational field as producing a change in the geometry of space and time. At one time it was even hoped that the rest of physics could be brought into a geometric formulation, but this hope has met with disappointment, and the geometric interpretation of the theory of gravitation has dwindled to a mere analogy, which lingers in our language in terms like "metric," "affine connection," and "curvature," but is not otherwise very useful. The important thing is to be able to make predictions about images on the astronomers' photographic plates, frequencies of spectral lines, and so on, and it simply doesn't matter whether we ascribe these predictions to the physical effect of gravitational fields on the motion of planets and photons or to a curvature of space and time. (The reader should be warned that these views are heterodox and would meet with objections from many general relativists.)

I could complement what Weinberg said, but... Well, I don't see why. I think it's pretty clear and awesome.

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    $\begingroup$ That was Weinberg's view in the 70s, but I think he has since changed his mind. And as Weinberg himself notes, this is a minority opinion. Probably more so today than when Weinberg wrote his book. If you had quoted from the contemporaneous Misner, Thorne, and Wheeler instead the view would have been very, very different. That book emphasizes geometric thinking throughout. $\endgroup$ Nov 18, 2015 at 16:56
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    $\begingroup$ Yes, I agree. I just think that Weinberg's view, as he himself said, is heterodox and very interesting. It might have changed over the years, but I don't see why the non-geometrical point of view can't be taken into account. As the quote says, "it simply doesn't matter whether we ascribe these predictions to the physical effect of gravitational fields on the motion of planets and photons or to a curvature of space and time", so I thought this point of view deserved to be mentioned. $\endgroup$ Nov 18, 2015 at 17:05

There is no %100 proof in science; at least not for good science. It's always a question of being the most accurate / descriptive / useful theory. For example, Newtonian gravity is 'true' to the extent that it is very effective in a huge diversity of situations. General Relativity (GR) includes all of the accuracies of Newtonian Gravity, and then also explains a huge variety of additional phenomenon where Newtonian Gravity fails. We think that there are places where GR is incomplete: when you need to also describe things on the quantum mechanical scales. There are also some quirks about the 'dark sector' (Dark Matter, and Dark Energy) that we don't really understand. But, for all intents-and-purposes, GR can satisfactorily explain all observed gravitational phenomenon, including a wide variety of 'Tests of GR' --- which, very importantly, no other theory is able to do.

At the same time, the description of gravity through General Relativity is intrinsically that of 'curved' spacetime. The 'metric' of GR is fundamentally, and inextricably, a description of the very geometry of 3+1 spacetime which, purely from that, describes all of the resulting gravitational dynamics. A GR description is effectively synonymous with a curved-space-time description. To my knowledge, this is also unique to GR. Thus, by demonstrating the accuracy of GR, the validity of viewing gravity as curved spacetime is demonstrated. As described in this first paragraph, this should still subject to the same interpretation that this is currently the best description of observable properties of the universe.

Tests of General Relativity (brief summary, see wikipedia for details)

  • "Perihelion Precession of Mercury" (dynamics) : Mercury's orbit is not perfectly closed ellipses, but instead 'precesses' (rotates slightly). The exotic dynamical effects have also been observed by space missions, and in the dynamics of stars moving near the massive black-hole in our galactic center.
  • "Deflection of light by the sun" (lensing) : the path of light is observed to be deflected by massive objects. This was observed in the light of single stars moving behind the sun, but has since been extended to examples of completely distorted or duplicated images of galaxies, or subtle statistical effects to large fields of distant galaxies. The cause of the deflection (as read by GR) is very literally curvature of spacetime --- causing all objects (even photons, without rest-mass) to be deflected.
  • "Gravitational Redshift" (light travel) : The frequency of light is 'redshifted' as it changes it's depth in gravitational potential wells. Similarly, the delaying effects of warped spacetime have long been observed, and is a very important component of how GPS works.
  • Binary Pulsar (Gravitational Waves) : The orbital decay of the 'Hulse-Taylor' Binary-Pulsar is consistent with the emission of gravitational waves to an incredible precision, and won a Nobel prize. Gravitational waves are, very literally, traveling ripples in space-time which are able to carry energy.
  • Cosmology : The expansion of space, and especially inflation, fit very nicely and naturally into the GR context --- because it describes the space-time itself, explicitly, instead of only the objects inside of it. The only alternative explanations we have for these observations are extremely convoluted - and require many different tools for different regimes (i.e. changing mass/light-speed etc may kind-of explain expansion observations, but you need something else to explain homogeneity and yet another thing for the horizon or monopole problem, etc). This is another example of a dynamic, flexible spacetime being employed.

In the very near future we expect to directly detect gravitational waves using Pulsar Timing Arrays and ground-based Laser Interferometers. This would be a 'nail in the coffin' for interpretation of gravity as spacetime.

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    $\begingroup$ Nobody is denying that General Relativity has been experimentally confirmed many times. Your answer does not explain how is the concept of curvature of space more accurate / descriptive / useful theory than let's say the concept of gravitation force or field filed. $\endgroup$ Nov 19, 2015 at 14:53
  • $\begingroup$ @daniel.sedlacek, I'm sorry I was off-focus. I've tried to update my answer. The crux is that GR is effectively synonymous with space-time curvature. Let me know if there's still anything unclear. $\endgroup$ Nov 19, 2015 at 15:14
  • $\begingroup$ I agree with daniel - the issue was not whether GR is accurate, but has to do with interpretation. Interpretations are important, because they guide the mind when innovating and finding new ideas, among others. From a GPS engineer's point of view, it's of course irrelevant, but from a researcher's PoV it might be instructive to try to see things in other ways sometimes. Having said that, I think the current graviton models are for conceptually modelling quantum gravity in the weak field limit. $\endgroup$
    – BjornW
    Nov 19, 2015 at 15:58
  • $\begingroup$ @BjornW, please let me know how I can better address that issue. $\endgroup$ Nov 19, 2015 at 16:00
  • $\begingroup$ I'm not sure, as there is no real quantized theory of gravity (so far :). I guess it's the premise of the question which was a bit optimistic. I noted that the OP added an edit to the question as well about this. $\endgroup$
    – BjornW
    Nov 20, 2015 at 17:02

There's a very real phenomenon called 'Gravitational Lensing', in which light is bent from its original trajectory by a massive enough cluster of matter (which curves the space-time around it). Moreover, it's bent by a different amount than predicted by a simply application of Newtonian ideas, as kindly pointed out by Rob Jeffries. Is this evidence enough? =]


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    $\begingroup$ You might add - and bent by a different amount than predicted by a simply application of Newtonian ideas. $\endgroup$
    – ProfRob
    Nov 19, 2015 at 15:30

One must distinguish between the common meaning of curvature (called extrinsic curvature) from the mathematical meaning of curvature used in general relativity (intrinsic curvature). Intrinsic curvature can sometimes be represented as extrinsic curvature, but generally speaking this is unhelpful, and it is unfortunate that popular accounts tend to focus on this kind of representation.

Intrinsic curvature does not imply that anything is "bent" in the usual sense. It means that the theorems of Euclidean geometry do not apply to space, and those of special relativity do not apply in spacetime (except in local approximation).

We can see the truth of this by recognising the daily fact that clocks on GPS satellites do not keep time with identical clocks on Earth. The laws of physics are the same on GPS satellites as they are on Earth, so the local speed of light is the same on a GPS satellite as on Earth. It follows that the metre is affected, and that the circumference of the satellites orbit cannot be exactly $2\pi R$, as would be the case for a circular orbit in Euclidean geometry. This is precisely what we mean by intrinsic curvature.

  • $\begingroup$ Somehow, I think that this was not suitable as an answer, this does not attempt to answer the question at all! $\endgroup$
    – PNS
    Jun 28, 2020 at 9:11
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    $\begingroup$ @PNS, I do not see how you can say that. The question cannot be answered without clarifying the distinction between intrinsic and extrinsic curvature, and the answer gives a simple demonstration of the existence of intrinsic curvature, which is what was asked. $\endgroup$ Jun 28, 2020 at 9:19
  • $\begingroup$ Maybe I am historically wrong, but it was already known from the Equivalence Principle that time slows down in gravitational fields. The curvature insight first occurred when considering tidal effects. So, this is not the best example to provide for intrinsic curvature. More accurate examples could be found in DilithiumMatrix's answer and Yuval Weissler's answer. $\endgroup$
    – PNS
    Jun 28, 2020 at 9:31
  • $\begingroup$ @PNS, the curvature insight was discovered through Einstein's exploration of how to mathematically describe the implications of the equivalence principle, which led him to study differential geometry and tensors. Historically, this is precisely the best example. More importantly, it is logically the best example, because it uses simple deduction from well established empirical evidence. $\endgroup$ Jun 28, 2020 at 10:22
  • $\begingroup$ Still feel that better examples can be found. Gravitational time dilation is an important consequence of GR, but not an explicit proof. Deflection of starlight, the perihelion of Mercury, those are much better ways to prove the curvature part of GR. $\endgroup$
    – PNS
    Jun 28, 2020 at 12:25

Science does not allow us to be sure of what things are, but rather of what will happen: scientific knowledge, and "truth", is about consequences, implications, and relations more than it is about "what things really are".

The purpose of building theories is to try and describe patterns of cause and consequence, so that we may extrapolate them to domains where we have not yet explored and/or better understand how that what we get when we do explore those domains relates to what we have already explored. That is, so that we can ask a question of the form "what will be the consequence if I do X?" and be able to have a trustworthy answer even without necessarily actually going and doing X (which may not be feasible).

Hence while it's common to hear it, ideas that there are "true" and "false" theories aren't really correct: there's only better and worse theories in terms of being able to cover a larger domain and make fewer incorrect inferences regarding those consequences - but no theory can be assured of capturing everything for the totality of all empirical investigations is only ever going to be finite. It is entirely possible (though we have no a priori reason to assume) that, say, the model of things as "spacetime" actually fails if you could somehow manage to "get over the cosmic horizon" - or even just if we were to set out suitably long into space now. It is possible that subatomic particles may really be little gnomes. It's just that there isn't anything unambiguous to be gained in terms of extrapolating the patterns of consequences we see.

You can also think of it as a form of "data compression" that we perform using our intelligence: a theory compresses a large sum of empirical data - perhaps imperfectly - into a small, cogent set of generating rules. Indeed, this is how proper, dumb, data compression algorithms work: they try to find patterns they can use to shrink the size of a piece of input data. And just as with data compression, the more fidelity and more data they can get while still retaining a reasonable size output, the better. But the compression is not unique: different algorithms may produce very different compressed outputs, and likewise there may be a variety of very different theories that we can use for "compressing" what we have now.

Hence, if there's anything that science does say about "what things are", it's that they are, in fact, compressible as such with such good fidelity.

So back to the question - does "space" get "bent" by gravity or not? Well, space - or more properly space-time - is a human construct, an intellectual construct, or a social construct: it's something that we humans made up to try and compress our empirical data and it works pretty well and lets us extrapolate those consequential relations. In that regard, its behavior is fully defined by the theory we construct, so the answer to your question is yes, gravity bends - or better - is a bending of - spacetime, because that is part of what "spacetime" as we've defined it, and found makes it useful in this regard, is. And it's also the most widely-applicable such construct we have so far - but not necessarily the most convenient or most useful: in day-to-day work, the simpler Newtonian system is entirely adequate.


The post by Dilithium Matrix is rather helpful and fairly informative if what you truly care about here is the epistemological (experimental) value of general relativity. No physicist should be too worried about the experimental falsification of general relativity given the many numerous experiments that have been conducted in which it passed with flying colors. If your concern was more about the coherency of the mathematics then any mathematician worth their degree would of found that discrepancy in the mathematics long by now.

What you really care about seems to be the interpretation of the theory in question which is closer to a philosophical question than it's a physics one (though physics and philosophy frequently overlap). There has been a long history among philosophers and physicist who have discussed the reality of space-time as well as the relationship of that discussion to more modern theories such as general or special relativity. There have been philosophers in regards to general relativity who have argued both ways in how to interpret general relativity whether there really is an existent spacetime that curves or if its a more complex set of relationships between material particles. The most I could glean from these discussions was that it was indeterminate whether we should interpret general relativity in such a way that spacetime exists and explains the motions of material objects or it does not exist but it's the physical relations between material objects that gives raise to this phenomenon.

Here is an encompassing book here which outlines that discussion both historically but also in regards especially to general relativity. Here is a post from stanford encyclopedia of philosophy which also covers this discussion up to even a more recent investigation into dynamics without getting too preoccupied with the mathematics.

Sincerely, a college freshman going on sophomore year


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