# Flow of time in an unchanging 4D universe

If the universe is a 4D manifold where every event has already happened (past, present and future) what constitutes flow of time in such a universe?

In above description of time (known as the block universe) the universe is compared to a film reel in which everything is already there. I could only think of two ways in which a sense of motion or unfolding of events can occur.

1. the frames of this movie (the universe) keeps changing which we perceive as events unfolding.

or

1. the frames doesn't change but we move along it in some direction thus perceive the dynamic unfolding of events in that direction.

So, how to make sense of unfolding of events or motion in such a static description of world?

Note: I'm not asking about the "arrow of time" which is usually explained by increase in the entropy.

I'm not too familiar with the block universe formulation, but in GR we think of a 4D Lorentzian manifold where spacetime points represent events. I'll try to give an example where this static 4D manifold can still appear to give those desired phenomena.

Take the example of where the manifold is globally hyperbolic, and we can foliate it into 3D spacelike hypersurfaces with the parameter $$t$$ (coordinate time), as is done in the ADM formulation. Given data on the hypersurface at a given $$t_1$$, you can think of the evolution in $$t$$ as how the configuration changes to the next hypersurface at $$t_2$$. This seems to give a well defined notion of events unfolding (configurations on the 3-space changing with time), despite the fact the full 4D spacetime manifold can be described as static. Obviously this isn't a unique way$$^1$$, because in general you have freedom of how you paramaterise your spacetime with $$t$$, but it gives a sense of a sequence of events taking place in time.

$$^1$$In cosmology, the way in which we make this 3+1 splitting can be defined in a nice way (making $$t$$ the cosmic time defined along the Hubble flow), which gives a satisfactory way for different observers to order events.

• are you saying there is an evolution of 3D space over time ( from t1 to t2 ) and the word static only applies to manifold as a whole ?
– user282467
Dec 13, 2020 at 11:28
• In a way, yes. The full spacetime manifold can be static, but it can still be seen as an evolution for the spatial 3-manifold in the time direction. Does this answer any of your original question? Dec 13, 2020 at 12:06
• If I think of 3-manifold evolving In time yes it makes sense. Thank you 🙂
– user282467
Dec 13, 2020 at 12:14
• @shashankaMg Bear in mind that there isn't a unique universal now. Even without spacetime curvature, different frames have time axes pointing in different spacetime directions, so they disagree on which events are simultaneous. Curvature makes it more complicated. Dec 13, 2020 at 12:45
• I'm aware of relativity of simultaneity and lorentz transformations .
– user282467
Dec 13, 2020 at 12:52

In the block universe concept there is no physical flow of time. The flow of time is a purely mental sensation that does not depict reality. This mental sensation occurs at the events on a human’s worldline where the human is conscious. Distortions to this sensation are also common, such as “deja vu” and “time flies” and various temporal illusions such as the synchronicity of auditory and visual stimuli.

Note, there is no need for the mental sensation to correspond to something physical. In particular, the existence of temporal illusions shows that our mental perceptions of time are not very physical, but involve a substantial amount of artificial editing to produce our subjective experience.

• Are you saying conciousness traverses along a human's wordline ? When everything is static how is that conciousness is able to do that ? And if everything is static how come there are sensations ( since sensations requires chemical reactions in brain ) ?
– user282467
Dec 13, 2020 at 13:19
• @shashankaMg asked “Are you saying conciousness traverses along a human's wordline?” No, I am most definitely not saying that at all. It is a sensation that simply exists at each event on the worldline where the human is awake and in normal health. Comments are not for discussion. New questions should be asked as separate questions.
– Dale
Dec 13, 2020 at 13:41
• All I'm asking is how there is sequential ordering of subjective experience from birth to death . If as you say it's a sensation that exists at each event how can there be a sequentially ordered subjective experience from birth to death ? And it would also mean that every event along the worldline is experienced all the time . Like do you experience same event again and again ? Since that event stays at the point for ever . A movie reel by itself won't make a movie , it's when you play it in a sequential order it can be seen as a movie .
– user282467
Dec 13, 2020 at 14:11
• Sequential ordering is not unique to time. You can have things sequentially ordered from left to right or from top to bottom. There is nothing about a sequential ordering that requires an evolving sense of time
– Dale
Dec 13, 2020 at 14:42
• But when doing calculations in GR, we almost always use evolution equations parameterized by some time coordinate. Dec 13, 2020 at 15:33

A fifth dimension for "historical time". :)

So, then all four dimensions of space-time are flowing in time!

Two places where this played out, in science fiction, were the Premonition episode of The Outer Limits, where a husband and wife were caught up in a kind of side-ways time stream along a fifth dimension that was nearly orthogonal to the flow of real time; and in Steven King's Langoliers. Both stories made tacit reference to the idea of real versus imaginary time, in which in the latter dimension, oscillatory processes became exponentially decaying. So it was difficult to start and maintain fires (in Premonition) and jet fuel lost much of its effectiveness (in Langoliers).

Since you would want the time flow to be some kind of universal reference, it should be invariant. Additionally, if it matches each of our experience of time flow, then it should match the proper time for the space-time geometry. So, the natural choice is to just add in the proper time ($$s$$) alongside the spatial coordinates ($$x,y,z$$) and real time coordinate ($$t$$) of spacetime.

In the Minkowski geometry of Special Relativity, $$dx^2 + dy^2 + dz^2 - c^2 dt^2$$ is an invariant which, when negative, equates to $$-c^2 ds^2$$, giving you the differential relation for proper time. So, if we throw in the coordinate then we can rewrite the line element as a constraint: $$dx^2 + dy^2 + dz^2 - c^2 dt^2 + c^2 ds^2 = 0.$$

So, now the time dilation itself can be expressed in terms of the coordinates: $$s - t$$. Notably, this continues to have meaning even in the non-relativistic limit $$c → ∞$$, provided we scale it up as $$u ≡ c²(s - t)$$.

The line element, expressed in terms of $$t, u$$ becomes $$dx^2 + dy^2 + dz^2 + 2 dt du + {du^2 \over c^2} = 0,$$ and the invariant $$ds$$ is now replaced by $$dt + {du^2 \over c^2}$$. In the non-relativistic limit, these become the constraint $$dx^2 + dy^2 + dz^2 + 2 dt du = 0,$$ and invariant $$dt$$, respectively.

For the non-relativistic world, real time and historical time become one and the same; while a vestige of time dilation lives on as $$u$$. In the non-relativistic limit, the $$u$$ coordinate for a body moving with velocity $$𝐫̇(t) = 𝐯(t)$$ can be expressed as $$u = u_0 - \int_0^t {1 \over 2} v^2 dt$$ which is also (up to an additive constant) the negative of the action per unit mass for the body.

The resulting geometry is known as the Bargmann Geometry. The symmetries which leave both the Bargmann metric and the linear $$dt$$ invariant fixed is the Bargmann Group, which is the central extension of the Galilei Group and is the symmetry group more correctly suited for the non-relativistic world. For instance, the mass $$m$$, momentum $$𝐩 = (p_x,p_y,p_z)$$ and kinetic energy $$H$$ of a body transform under a change of velocity $$𝐯$$ as a Bargmann 5-vector $$(m,𝐩,H) → (m, 𝐩 ‒ m𝐯, H - 𝐩·𝐯 + ½mv²),$$ and combine with the coordinates to produce an invariant 1-form $$m du + p_x dx + p_y dy + p_z dz - H dt.$$

We need only expand Hamilton's equations to handle the more general case where $$H$$ not interpreted as the Hamiltonian and also includes the potential energy, $${\partial H \over \partial m} = {du \over dt}, {\partial H \over \partial 𝐩} = 𝐯 = {d𝐫 \over dt},$$ $$-{\partial H \over \partial u} = {dm \over dt}, -{\partial H \over \partial 𝐫} = 𝐅 = {d𝐩 \over dt}.$$

For instance, in the the presence of a conservative force satisfying the equivalence principle, given by a potential per unit mass $$U$$ (e.g. $$U = -{GM \over r}$$ for a gravitating body of mass M), we can write the Hamiltonian as $$H = {p^2 \over 2m} + mU,$$ and convert it to a quadratic constraint $$p^2 - 2mH + 2m^2 U = 0.$$ Applying Hamilton's equations, we get $$2𝐩 - 2m {d𝐫 \over dt} = 𝟎, -2H - 2m{du \over dt} + 4mU = 0,$$ $$2m{d𝐩 \over dt} + 2m^2 {\partial U \over \partial 𝐫} = 𝟎, -2m{dm \over dt} = 0,$$ from which we obtain (with the aid of the quadratic constraint) $$𝐩 = m {d𝐫 \over dt}, H = m\left({1 \over 2} v^2 + U\right),$$ $$\left|{d𝐫 \over dt}\right|^2 + 2 {du \over dt} - 2U = 0.$$ If we multiply the last of these equations by $$dt^2$$, we obtain the following metric constraint $$dx^2 + dy^2 + dz^2 + 2 dt du - 2U dt^2 = 0,$$ which may be identified as the 5D curved space-time metric for the force associated with the potential $$U$$ per unit mass. In fact, the geodesics for this metric produce the orbits arising from $$U$$.

In the Relativistic case, the corresponding geometry does not have any name that I am aware of. Both the Bargmann line element and its relativistic version are metrics for a 4+1 dimensional geometry. In the non-relativistic case, both $$s$$ and $$t$$ are null directions with respect to the metric, while in the relativistic case, $$t$$ is time-line, while $$s$$ is space-like. Thus, the relativistic and non-relativistic versions have the same 4+1 dimensional geometry in common, but differ in that the "proper time" coordinate $$s$$ is null in the non-relativistic world, but spacelike in the relativistic world; the same applies to the coordinate $$u$$.

The relativistic version of the Newtonian gravity metric is equivalent to the Schwarzschild metric. The line element for both can be written as members of a 1-parameter family of metrics: $$dx^2 + dy^2 + dz^2 + 2 dt du - {\alpha du^2} - 2U dt^2 - {2 \alpha U \over 1 + 2 \alpha U} dr^2 = 0,$$ with the proper time invariant $$ds = dt + \alpha du$$, the potential per unit mass $$U = -{GM \over r}$$ and $$r = \sqrt{x^2 + y^2 + z^2}$$. For Newtonian gravity, $$\alpha = 0$$, while for the Schwarzschild metric, $$\alpha = {1 \over c^2}$$.

So, one way of putting all of this together is to think of the 5th dimension as the arena for alternate time-lines. At different layers are different versions of the 4-dimensional spacetime, which we think of as flowing in the $$s$$ direction - a direction for absolute time or historical time, while the $$t$$ coordinate describes the coordinate time, or real time ... all of which is just like Lorentz conceived of it, before Einstein came along. A time flow is embedded into the 4-dimensional manifold by associating a "now" at each $$s$$ on the 4-dimensional manifold. So, the combined geometry describes a moving "now", rolling up the 4-dimensional continuum, as we move from layer to layer up in the $$s$$ direction.

For Relativity, the canonical 1-form can be written in different ways, depending on which two of the three coordinates $$(s,t,u)$$ are chosen: $$p_x dx + p_y dy + p_z dz - H dt + m du,$$ $$p_x dx + p_y dy + p_z dz - H ds + M du,$$ $$p_x dx + p_y dy + p_z dz - E dt + mc^2 ds,$$ where $$M = m + {H \over c^2}$$ is the "moving mass" and $$E = Mc^2 = mc^2 + H$$ is the "total energy". The mass shell invariant can be written as $$p^2 - 2MH + {H^2 \over c^2} = p^2 - {E^2 \over c^2} + m^2 c^2 = 0,$$ and is supplemented by the linear invariant $$m = M - {H \over c^2}$$.

This geometry, strictly speaking, goes beyond Relativity since the symmetry group associated with it not only includes the Poincaré group, but an extra generator corrsponding to translation for the $$u$$ coordinate. Its representations differ from those of Relativity in that they also allow for (1) non-zero values for the quadratic invariant, (2) linear invariants that differ from the mass $$m$$ and (3) non-zero rest frame values for $$H$$ (more generally: they make the zero point for $$H$$ relative, as it is in the non-relativistic world). If $$H$$ has a rest-frame value of $$U ≠ 0$$, then the two invariants will become: $$p^2 - 2MH + {H^2 \over c^2} = -2mU + {U^2 \over c^2},$$ $$M - {H \over c^2} = m - {U \over c^2}.$$

Another notable difference from the representations in Relativity is that while the vacuum is characterized by $$(E,𝐩) = (0,𝟬)$$, in the expanded geometry, this condition becomes $$(M,𝐩) = (0,𝟬)$$, while $$H = U ≠ 0$$ is allowed. There is extra room provided for a notion of vacuum energy, which is not accounted for properly or at all, in the representations of the Poincaré group.

Another interesting feature of the flat space metric is that when it is written in terms of $$s$$ and $$t$$, rather than $$t$$ (or $$s$$) and $$u$$ $$\Re\left(dx^2 + dy^2 + dz^2 - c^2 (dt + i ds)^2\right)$$ it becomes a metric for an expansion of Minkowski geometry with complex time $$t + is$$. So, this may also provide the basis for a geometrical interpretation of complex time.

• Interesting, but I suspect that it's more likely to confuse or mislead the OP than to help them. ;) Dec 13, 2020 at 12:39
• Yeah you are right I don't understand anything beyond the basics of special relativity 😂
– user282467
Dec 13, 2020 at 12:57
• I also found this answer quite confusing, specifically how it links to the apparent flowing of time in 4D spacetime, without needing some embedding in a higher dimensional spacetime? But this might just been me misunderstanding the content I'm less familiar with - not the fault of the answerer. Dec 13, 2020 at 13:09
• @PM2Ring Wow, that's a lot of unnecessary self inflicted censorship ;) Dec 14, 2020 at 23:08