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Definition of a block universe - The idea that the whole universe exists simultaneously and time doesn’t flow.

For those who favor this kind of theory (the few of you), what is the role of the laws of physics in such a universe? Are the laws of physics just a reflection/projection determined by the blocked universe? Are there other superior laws that created the block? or was the block created by the laws of physics too? Is it possible to have a glitch in such a block?

I understand that there can be only theories and nothing more, but I am interested in any kind of references.

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  • $\begingroup$ I had to look up "block universe" and got apparently contradictory answers. Past, present and future co-exist v past and present do and future doesn't exist. Unless you can travel through forward through time, what difference would either make? Either way, how would either change the role of the laws of physics? $\endgroup$ Feb 6, 2021 at 23:56
  • $\begingroup$ @Robbie Goodwin Maybe there are some other external superior laws that have created the so call “block” with the laws of physics in it? In the same way that the laws of Biology depend on the laws of chemistry and the laws of chemistry on the laws of physics. Maybe the laws of physics depend on a much more superior set of laws that we are restricted from knowing or understanding? I meant for it to be a soft question, and I apologize if non practical $\endgroup$ Feb 7, 2021 at 0:14
  • $\begingroup$ Thanks and isn't that more about metaphysics, or philosophy? Either way, what difference might a block universe make? Why not ask simply "What is the role of the laws of physics?" (Come to that, why not just ask "What is the role of physics?") $\endgroup$ Feb 7, 2021 at 0:20
  • $\begingroup$ @Robbie Goodwin most of the articles that I have read on the subject came from physicists (more than philosophers), While they are all speculative, they are pretty interesting and they also help expand my imagination. It is much easier for me to imagine “other superior” laws in a “block” universe than in “regular” universe. So again, I apologize if my question isn’t practical enough $\endgroup$ Feb 7, 2021 at 0:50
  • $\begingroup$ Thanks Isaac You seem to be looking for an extended discussion, in which case someone might link you to a chatty site but it wouldn't belong on SE. If you must risk the wrath of any official noticing you're looking for an extended discussion, why is it easier to imagine “other superior” laws in a “block” universe and how is that different from the “regular” one? $\endgroup$ Feb 7, 2021 at 22:28

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A block universe is simply one in which all the laws of physics are deterministic and reversible. In such a universe, complete and precise knowledge of the configuration of the universe at one point in time (or across one "slice" of the block) contains enough information in principle to determine the configuration of the universe at all past and future times. I emphasise "in principle" because there is nothing that says such a calculation has to be even remotely feasible in practice.

In a block universe there can still be an arrow of time, and so a flow of time in a psychological sense. If the universe starts in a state of low entropy (as our own universe did) then its entropy increases over time due to the second law of thermodynamics, and because of this it is only possible to construct memories in one direction (memories of what we call the "past") and not in the other (we cannot remember the "future"). We could, in principle, predict the future, but an exact prediction would require a level of complete and precise knowledge which would make it unfeasible. Of course, we make approximate predictions of the future all the time - if I go for a walk on a sunny day I predict that is unlikely to rain, so I do not take an umbrella. I also predict that an earthquake, a tidal wave, a volcanic eruption or a shower of frogs are also very unlikely events in the next hour or so.

In a block universe, whether you regard the laws of physics as determining the timeless configuration of the universe, or you regard the configuration as determining the laws of physics is a philosophical question. Like the chicken and the egg, these two aspects of the universe are inextricably linked.

Doubts about whether our universe is a block universe do not arise because it is illogical - the concept is entirely self consistent. Instead, they arise from our observations of the laws of quantum mechanics. The non-reversible nature of quantum measurement (you cannot reconstruct the previous state of a quantum system once it has been measured and its state has "collapsed" into an eigenstate of the measurement operator) seems to rule out the possibility of our universe being a block universe - if you grant the physical reality of wave function collapse.

However, there are interpretations of quantum mechanics, such as the "many universes" interpretation - where wave function collapse is an illusion due to our restricted knowledge - we only have knowledge of one universe from an immense collection of possible universes. The evolution of the collection of all possible universes in a "many universes" scenario could still be deterministic and reversible, so this could still be a block "mega-universe", although we can only ever have knowledge of an infinitesimally small part of it.

Two well known books on this topic are Julian Barbour’s The End of Time (pro block universe) and Lee Smolin’s Time Reborn (anti).

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  • $\begingroup$ So in theory the ideas of a block multi universe can be promoted (being caution not to use the word “proved”) by our restricted knowledge of the laws of physics, but our restriction means that we probably can never prove it, since we don’t have access to all the knowledge of the laws of physics, nor to the possible knowledge of other “possible superior” laws. $\endgroup$ Feb 6, 2021 at 22:53
  • $\begingroup$ When I mention other “possible” laws I am referring to how for example the laws of biology depend on the laws of chemistry and the laws of chemistry depend on the laws of physics, maybe the laws of physics depend on some laws that we don’t have access to? $\endgroup$ Feb 6, 2021 at 22:59
  • $\begingroup$ @IsaacBrenig We know that our current understanding of physics is not complete because there are fundamental incompatibilities between general relativity and quantum mechanics. The search for a viable theory of quantum gravity and for a “Theory of Everything” (en.wikipedia.org/wiki/Theory_of_everything) are active areas of research in theoretical physics. $\endgroup$
    – gandalf61
    Feb 7, 2021 at 9:51
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    $\begingroup$ @IsaacBrenig: don't think about it as of something that needs to be proven, but rather as of a way to describe the universe - the question then is how well this conceptual description fits reality, and where it breaks down. To get a feel for the idea, consider this: a wave can be described by a traveling sinusoidal curve, OR you can imagine it extending out in time to form this sort of undulating pattern, and that we're "along for the ride" on a time slice that travels into the future. The laws just describe relationships between things in both cases. $\endgroup$ Feb 7, 2021 at 14:36
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    $\begingroup$ @IsaacBrenig - sure, but, a few things. At the moment, we don't know what the deeper, more fundamental laws are; when these are discovered, they'd likely be studied within physics (the scientific discipline), and would themselves be considered laws of physics - unless a new discipline/science splits off. These divisions (physics/chemistry/biology) are in a sense arbitrary and are a historical accident - they reflect how humanity went about understanding the world throughout history. $\endgroup$ Feb 8, 2021 at 2:07
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What is physics? Physics is the discipline that studies numerically nature and uses mathematical models, the theories, in order to describe the data and , important, predict new situations. Laws (also postulates, principles) of physics are extra axioms to the mathematical axioms in order to pick from the infinity of mathematical relations in the models, the ones that are relevant to fitting the data.

For any new way of looking at data, as is this block you are discussing, the existing laws have to be incorporated so the new theory would fit the data, and in order to be a legitimate new theory it should also make measurable predictions, otherwise it will be just a complicated way to fit the same data.

Take a diamond lattice. It can be geometrically mapped, no laws obvious. Where are the laws? They are in the theory that allows the study of lattices, built in the mathematical model that finally gives the lattice points.

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  • $\begingroup$ As for my understanding: there might be superior laws that “created” the laws of physics, But not only it can’t be proved, it also can’t be theorised, since the definition of a physics theory must include the laws of physics ? $\endgroup$ Feb 6, 2021 at 22:14
  • $\begingroup$ A physics theory must have laws. As with mathematical axioms, it is possible that a new formulated theory to describe the data could have these laws as theorems and have new laws for axioms. If the data is described by both we speak of one theroy being and interpretation of the other. A theory to be new, would need to model the same data and predict new effects to be discovered.. It could have completely new axiomatic statements/laws , but the original ones have to be there, in the of the new theory, because the old theories should be embedded in the new because it should fit the old data. $\endgroup$
    – anna v
    Feb 7, 2021 at 5:49
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If you think of the block universe as the name suggests, as a "block" of unchanging structure, then the "laws of physics" are simply a compressed representation of the patterns that are etched into that block. Their existence means that the Universe's pattern is, however apparently complex, one that is "just so" as to permit an efficient, small, representation.

It's like if you take an image of the Mandelbrot set. The picture does not change, but that lack of change does not negate that one can describe every detail thereof simply by writing down the following formula:

$$M := \left\{ c \in \mathbb{C}: \exists L \in \mathbb{R} \forall n \in \mathbb{N} |(z \mapsto z^2 + c)^n(0)| < L \right\}$$

You could say that the existence of laws of physics, in effect, is a statement about the Kolmogorov complexity of the universe: the length of the shortest possible text required to completely describe it in some description language. In some sense, it is "simpler" than just an arbitrary, generic block of random patterns, for which no other way exists to represent it in the language than to just painstakingly specify each and every single point.

That said, it is not necessarily absolutely so: we could imagine that parts of it were in effect "chosen at random", while other parts are not - that is what non-determinism would mean in a block universe context. In that case, the actual descriptive complexity will be that much higher, but the existence of "partial" laws will nonetheless drive it down by a factor, i.e. that many fewer zillions of bits, than the case with no laws at all, i.e. a fully, or Kolmogorov-random, universe.

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Laws of physics arise from noticing that there are patterns in nature that appear to recur reliably. For instance, Newton's laws of motion and quantum mechanics. These suggest that the universe is not an arbitrary, eternal collection of blocks, but an emergent phenomenon that arose out of some basic principles that produce these patterns.

What would this mean in the context of a block universe? It means that there are rules governing how the could be constructed out of all the blocks. Sort of like the workings of Lego blocks and Lincoln Logs: they can only interlock in certain ways. These rules become visible to us as the patterns that we notice, and physics is the process of discovering these patterns. Perhaps if we keep investigating we might eventually be able to figure out how the blocks themselves are design and why they have these rules. This is the holy grail of physics: a "theory of everything".

But since we're part of the universe, this becomes incredibly self-referential. All our thoughts and actions are encoded in these blocks. How much sense does it make to talk about "discovering" things when the state of having this knowledge is already in the blocks? There's little room for the concepts of free will, randomness, and unpredictability.

What we view as the arrow of time is just a dimension in the block universe. The second law of thermodynamics, which says that entropy always increases in the "forward" direction of time, could be a consequence of the rules of block construction mentioned above.

How likely is it that some basic rules for interlocking blocks could produce the emergent effect of detecting and understanding the patterns produced by those rules? This is not unlike the question of how likely it is that the basic constants of physics are such that we get a stable enough universe to produce stars, planets, and life. We don't know how this happened, and a common, but not very satisfactory, solution is the anthropic principle: if things weren't the way they are, we wouldn't be here asking the question.

Doing physics essentially assumes that all these patterns really exist, and they have some cause. But it could also be an illusion, like the Matrix. Or like biblical literalists who believe that the Earth is only a few thousand years old, and that all the geographic and archeologic evidence scientists use to prove its age were created by God to make it appear older.

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There is a possibility that the "flow" of time may primarily be an analogy traceable to medieval water clocks: Einstein does appear to have consistently regarded time as a dimension comparable to the spatial ones, although the directionality of passage through it is limited to the familiar past-to-future one through its widely-accepted relation to Boltzmann's 2nd Law of Thermodynamics (the only basic law of physics that's not consistently reversible), whose "one way" directionality's derived from Clausius' observation that heat can never pass (or "flow") from a cool object to a warmer one.

The validity of Einstein's "spatialization of time" may, however, be limited to 1905's Special Relativity, which was not a theory of gravity: 1915's General Relativity was, but, as any earthling lifting a needle from a desktop with a toy magnet can see that the toy's power easily overcomes the gravitational strength of the entire earth's mass, the distortions of time's passage by the gravitational field (which has been numerically quantified over heights above the earth's surface at least as low as the height of a three-story building) are extremely difficult to take into account over such distances as the width of our astronomically observable region: Consequently, such quantum physicists as Rovelli see time more simplistically as "change". Such great relativists as Vilenkin have pointed out that "time and simultaneity are not well-defined in GR".

This is not to say that perceptible effects equating to a reversal of time, even within three-dimensional space, may not occur with a variation in scales that would have to extend across a sufficiently colossal spatio-temporal range: Poincaré recurrence (whose mathematical consistency was verfified in the 1920's) remains the nemesis of such quantum physicists as Carlo Rovelli and Laura Mersini-Houghton, although it may be most plausible in such cosmological models as Nikodem Poplawski's "Cosmology with torsion", described in numerous 2010-2021 papers whose preprints can be found by his name and are freely visible on Cornell University's Arxiv website. (The material interactions hypothesized by Poplawski depend on the 1929 Einstein-Cartan Theory worked out by Einstein in collaboration with the mathematician Cartan, whose formalism remains unfamiliar to many physicists, although that theory, developed within a few years after the discovery of particulate spin and elaborated by Sciama and Kibble some decades later, is considered to reduce to GR in vacuum.)

"Poincaré recurrence" is well-described in the Wikipedia article by that name. Rovelli's viewpoint, basically seeing time as illusory, is beautifully described in his 2017 pop-sci book titled "The Order of Time". Vilenkin's relativistic stance on time is described through footnotes in his online essay titled "The Principle of Mediocrity": Like Poplawski's model (initially described as an "alternative" to cosmic inflation, but more generally seen as a version of it), it depends on divisions of the universe into a multitude of causally-separated localities.

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This:

"The idea that the whole universe exists simultaneously"

is not what the block universe is. Rather, what it is, is that there is simply the absence of any distinguished point in time corresponding to "now", and of any distinguished point in space-time corresponding to "here and now"; so that all points are equally when and where they are.

In other words: a Universe without a Cursor.

That corresponds to the view of space-time as a timeless four-dimensional continuum; so that verb tenses are also inapplicable (since they are all temporal). So, really, you can't use verbs at all either, and everything is a noun or noun phrase. In other words: academic-speak (e.g. where all the "X-ing"'s are turned into "the X-ion of").

In science popularizations (and even amongst professionals in the field), the dichotomy between "block" and "moving" time is presented as one between Relativity versus Quantum Theory, with the idea being that Relativity is on the "block time" side of the schism, while Quantum Theory is on the "moving time" side. But that's wrong. That's not where the schism is, at all.

    The schism is entirely internal to Quantum Theory
    and exposes a large gap in Quantum Theory

as you're about to see; and as you're about to see, the whole dichotomy and schism is a false dichotomy:

    You can have both block time and moving time.

When the above-mentioned gap is exposed and resolved, that's exactly what the result is.

The gap will also happen to be where your own question comes into focus.

So, if you were to ask me whether time is "all there" or is "moving", I would answer: "both, because they are referring to two different things and to two different senses of 'time'" and that:

    The whole timeless four-dimensional continuum is,
    itself, moving in time.

The dichotomy between block time versus moving time corresponds to the two views of time presented, respectively, by the Heisenberg Picture and the Schrödinger Picture in Quantum Theory. In the Heisenberg Picture, a state represents the entire history of a system and is timeless, while in the Schrödinger Picture, states change in time, represent a system at a given point in time, and change continuously in time.

The Heisenberg Picture in Quantum Theory agrees with General Relativity with respect to the question of how to view time. In place of the "Schrödinger Equation", which expresses the evolution of a quantum system as an unfolding of the system in time, one has the "Heisenberg Equation", which actually, and ultimately, treats all the dimensions of space-time equally and expresses the evolution of the system as an unfolding in space-time.

So, as a result, what we find is that the widely-popularized schism is in the wrong place: ...

    It's not:
            Moving { Quantum Theory } ⇔ Block { Relativity }
            Quantum Theory Versus Relativity schism
    but is:
            Moving { Schrödinger } ⇔ Block { Heisenberg, Relativity }
            Schrödinger Versus Heisenberg schism

The way this shows up most clearly is as follows:

    What is the Heisenberg Picture version of "Wave function collapse"?

In particular, what is the Heisenberg Picture version of the Born Rule? There's a gap there! In fact, the gap is much larger:

    There is no Heisenberg Picture version of Measurement Theory!

There are attempts to formulate one - even a recent attempt by Deutsch et al. to write out a Heisenberg Picture version of Many Worlds - but none that has received wide-spread acceptance. And even Deutsch's attempt at resolution is showing some of the features of what I'm about to describe below.

The main problem is that a Heisenberg Picture state doesn't change, at all. So, what's changing when there is any kind of "measurement"? That's where the gap is exposed.

To even so much as express the result of a quantum measurement in the Heisenberg Picture, you need extra infrastructure that is not present in any formulation of Quantum Theory, though there is something that is similar to what we need in Quantum Field Theory.

The critical part of the Born Rule (or of whatever replaces the Born Rule, in whatever other quantum theory interpretation you wish to talk about, be it Consistent Histories, Decoherence, Many Worlds, etc.) is that a causal ordering is placed on measurement outcomes, so that any subsequent measurement done on a system after a given measurement receives the "post-collapse" result of that measurement.

Merely to express the Born Rule in the Heisenberg Picture, you need to first specify the set of all measurement events; that means a list of places and times where measures are done, plus what each of those measurements is.

Second, you need a prescription of the causal ordering between the elements of that set. The measurement set has a temporal logic in it, that specifies the "before-after" relation between measurements as a partial ordering of the set.

Third, you need to consider all the possible ways to partition this set into "before" and "after" subsets, such that no measurement in the "after" subset lies before any measurement in the "before" set. Each such partition $(B, A)$ into a Before set $B$ and After set $A$ may be referred to as a ... [drum roll] ... [big reveal] ... a Now.

The "Now"'s, themselves, can be partially ordered. So that of $(B_0, A_0)$ and $(B_1, A_1)$ are two Nows, then we say that $(B_0, A_0)$ is "before" $(B_1, A_1)$, if $B_0 ⊆ B_1$ or, equivalently, if $A_0 ⊇ A_1$, where we note that $B_0 ∪ A_0 = B_1 ∪ A_1$ and $B_0 ∩ A_0 = ∅ = B_1 ∩ A_1$.

In non-relativistic theory, this is a linear ordering provided that no two measurements in the set are simultaneous. If so, then the linear ordering if a sub-ordering of that given by the time, itself.

In Relativity, however, it is not a linear ordering, though it has linear sub-orderings. Each maximal linear sub-ordering may be identified as a sequence of measurements - or more simply as ... [drum roll] ... [big reveal] ... an Observer.

The final completion of the gap consists of generalizing the notion of "state" in the Heisenberg Picture. Instead of just one state, you have an entire set of states indexed by all the Now's. That is: the Heisenberg state is generalized to a state-valued function of the set of Now's.

We'll refer to these functions as Temporal Heisenberg states. Thus, a Temporal Heisenberg state assigns an ordinary Heisenberg state to each Now, with a further restriction to be mentioned below. In addition, we'll say that, for a given Now $(B, A)$, the Temporal Heisenberg state $Ψ$ "at" the Now is the Heisenberg state $Ψ(B, A)$.

Finally, with all that infrastructure in place, you can express the Born Rule. If the set is discrete and has a discrete topology, then one can refine the Before-After relation to an Immediately-Before-After relation. Two Now's are said to be immediately before or after one another, if there is no third Now that lies between them in the before-after relation.

For such a relation, the respective sets take the form: $B_1 = B_0 ∪ \{ m \}$ and $A_0 = A_1 ∪ \{ m \}$, where $m$ is a single measurement. At the place and time of $m$, you transform the temporal Heisenberg state at $Ψ(B_0, A_0)$ to a state $ψ$ in the Schrödinger Picture. Then, you use the Born Rule to write out the results $ψ'$ of the measurement $m$, along with the associated probability of each such result. Finally, transform these back into the Heisenberg Picture, as the corresponding states $Ψ'$, and this yields a probabilistic transform to the state $Ψ' = Ψ(B_1, A_1)$.

The central requirement for a temporal Heisenberg state is that all state transitions at Now's that are immediately before-after one another, with $m$ being the measurement that changes from a "before" to an "after" measurement, should be transitions that are consistent with the Born Rule applied to $m$.

That's very similar to the Consistent Histories formulation of Quantum Theory, with the key difference being that we have all the extra infrastructure in place.

In the case where the measurement set is not discrete, then one has to deal with continuous measurements ... as well as the related phenomena such as the Quantum Zeno Effect and the Quantum Anti-Zeno Effect.

So, where does the likeness of this show up in Quantum Field Theory?

When Quantum Field Theory is done within the framework of perturbation theory, the states of a field, and operations carried out on the fields, are expressed in terms of the states of the free fields (possibly with the addition of states corresponding to bound systems, like molecules). In the Schrödinger Picture, the evolution of a quantum system is given by a Hamiltonian $H$. Within the context of Quantum Field Theory, a split is made to it, to extract out the Hamiltonian $H_0$ that corresponds to free non-interacting fields, with the remainder $V_S$ being the "interaction": $H = H_0 + V_S$.

The evolution of the fields is then expressed in the Heisenberg Picture! But, there's a twist. The Heisenberg Picture is that for the free fields, i.e. that whose evolution is governed by $H_0$, not by $H$. In the Heisenberg Picture, the interaction term $V_S$ is, itself, transformed into the corresponding Heisenberg Picture version $V$ whose unfolding in space-time is given by the Heisenberg equation. And, again: this equation treats all space-time coordinates the same, so they're actually partial differential equations of the same form as those in the corresponding classical theory.

The Schrödinger Picture evolution of the system under $H$ is then expressed in terms of a side-ways evolution of $H_0$-Heisenberg Picture states; this evolution is given (asymptotically) by the S-matrix. The resulting picture is called the Interaction Picture.

In effect, the Schrödinger evolution under $H_0$ is with respect to "block time", because it's all been rendered in the $H_0$-Heisenberg picture; while the sideways evolution in the Interaction picture with $V$ is with respect to "moving time".

Also, in effect, the Interaction Picture treats the interaction $V$, itself, as continuous measurements on the system. The asymptotic form of the S-matrix is actually a limit form that is used (in practice) to encode the before-after results of the Interaction Picture state following a single interaction event. These are the physical forms of what are depicted in the vertices of a Feynman Diagram. (Footnote: Not all vertices or lines in a Feynman Diagram are physical.)

Each such event corresponds to a measurement, like $m$.

This raises the interesting idea that $V$, itself, might be represented as a kind of continuous measurement; and, indeed, that all measurements in quantum systems are formed by way of quantum field interactions in a similar way.

Now, in the big picture presented above, you may have noticed that I put Relativity all on one side of the schism. Is it actually all on one side, or does the internecine "block time" versus "moving time" schism of Quantum Theory also arise in some fashion in Relativity? If so, then what does the "moving time" version of Relativity look like?

This question is closely connected to the question of precisely what is the "non-relativistic limit". By way of an example, consider the Schwarzschild geometry, which is given by the line element $$-α ds^2 = \frac{dr^2}{1 + 2αV} + r^2 \left((dθ)^2 + (\sin θ dφ)^2\right) - (1 + 2αV) dt^2,$$ where $α = (1/c)^2$ and $V = -GM/r$ is the gravitational potential (per unit mass), given by the central object, which has mass $M$. $G$ is Newton's gravitational coefficient.

I present the line element, here, as a line element not for length but for proper-time, denoted as $s$.

The proper time is the closest equivalent, in Relativity, that we have to a notion of a personalized "moving" clock time, in contrast to the "block time" $t$ that is treated as a coordinate for one of the space-time dimensions. In the non-relativistic limit, the "time dilation", which is given by the difference $s - t$ vanishes; i.e. $s - t → 0$ or, equivalently, $s → t$.

In non-relativistic physics proper time and coordinate time coincide, and there is no time-dilation. In that setting, there is a conflation of "moving" and "block" time since $s$ and $t$ now coincide.

Nonetheless,

    there is still a non-relativistic version of time-dilation!

Consider, instead, the version of time-dilation that is scaled up by a factor of $c^2$, and define $u = c^2 (s - t)$. Its non-relativistic limit is well-defined and is not 0. If you substitute this into the line element for the above-mentioned geometry, the result is that in it place you get both a linear invariant and a quadratic invariant: $$ds = dt + α du,\\ \frac{dr^2}{1 + 2αV} + r^2 \left((dθ)^2 + (\sin θ dφ)^2\right) + 2 dt du + α du^2 - 2V dt^2 = 0,$$ with the quadratic invariant set to 0.

In the non-relativistic limit, as $α → 0$, these reduce, respectively, to the following invariants (and invariant condition): $$ds = dt,\\ dr^2 + r^2 \left((dθ)^2 + (\sin θ dφ)^2\right) + 2 dt du - 2V dt^2 = 0.$$ The quadratic invariant can, equivalently, be written in terms of Cartesian coordinates as $$dx^2 + dy^2 + dz^2 + 2 dt du - 2V dt^2 = 0.$$

That's actually a curved space-time geometry that embodies Newtonian gravity. the geodesic equations are one and the same as the equations for orbital motion of a body around the mass-$M$ gravity center.

The relativistic version of this is a geometry that represents a deformation of the Newtonian case, with $α$ being the deformation parameter. The term $2 dt du$ is present in both the relativistic and non-relativistic cases and embodies Galileo's principle of relativity. The Euclidean part of the line element $dx^2 + dy^2 + dz^2$ comes out of Euclidean geometry. The term $-2V dt^2$ is the Newtonian gravity term, while the result of the subtraction $$\frac{dr^2}{1 + 2αV} + r^2 \left((dθ)^2 + (\sin θ dφ)^2\right) - \left(dx^2 + dy^2 + dz^2\right) = -\frac{2αV dr^2}{1 + 2αV},$$ is a deformation of the Euclidean part of the geometry that, I believe, also happens to be the term from which the General Relativistic precession of the Mercury orbit can be derived.

The geometry that emerges from this is a 5-dimensional geometry that has a 5-dimensional metric and an invariant direction. That invariant direction corresponds to proper time and plays the role of "moving time" in Relativity. So, you have two different senses of time, here: one embodied by the coordinate time $t$, and the other by the proper time $s$.

Perhaps there is a way to generalize this geometric picture so as to be able to accommodate the formulation provided for Temporal Heisenberg states.

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