Einstein’s general relativity is hailed as being the supreme theory when it comes to  describing the macroscopic world. However, mathematicians have a modified theory that could more closely describe the universe, as well as removing some of the big problems posed by general relativity.

One of the popular phrases used to describe the nature of space and time through the eyes of general relativity is as follows;

“Spacetime tells matter how to move,

matter tells spacetime how to curve,”

Imagine the old bowling ball analogy for general relativity and spacetime. Spacetime is depicted as a rubber sheet whilst an object such as a bowling ball, often representing a planet or star (though it can be any object with mass) then deforms this rubber sheet when it is placed on top. This simple analogy is meant to represent the curvature of spacetime as a result of the mass of an object. The two key words here being; mass and curvature. Mass produces curvature. The larger the mass of the matter in question, the greater the curvature of the surrounding spacetime. A black hole for example, one of the heaviest objects in our universe, produces a curvature so large, a warping so great, of the surrounding spacetime that phenomena like no where else occur – see here and here. In fact, at the centre of the black hole where bending of the spacetime is so great due to the highly concentrated mass, the curvature is mathematically described as infinite – and our understanding of the nature of spacetime here breaks down completely. Such an extremity causes us a problem, we do not like it when our maths doesn’t work as a descriptive language for nature. These points of infinite curvature, called singularities in the nomenclature, create problems in the standard theory of general relativity. Take home message 1; general relativity only includes the mass of matter as a property affecting the behaviour of the surrounding spacetime and it does so by causing it to curve. If there was no matter in the universe, or if all matter was massless (for example photons) the surrounding spacetime would be flat – imagine a sheet stretched out completely taut. Take home message 2; the theory of general relativity causes us problems, which come in the form of singularities at points of extreme curvature.

An alternative theory, Einstein-Cartan theory, holds a possible key to these problems. Matter in our universe has two fundamental properties; mass and spin. Spin is a funny little characteristic of matter, it is not related the spinning of a particle in actual space, so don’t imagine a spinning top on a table but instead is the intrinsic angular momentum of a particle. If that doesn’t mean much to you don’t fret, but for the purpose of this article humour me and the particle physicists of the world and accept that spin is a fundamental characteristic of matter. See here for more reassurance. Einstein-Cartan theory, claims that both these fundamental properties of matter affect the nature of the surrounding spacetime, whereas general relativity only incorporates mass. Whilst mass created a curvature of the spacetime, the spin creates an effect called the torsion of spacetime – unfortunately a similar analogy to the bowling ball does not exist for a visualisation of this torsion (that I know of). The mathematics of Einstein-Cartan theory, with the inclusion of spin and the resulting torsion into the framework, has some very interesting implications. In standard general relativity, the black hole situation described above creates a singularity at the centre due to the infinite curvature of spacetime. However, when torsion is also in play at such extreme matter densities, the torsion field creates a repulsive force that pushes outward against this extreme warping. Instead of a singularity at the centre, in Einstein-Cartan theory the interaction between torsion and curvature creates a wormhole or Einstein-Rosen bridge, at the centre of the black hole. The wormhole creates a passage to a new, growing universe on the other-side of the black hole! The same can be said of the situation at the beginning of the universe. In standard general relativity, the big bang represents a singular point, of infinitely dense matter, from which the universe then somehow comes into existence. However in the Einstein-Cartan formulation, the torsion again creates an outward-type repulsion at these  such points of extreme curvature and density, forbidding them to occur. The big bang is then replaced by what is known as a big bounce scenario in cosmology.

The nice thing about Einstein-Cartan theory is that, because the extra features only come into play in extremely high matter density regimes, like the centre of a black hole, the tests which probe astronomical phenomena within our experimental reach still agree with the predictions from general relativity. For example, the perihelion of Mercury, a key test of general relativity, would still be true if working within the Einstein-Cartan formulation. Therefore, we can see the theory not as an opponent to general relativity but a slightly revision, extending its validity and in so doing presenting resolutions (and very exciting ones at that) to our previous trouble points. On the other hand, this is also unfortunate because given our nearby surroundings and inability to accurately probe quantum phenomena, we are not in a position to experimentally test the predictions of Einstein-Cartan theory. The experiments that we can conduct are ones where mass is the heavily dominant player affecting the behaviour of spacetime and spin effects remain physically hidden to us.

What we can say, without even commenting on the mathematical elegance of the theory is that Einstein-Cartan simply seems more wholesome ontologically. Why include one fundamental property of matter (mass) and not the other (spin)? The theory seeks to resolve this omission and unlock the missing results even if they cannot be physically probed. A curveball theory, where curvature is no longer the only behaviour of spacetime.

## 15 responses to “Not your “general” relativity”

1. Really exciting stuff. I didn’t know about ‘big bounce’ or torsion or even Cartan’s extension of general relatively for that matter. For experimental testing, I could only imagine someone like Cooper falling into the abyss and transmitting the much required quantum data 😛

Btw, could you talk about rotating black holes on RTU? The dynamics is vastly different in that one. Also which field are you doing your PhD in?

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• Thank you very much! Yes maybe interstellar could have a sequel where they take into account torsion in gravity 😉
Great recommendation on rotating black holes – I’ve been meaning to do a more detailed post on them for a while. In the meantime, if you check out ‘Black Holes #3’ there are some references to their unusual properties there.
Finally, I was doing a PhD in Astrophysics with a research focus on exoplanets but now I am trying to move into a position in Theoretical Physics as it is gravity, general relativity and black holes that I love to study.

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2. Small point: I think you meant ‘taut’ not ‘taught’…

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• I did! Thank you for your keen eye, I have corrected.

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3. You didn’t happen to read the recently published book by Michael Wall called “Out There” did you? Even the quote you used is referred to in that book. Just curious. It doesn’t diminish you blog, but there are no coincidences.

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• Ah no I have not read it. The quote is a very famous one in general relativity made by the leading theoretical physicist John Wheeler. I’ll check out the book if you recommend it. Thanks for reading the blog!

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4. Small theoretical point: you only have singularities in theories that assume the field is infinitely differentiable. If the structure of space is quantized, doesn’t the problem go away?

The irony of Einstein’s career is that he won the Nobel prize for noticing the behavior of two fields (water and electromagnetic radiation) could only be understood by dropping the assumption of infinite differentiability. (The citation was for Brownian motion and the photoelectric effect.) He then went on to spend the rest of his life building general relativity, which is a theory of infinitely differentiable fields.

When I pointed this out to a working theorist, he said that adding discrete structure to general relativity and quantum mechanics (also a theory of infinitely differentiable fields) might be fruitful, but was just not “fashionable.” It seems obvious to me, however, that the easiest and most natural explanation for fermion generations must be an excited state of smaller elements.

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• Attempts to quantise spacetime are at the forefront of many theoretical physicists minds right now so I would say it is very fashionable! Such a road is hoped by many to lead to a theory of quantum gravity. If you’re interested I can link you to some papers working on such a quantisation.

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• I am aware of theories of quantum gravity. Just to be clear, I am thinking of something different: a theory in which hidden structure interferes with the smoothness of the manifold. This is not structure at the Planck scale, but at the granularity of the weak interactions – in other words, something that would help to explain well-known anomalies in quantum phenomenology.

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• I recommend it – thanks for reading!

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6. Your writing is always a good introduction to these topics, Mekhi 🙂

Certainly enough to leave some mouths gaping at a dinnertime conversation! (Though I hope to get a deeper understanding of these topics eventually)

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