It is often claimed that the "block universe" of general relativity makes it impossible for us to have free will. This is nonsense, for two quite different reasons; one is well enough known, the other seldom appreciated.

Relativity famously describes not just a space of three dimensions but a spacetime of four, in which time is the extra dimension. Each of us traces a "world line" or timeline through it, from birth to death. Any point along that timeline has equal validity in the block spacetime, the theory has no "moment of now" to mark where anybody is on their timeline. Even if several observers, moving in different ways, tried to synchronise themselves, they could not agree among themselves any exact moment common to all of them (this is why the theory is called Relativity). Indeed, every particle of existence follows its own track through spacetime, according to the laws of physics.

Since, the argument goes, we are made entirely of such particles, we too must slavishly follow those laws. But by definition free will ignores such deterministic laws, therefore it cannot possibly exist. Consider for example if you made two identical bombs atom for atom, right down to the quantum state of every single atom, and detonated them both; they would fly apart identically, fragment for fragment. Of course it is impossible in our present state of technology but the principle is there. If we could describe our brains with the same precision, every thought would be totally determined by those same laws. We may not like it, but we are all physical zombies.

But that argument is fatally flawed. One reason is well enough known, though not accepted by the deniers. Their argument assumes that the laws of physics are fully deterministic, purely mechanical. But quantum physics challenges this. The question hinges on whether the famous quantum randomness is just a consequence of the imperfections of measurement, or whether it is inherent in the nature of the quantum itself. The argument has raged for a century and opinion is still to some extent divided. However, over the last few decades the inherent quantum randomness of nature has been suggested more and more strongly by experiments of ever-increasing sophistication; the more you rule out the ignorance of the experimenter and the disturbing of the subject by the act of measurement, the more you are left with a pure and impenetrable randomness underlying the quantum phenomenon itself. The majority of mainstream physicists many years ago reached the conclusion that at the quantum level, reality behaves in a way utterly inaccessible to physical investigation. The diehard deniers fall back ever more desperately on claims of some hidden underlying deterministic mechanism, and they make enough pseudoscientific noise to convince many of the less knowledgeable. But we now understand that these claims are just speculative, even metaphysical interpretations. If we stick to the laws of physics, we do not bring those interpretations into it; scientifically they are worthless.

Returning to the relativity argument, we can now understand that our world line is governed not just by the forces of relativistic mechanics, but also by the forces of quantum randomness. At the deepest quantum levels it is not determined at all. The famous butterfly effect means that random quantum fluctuations will, at critical moments, tip us this way and that so that out line changes its path. It must be emphasised that to the physicist, these changes are not determined by the laws of physics, they are random; physics has absolutely no way to explain them individually. But might free will be able to intervene, in ways undetectable to the physicist?

True, free will is an equally metaphysical concept. And that is the whole point, it is equal in scientific stature, or rather its lack of any such stature, with mechanistic determinism. If we wish to explain the individual quantum events which steer our thoughts, we have no choice but to resort to metaphysics. And here, free will is the equal of hidden variables. There are other interpretations and some, such as the transactional model or even Everett's parallel worlds, offer beguiling rationalisations. But none is any more scientific than any other – if they were, they would not be mere interpretations. It is a steadily fading hope by the deniers, that some hitherto unknown deterministic mechanism might remain to be discovered. One can seldom prove a negative, but unless and until such a mechanism turns up, your preferred interpretation is a matter of belief or even faith, not of science.

I said that there is another reason for the failure of the deterministic argument. This is a more subtle one, mathematical in origin while also leaning as much on philosophy and the use of language as on physics. Quantum theory is once more to blame, for illustrating the issue at least. But first I will introduce some basic mathematical and philosophical ideas.

The mathematical idea of a transformation arose in geometry. The perspective distortion of object seen in photographs is an example of a geometrical transformation. Something as simple as moving your coffee mug from the kitchen table to your office workstation is another, called a translation. The key thing about transformations is that the before-and-after mathematical descriptions, such as the coordinates of points in space, both describe exactly the same object, just from different points of view.

Some transformations are more radical. For example a perspective drawing is an example of the more general projective transformation. It may be further transformed into what is known as a dual figure, under a different kind of projective transformation. Such a projective duality was invented by James Clerk Maxwell (better known for his equations of electromagnetism) to work out the structural forces in trussed girder constructions; he simply drew up the dual figure and measured the lengths of the lines.

Mathematical transformations go way beyond geometry. The Fourier transform is a common example, and among its applications is sound waves. Sounds are typically described mathematically in either of two ways. A microphone will capture the physical movement of the wave, which you can replay on a screen to watch the wiggles as they evolve over time. If you replay the lower frequencies through a large loudspeaker, you can touch it and feel the vibrations. Each sound has a characteristic shape. Low-pitched bass sounds have a long, low-frequency wave, while high-pitched treble sounds have a short, high-frequency wave. You can also describe the sound by listing all the frequencies, from low to high, together with the loudness or amplitude of each. This is known as a sound spectrum. It is just another way of describing the same sound, and is in fact how our ears work; tiny hairs of different lengths respond to each frequency in our ears, and pass the resulting spectrum to the brain. The Fourier transform is used to change from one description to the other.

We might now ask which of these mathematical descriptions is the "right" one? Which one describes the true nature of a sound? The answer is that neither is any less true than the other. For example our vocal cords create speech as a time-varying wave, while our ears hear it as a spectrum of frequencies. The reality is that both are simultaneously true. The mathematics of sound may be described as a schema comprising the time and frequency domains in a dual relationship. That is as far as the mathematician can go in helping the physicist. Philosophically, we may conceive of a core reality, of which the two models are partial representation. This idea of a schema has been expressed by theoretical cosmologists concerned with something called AdS/CFT duality, in the hope that some model underlying two dual approaches to cosmology might be discovered. I believe that this hope is unfounded; any underlying model can only ever be an interpretation of the schema; the underlying reality, the nature of the cosmos, is that both models are equally true. If we ever did discover another model, be it for sound or the cosmos, it would merely establish a "triality" of equivalent descriptions. In fact there are many such models for sound, known broadly as wavelet theory. The nature of sound is that all are true, it is only the nature of mathematical notation which presents them as somehow different one from another. I believe the same to be true of the cosmos.

One feature of transformations is that they are no respecters of dimensions. The Fourier transform swaps the time dimension (or distance travelled) for a frequency dimension. But it gets worse. The frequency spectrum of a sound has a phase associated with each frequency, as well as a magnitude. This representation can also be treated as a complex number comprising a "real" magnitude and an "imaginary" amplitude at right angles to it. That takes two dimensions. Add the frequency range and you have a three-dimensional "sound space" which the sound occupies. Yet the original wave at the larynx or microphone had only magnitude and time, it was two-dimensional. The hologram is another example, where a three-dimensional scene may be captured on a two-dimensional photographic plate, and then restored to three when required by shining a laser through it. Dimensions may come and go at the twist of an equation.

A more relevant example from modern physics is the twistor space introduced by Roger Penrose. It is advanced mathematics, but with a little suspension of bafflement you can get a feel for it. It begins with Einstein's observation that light does not experience time. An astronomer may see a photon of light which was created in a star a million years ago, travelled across space for all that time, struck their camera and died. But relativistic time dilation means that from the photon's point of view, travelling naturally at the speed of light, it experienced no time at all; it was born, occupied a million lightyears of space, and dies at the camera all in one fleeting instant. The mathematics of relativity, known as Minkowski spacetime, calculates the interval between its birth and death by (very roughly speaking) adding up the spatial distances in each direction and then subtracting the time taken. It turns out that whether you are the astronomer or the photon, that spacetime interval turns out to be zero. Such intervals are among the few invariants in relativity and therefore have a special significance for the underlying reality. This it twistor space's starting point. Penrose applied what is essentially a Fourier transform to Minkowski spacetime, so that a photon is represented as a single point (loosely analogous to a frequency in the audio spectrum). Any given point in physical space is smeared out across all the photons that ever passed through it - or will pass through it. Any particle is now represented as something called a twistor.

Twistor theory all seems a rather abstruse intellectual game until you discover that by using twistors, the particle interactions that occur during nuclear reactions and in atom-smashers suddenly become easier to calculate. Instead of perhaps twenty or a hundred pages of densely-written and extremely difficult equations, the whole thing drops into place on one side of paper. Its elegance and beauty are stunning. There is very evidently a fundamental truth to twistor space. And yet other calculations, such as for slow and massive particles, prove intractable for twistor theory and must be dealt with the old-fashioned way. Here we see another example of the reality being approached via a duality of forms, linked by a mathematical transformation.

Now at last we can revisit block spacetime with some understanding of how it relates to free will. The first thing to note is that block spacetime is just one mathematical description, among many others, of the cosmos. AdS/CFT and twistor space have introduced some examples of others that have been investigated. In each model, some truths become clear and almost self-evident, while others become darkly obscure.

Yes it is true that some supposed spacetime demigod could stand over their block universe and declare, "look, there's Einstein's timeline; born over *there*, travelled across *these* regions and died just *here*." And historians often do just that. But put yourself in Einstein's shoes, sitting in a cafe and wondering what to order for lunch. A suitable mathematical transformation from Minkowski spacetime to Euclidean space-and-time will do nicely. Einstein is trapped firmly in his moment of now; he may be carrying a somewhat imperfect memory of the past along with him but has no inkling of what the future may hold, it is up to him to make up his mind. He does so, and orders his lunch.

We now return to the demigod, who is busy pointing out the little twitch in Einstein's timeline where he placed his order and the waiter dashed off to the kitchen with it. "Look", the demigod cries triumphantly to his demigoddess colleague, "That's where the laws of physics made the inevitable decision and Einstein was left with the false impression that he had chosen freely." "Nonsense!" she retaliates, "That is where Einstein's brain leveraged quantum randomness to slip his decision in without breaking any law of physics, causing the twitch that you just pointed out. At the time he made it, the future half of your precious block universe did not even exist. Einstein's free choice helped shape it as it grew. You can deny him free will in our plenum time, but that is not the the same dimension of time he lived through; in his kind of time, he could easily have exerted free will."

The High Deity standing behind them sighs and steps forward. "She makes some valid points. But have neither of you grasped it yet, my children? Both of you are interpreting the facts set out before you. Both of your descriptions of reality are right as far as they go, but the ultimate truth embraces both – and more – while your interpretations are nothing but the empty speculations of your limited minds. The term "free will" implies different things depending on your point of view. In Euclidean space Einstein may quite possibly have free will, when transformed into block spacetime he has expended all that he might ever have had, that is all there is to it. You may think of it as the free will transformation, if it helps. You yourselves have the possibility of free will even here in the M-plenum, a luxury denied to Einstein.

"Tomorrow we will consider whether valid phenomena might exist which can subvert quantum randomness in ways undetectable by human physicists – and perhaps even by minor deities. How much free will might one be able to slip in before the physicists started to notice? Might the clues be already in there for them, if they only thought it through properly? Please make sure you read Ternalism and Towards a Theory of Qualia beforehand."

Updated 6 Apr 2022