Does the fact that the general theory of relativity predicts singularities imply that it's wrong?

Question

Since the general theory of relativity predicts the existence of singularities in spacetime and (I think) singularities do not physically exist, does that mean that GR is wrong under that aspect and needs a reformulation?

Answer 1

No, because in physics we only try to reformulate our theories when they are contradicted by experiment*. However, a singularity, may not correspond to anything in physical reality.

To take a very simple example, suppose I tell you that a certain particle moves according to the trajectory:

$$ x(t) = \frac{1}{t-5} $$

Now, as someone who doesn't know anything about the actual physical situation, you may immediately jump and say: "At $t=5$ the position of the particle is undefined, so your description must be wrong!". But after a bit of investigation, you may find for example that it's a particle that can never exist for more than $4$ seconds.

So, the mathematical singularity you've found in the equation doesn't allow me to conduct any experiment to show that my description of reality is wrong in that case. Therefore physically it is irrelevant.

Analogously, the well known black hole singularity according to which at the center of a black hole, curvature becomes "infinite", etc. may not really correspond to anything physical. It may for example be the case, that under no physical circumstance can anything be observed at that point, and that there is no physical way to obtain any information about what occurs at that specific spacetime point.

So, as long as we don't have any way to obtain experimental data and measurements, which are always finite values of course, of some kind about that point, we really can't know if the singularity which manifests in the equations of GR, means that the theory is "wrong".

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^*_{That's not entirely accurate. Some people like to try and reformulate a theory of nature in a different way just to see if they can, or just because they want to find a more efficient way to carry out calculations, etc. What I really mean here and said in fewer words, is that there's no inherent need to reformulate a theory as long as it agrees with experiment.}

Answer 2

I would like to quote what Penrose said about the singularity theorem in his interview for the European Mathematical Society on page 19:

“What it really showed is that the space-time could not be continued, it must come to an end somewhere – but it doesn’t say what the nature of that end is, it just says that the space-time cannot be continued indefinitely.”

And further:

“What would be the most striking physical implications of the singularities here? What the singularities tell us is that the laws of classical general relativity are limited. I’ve always regarded this as strength in general relativity. It tells you where its own limitations are. Some people thought it was a weakness of the theory because it has these blemishes, but the fact that it really tells you where you need to bring in other physics is a powerful ingredient in the theory. Now what we believe is that singularities are regions where quantum theory and general relativity come together, where things.”

Another famous physicist, R. P. Kerr, in his paper Do Black Holes have Singularities?, said: “The author’s opinion is that gravitational clumping leads inevitably to black holes in our universe, confirming what is observed, but this does not lead to singularities.”

Einstein distinguished between two types of theories: principle theories and constructive theories. General relativity (GR) belongs to the latter. He said: "The main appeal of a principle theory lies in its logical consistency. If a single conclusion drawn from it turns out to be wrong, it must be abandoned", or in words of R. P. Kerr: "If the theory predicts singularities, the theory is wrong!" Penrose's interpretation of singularity, as quoted above, would degrade Einstein's principle theory to a constructive one. I believe that Einstein would not agree with this opinion.

So if we assume that GR is correct, then the current conclusions about the singularity should be wrong. Of course, this would be a matter of proof.

It is a pity that this interesting question has been closed. As I understand it, the question is not opinion-based.

Answer 3

It is a conjecture (or hypothesis, depending) that all singularities must be “censored” with an event horizon or the like. A spacetime that has an event horizon in front of the singularity is “okay”(-er) because the singularity can’t causally affect the rest of the spacetime.

The invalidity of relativity would be extra confusing given that we’ve confirmed it in every which way from

• measuring time dilation due to motion and gravity at very small scales, to
• observing that we can’t get particles to pass $c$, to
• observing gravitational lensing from distant galaxies, to
• observing a finite speed of light, to
• measuring gravitational waves propagating through space, to
• taking a picture of a black hole, the singularity-having thing itself

which are all things that relativity predicts, among many other things.

Answer 4

All models are wrong, but some are useful

George Box

We already expect that General Relativity is "wrong" in the sense that it does not have unlimited applicability. That does not prevent it being an astoundingly accurate theory that has made predictions covering an extremely wide range of phenomena at different scales. That is what elevates it to the status of a theory rather than "just" being a model with which we can make calculations.

One limit to its applicability is when we get down to scales where, for example, the Schwardzschild radius is comparable to the uncertainty in position that might be associated with the Heisenberg uncertainty principle. We do not yet have a theory of quantum gravity, which could be "more correct" than General Relativity (but would still probably not be "correct" in the way that I think you would like to define it).

Fortunately, from the point of view of the singularity problem you highlight, nature seems to find a way to shield such problems from observation with event horizons. That means we cannot yet say we have actually found the limits of applicability of General Relativity or falsified it under such circumstances.

Answer 5

Isaac Asimov's The Relativity of Wrong is highly relevant to this question.

If you live in an absolute world where a theory is either right or wrong, and anything that isn't completely right is wrong, then General Relativity is wrong. But that's already well-known, since it's incompatible with quantum mechanics and we'd expect the theory to fail whenever quantum effects become significant (like near singularities).

But in real life, the idea that "anything that isn't completely right is wrong" is not justified. To quote an example Asimov gave: what is $2+2$? The expected answer of course is $4$. But if someone had said "an integer", would they be wrong? What if they said "an even integer" - would they be wronger (or righter) than the person who said "an integer", or are both of them just flat out wrong because they didn't say $4$?

This section from the last part of Asimov's essay hits the nail on the head:

Newton's theory of gravitation, while incomplete over vast distances and enormous speeds, is perfectly suitable for the Solar System. Halley's Comet appears punctually as Newton's theory of gravitation and laws of motion predict. All of rocketry is based on Newton, and Voyager II reached Uranus within a second of the predicted time. None of these things were outlawed by relativity.

In the nineteenth century, before quantum theory was dreamed of, the laws of thermodynamics were established, including the conservation of energy as first law, and the inevitable increase of entropy as the second law. Certain other conservation laws such as those of momentum, angular momentum, and electric charge were also established. So were Maxwell's laws of electromagnetism. All remained firmly entrenched even after quantum theory came in.

Naturally, the theories we now have might be considered wrong in the simplistic sense of my English Lit correspondent, but in a much truer and subtler sense, they need only be considered incomplete.

Relativity is wrong in the absolute sense, but it's much truer to consider it incomplete, and in the domain where it is valid it is an extremely good theory.