Science and the Metaphysics of Time

Continuing the discussion from Gravity is Not Intuitive:

So, I’m going to take @PdotdQ’s invitation to explore this topic a little bit for those who are interested (since I find it quite interesting).

The basic question here is: what does science tell us about the nature of time? There’s a big philosophical debate about what the temporal aspect of reality is like, and the accepted belief among most scientists and philosophers who think about this issue seems to be that science has come down pretty decisively on one side of the matter. But that conclusion may be premature.

One focus of that big philosophical debate is about, essentially, what is now? Is the present moment an objective feature of reality - and therefore the passage of time happens really and objectively? (This thesis is called the A-theory of time.) Or is the present moment merely subjective - there is nothing metaphysically privileged about the present, and now is merely an indexical word like here? (This is the B-theory of time.)

Science steps into this debate in a major way with Einstein’s theories of special and general relativity. The reason for this is that, with a metaphysically privileged present, the A-theory requires that there is absolute simultaneity - when something is happening here and now, there is an objective fact of the matter about whether something that is happening somewhere else is also happening now, or whether instead it is happening earlier or later.

But Einstein’s relativity theories imply the relativity of simultaneity - or at least, if there is such a thing as absolute simultaneity, relativity theory implies that we can never ever detect it. Absolute simultaneity is superfluous, according to relativity physics (and a strand of philosophy that was popular, back in Einstein’s day), and thus we should discard it from our theories. So it is commonly believed that science says B-theory wins, A-theory loses, and that pretty much settles it.

This is a case where I think the common belief is, in fact, incorrect.

First, the inference from “we can’t detect it” to “we must discard it from our metaphysics” isn’t a given and has to be justified. The philosophy that was popular in Einstein’s day - called “logical positivism” - took it to be justified because this philosophy actually held that statements were meaningless unless they could be logically or empirically verified. Unfortunately for logical positivism, that criteria of meaning can’t be logically or empirically verified, so it’s a little self-refuting. So belief in absolute simultaneity may still be justified, after all.

Second, absolute simultaneity is not actually in contradiction with relativity physics - all that the physics implies is that absolute simultaneity (if it exists) is not the same as apparent simultaneity (simultaneity operationally determined by synchronizing clocks via light signals). Absolute simultaneity can be represented in relativity physics as a particular slicing (or foliation) of 4-d spacetime into many 3-d spaces (the leaves of the foliation) ordered in time. We can’t detect exactly what that slicing is, but it might still be there.

This means that physics alone can’t decide between the A-theory and the B-theory - it can only provide one more philosophical consideration that has to be weighed against others.

Third, relativity physics may not be the last word from science on the nature of time. Though it isn’t usually recognized as such, there’s a good case to be made that Bell’s theorem does more than just show that quantum mechanics is weird - it shows that quantum mechanics is in outright contradiction with relativity physics, and that there are causal influences in nature that must violate the speed-of-light limit in order to produce the (experimentally well-confirmed!) predictions of quantum theory.

So it may very well turn out that quantum physics - and a quantum theory of gravity, in particular - will provide a natural slicing of spacetime into 3-d spaces evolving over time which is simply hidden at the observational level. (Pilot-wave theories are a candidate for such.) Which would flip the support of science from the B-theory to the A-theory; or at least turn the focus of the debate back towards more traditional philosophical considerations.

So that is a little introduction to this topic. I would enjoy discussing it here if anyone wants to do that. Or if (for some reason) you really just want to read more of what I have to say about this subject, I do have some posts about it on my blog (specifically, this post and the three immediately following it).


Thank you @structureoftruth for taking the time to write this post.

I have a some followup questions on this that perhaps are already addressed in your blog:

  1. Aside from the “what is now” question, which is interesting by itself, is there a larger philosophical context to whether someone is an A or B theorist? By this I mean: are there philosophical arguments that has in its axioms the A or B theory? And are there philosophical positions (perhaps something that is still popular today, unlike logical positivism) that implies the A or B theory?
  2. How does the A and B theory relate to the block/growing block Universe and presentism/eternalism? To someone who does not follow philosophy, they all seem very similar.
  3. I think I understand the position of B theorists - they believe that the entire spacetime is ontologically real. However, what exactly do A theorists believe to be ontologically real/not real? Do they not believe that spacetime is real, and only a spacelike slice of it is real?

Probably the most complete description of A theory with regard to metaphysics anywhere on the internet. I don’t agree with Craig, but I feel like he is more respected for his work in this area than almost any other.

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I reject A theory on theological grounds. Seems to make God “smaller” than time itself, almost like a creature. Although I would say God’s energies can exist temporally and IN TIME while his essence exists wholly outside of time.

From Orthodox philosopher David Bradshaw:

“I would suggest that such a view offers a way of retaining the concept of divine imagination, and with it a properly robust understanding of creation, without requiring a full movement to open theism. Since God is both temporal and eternal, He can be said to grow in knowledge although, in another sense, the knowledge is always present to Him. Dionysius himself expresses a similar view in discussing divine knowledge in chapter 7 of the Divine Names. There he writes:
The divine mind embraces all things by its transcendent knowledge of all, having precontained (proeilēphōs) within itself, as cause of all, the knowledge of all. Before angels came to be He knew them and brought them forth, and so also all the others, knowing them and leading them into being from within and, so to speak, from their very source. This, I believe, is what Scripture means when it says, “You who know all things before they come to be” (Dn. 13:42 LXX). The divine mind does not know by learning of beings from beings, but precontains the knowledge and understanding and substance of all from itself and within itself as cause. It grasps them beforehand, not attending to each separately, but knowing all in its single embrace as cause, just as light precontains within itself as cause the knowledge of darkness, knowing darkness from no other source than from light.45”

Look up his article on Divine Freedom in Philosophical Theology and the Christian Tradition: Russian and Western Perspectives.

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Sean Carroll published a paper discussing his ideas along your discussion above. I don’t understand it well but it sounds intriguing.

Thank you for the link, I will read through some of them. Usually I have a poor view of Craig’s arguments, but I hope I can leave my prejudice behind.

I am not sure I agree. I believe that God sustains the Universe, and thus spacetime itself is sustained by Him. In this view it does not seem sensible that He is limited by a thing that He sustains. Bradshaw seems to have a problem with nonpredictability+nondeterminism instead of the A or B theory of time. We might have started with different axioms on this disagreement.

This paper does not give one an absolute simultaneity, and therefore does not support either the A or B theory (which is a philosophical idea anyway, and maybe cannot be answered by physics). This paper shows that defining a metric on a factorized Hilbert space through its mutual information gives a spatial metric that looks similar to those in relativity. The resulting time-slice in the emergent spacetime depends on the particular time coordinates that were chosen to define the states in the Hilbert space - different time coordinates give you different 3+1 splits in the emergent spacetime.


Yeesh. You’re really smart…

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I hope you are not referring to me with this comment, as I am ignorant about most things. Everyone here is smart in their own ways.


Defintely referring to you. You are able to break incredibly difficult concepts down, explain them and critique them in very few words. That’s really something I admire.


I’m sure there probably are other philosophical positions related to one’s philosophy of time - for example, actualism (the position which says that only actual entities, not merely possible entities, exist) is to modal logic what presentism is to temporal logic, and those two positions may be seen to “fit together” better than contrary positions. My interest in the philosophy of time comes more from the scientific and theological sides, though.

In addition to the question of whether or not there is an objective present (A-theory vs. B-theory), you can ask the question what times exist? And the common answers are presentism (only the present exists), eternalism (all times past, present, and future exist), and the growing block theory (the past and present exist, but the future doesn’t).
Presentism and growing block both require A-theory, while B-theory requires eternalism (but eternalism is also consistent with the A-theory: the combination of A-theory and eternalism is sometimes called the moving spotlight theory). I think there is good reason to believe that presentism is the most coherent position if the A-theory is true, which often means that the A-theory vs. B-theory debate becomes synonymous with the presentism vs. eternalism debate.

There’s a range of positions. Some A-theorists would say only the present space-like slice of spacetime exists. Some would say the whole spacetime exists, but only the present slice of it is filled with actual physical things. Dean Zimmerman’s paper “Presentism and the Spacetime Manifold” (if I’m recalling the title correctly) explores a couple of options.

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I have never studied this in detail. But Craig is famous for arguing for the A theory, which is important for his Kalam Cosmological Argument. The second premise of the KCA is “The universe began to exist.” Craig has recently started using the Borde-Vilenkin-Guth theorem to argue for this (which, as several have pointed out, only applies to classical gravity, so might not be true for quantum gravity), but before that he argued using philosophical arguments that the past has to be finite. This is formulated as follows (from Craig & Sinclair, Blackwell Companion to Natural Theology):

  1. An actual infinite cannot exist.
  2. An infinite temporal regress of events is an actual infinite.
  3. Therefore, an infinite temporal regress of events cannot exist.

Premise 1 is where the A/B-theory issue comes in. Apart from findings in cosmology, some people might argue that the universe had no beginning - that there is an infinite, temporal regress of events stretching back into the past. Craig has argued that even if this were true, this would only be a potential, not actual infinite. Following Aristotle, he argues that only potential, not actual infinites can exist in the real world. But if B-theory were true, then the past, present, and future all actually coexist at the same “time” and are ontologically real (eternalism), and that would mean that an actual infinite regress of events into the past can exist, violating premise 1.

Although in his article Craig also has short arguments for why actual infinites still do not exist even on B-theory, it is much easier for him to defend A-theory. So he has done some amount of work arguing that A-theory is not precluded by special relativity. I have not dug too deep into this, but he seems to defend Lorentz Ether Theory, which is mathematically equivalent to conventional SR (and thus not resolvable by experiment) but posits an undetectable aether. (I wonder whether LET conflicts with GR?)


Thank you for the clarification. I think I understand now, it is also very helpful that they name their theories with creative names like “growing block” and “moving spotlight”!

I see, perhaps I should read the Zimmerman paper, but it’s 146 pages long!

After reading the explanations, I think I am intuitively an A-theorist. I do not have any good reason for this, but I find the idea that the future is already “there” abhorrent.

However, as a relativist, the A-theory does seem quite unnatural if seen from the lens of relativity, so I am also sympathetic to proponents of B-theory. The reason I asked what A-theorist actually believe is because this position:

is very unnatural from the GR perspective. Note that a (3+1) splitting (which can only be done for subsets of very special spacetimes anyway), will not save you here. While it is true that the 4-metric can be written as a spatial metric and its canonical momentum, the momenta depends on the extrinsic curvature that cannot be defined without first embedding the spatial slice inside a 4-spacetime.

Perhaps there is an A-theory position in Zimmerman’s paper that is more palatable for me. I will take a look, thank you for referring the paper.

@dga471 thank you for mentioning this, this is the kind of philosophical consequence of A/B theory that I am interested in!

Why does he say that:

Did he assume that the past and/or future extend infinitely?

This is interesting. As you know, Lorentz Ether Theory is very not-in-vogue amongst physicists. Do you have a link to the articles in which he defended LET?

No. He is only assuming that if the past extended infinitely back, that would lead to absurd conclusions, namely that an actual infinite would exist, which is impossible. (He argues that actual infinites cannot exist on other grounds as well, such as Hilbert’s Hotel.)

But most versions of B-theory assume that actual infinites can exist. I guess you could hold on to B-theory and still believe that the past has to be finite based on other reasons.

He does talk about this at length in his academic book A Tenseless Theory of Time: A Critical Examination, but there are no online articles I found which are dedicated to defending LET. He does, however, make passing references to this regularly: e.g. here or here (where he argues that tachyons, if they existed, would make more sense within LET).

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The “undetectable aether” is really just an ontologically privileged reference frame - there is no need for some kind of weird substance over and above the things like electromagnetic field behaving according to the laws of physics in that reference frame to reproduce all of the relativistic behaviour.

The correlate of the privileged reference frame in GR is just a privileged foliation of spacetime, so Lorentzian relativity works in GR too.


My understanding of the 3+1 decomposition of GR (based on this resource) is that the extrinsic curvature can be written entirely as a 3-d tensor, so it acts just as an additional variable on 3-d space rather than depending on a prior embedding in 4-d spacetime (though of course it constrains how that embedding is done when the 4-d spacetime is built from the 3-d slices). Is that incorrect?

And while I understand that spacetimes which can be globally sliced up in this way are “very special” in the sense that they satisfy a very strong causality condition, I also am not aware of any “real world situations” which are good candidates for demonstrating a violation of that causality condition. (Black holes are the closest that I’ve found, but see here for a discussion on why they aren’t decisive against A-theory or presentism.) Are there any spacetimes that are commonly regarded as possible models for our universe which cannot be foliated in the necessary way?

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Of course the extrinsic curvature can be written as a 3-tensor. After all, it is defined on the spacelike slice. However, this 3-tensor is not embedding invariant, i.e. different embedding of the spatial slice into the 4-D spacetime produces different extrinsic curvature. This is in contrast to the intrinsic curvature, which can be defined without any notion of embedding.

First, claiming that all “real world” situations can be sliced up this way is equivalent to the claim that the Strong Cosmic Censorship Conjecture (SCC) is true. This is a large claim that is still a subject of active research. Any spacetimes that violates the SCC conjecture cannot be split in a 3+1 way.

The prototypical counterexample to the SCC is the maximally-extended rotating black hole. I am at home right now and do not have access to the paper you linked, but from the abstract they are talking about Schwarzschild black holes (nonrotating), which do not violate the SCC.

Other examples include the spacetime resulting in gravitational wave collisions.

New today!

But from a 3+1 perspective, couldn’t you say that the extrinsic curvature tensor plays a role in determining the embedding (once you build up 4-d spacetime from the evolution of the 3-d space) - rather than the embedding determining the extrinsic curvature?

They begin with a discussion of Schwarzschild black holes (in response to another paper), but then move on to talk about the maximally-extended Kerr metric, which they agree is much more troublesome for a presentist/A-theory position. The defense they suggest in that case is basically: we don’t know if these idealized solutions are physical. But yes, it is very much an open research question.

I hadn’t heard that gravitational wave collisions may result in SCC-violating spacetimes. That is interesting, I will have to look into it!

Are you speaking purely mathematically? Purely mathematically of course you can do this. However, put yourself in the shoes of a physicist who only believes in the reality of the spatial 3-slice. For this physicist, the extrinsic curvature tensor has no physical meaning, and indeed cannot be defined besides as a formula of a seemingly random combination of variables.

Nevertheless, this combination of variables appears all over the place, but they do not know why this combination of variables is important. Another physicist would come and show them that this puzzling combination of variables maps exactly to an extrinsic curvature if you just allow the existence of a 4-spacetime. For some reason the first physicist refuses this interpretation.

In this situation, while experimentally I cannot distinguish between the 3-space and the 4-spacetime theories, it becomes clear that one of them is more natural than the other.

Let me add something from the recent results of various groups:

  1. There is evidence that the volume of spacetime beyond the Cauchy horizon of a Kerr or a Reissner-Nordstrom spacetime goes to ~Planck volume under perturbation. This means that generically the validity of classical general relativity only works up until the inner horizon, and quantum gravity is required to fully explore the problem.
  2. A similar behavior does not happen for black holes in an expanding spacetime (i.e. black holes in our real universe), which can produce nonglobally-hyperbolic spacetimes.
  3. Various direct collapse scenarios now exists that can produce naked singularities out of well-behaved initial data. Note that naked singularities tautologically violates the SCC. Astronomical testing of whether these naked singularities exist in real life is currently ongoing through a mixture of electromagnetic and gravitational wave observations.

Didn’t you say it is related to the conjugate momenta for the 3-d spatial metric? Also, from those notes on the 3+1 formalism, “the extrinsic curvature tensor measures the failure of a geodesic of [the 3-d metric on the hypersurface] to be a geodesic of [the 4-d metric of spacetime].” Put another way, it measures the tendency of objects to deviate from spatial geodesics - that is, it shows what gravity does aside from distort the geometry of space. So it gives information about how the spatial metric evolves in time and about gravitational forces. That seems to be meaningful even from a 3-d perspective.

I won’t contest that the equations of GR are more naturally and elegantly written in 4-d language - you can’t really beat the Einstein field equations for elegance! I’m just saying that’s not the only consideration when it comes to interpreting the physical (or metaphysical) meaning of those equations.

Thanks for these!

I’m afraid I still have not fully comprehended the conditions required for the kind of foliation that an A-theory interpretation of GR implies… are there any resources you would recommend to learn about that?