Does the Block Universe of Physics Mean Time is an Illusion?

[RUTA]

I have written many Insights (and coauthored an entire book) explaining how the puzzles, problems, and paradoxes of modern physics can be attributed to our dynamical bias and resolved by rising to Wilczek’s challenge [1]:
A recurring theme in natural philosophy is the tension between the God’s-eye [4D] view of reality comprehended as a whole and the ant’s-eye view of human consciousness, which senses a succession of events in time. Since the days of Isaac Newton, the ant’s-eye view has dominated fundamental physics. We divide our description of the world into dynamical laws that, paradoxically, exist outside of time according to some, and initial conditions on which those laws act. … The God’s-eye [4D] view seems, in the light of relativity theory, to be far more natural. Relativity teaches us to consider spacetime as an...

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[Stephen Tashi]

Of course, the corollary to the fact that the laws of physics cannot negate the reality of our dynamical experience is that laws of physics cannot explain the reality of our dynamical experience either — again, because they are co-fundamental in this view. This has been acknowledged by Max Planck [10]:

As I understand the article, it asserts:

1) Because our consciousness and some of our perceptions are not illusions, it follows that no physical theory demonstrates that time is an illusion. (That is a concrete interpretation of "the laws of physics cannot negate the reality of our dynamical experience".)

and

2) Physics cannot explain "the reality" of our experience. From the reference to Planck, I gather this asserts that physics cannot the explain the experience of consciousness.

Assertion 1) is self evident if we assume our conscious experience is , on the whole, not an illusion. Assertion 2) is plausible, but unproven one way or the other.

Would neuroscientists agree with assertion 2)? They study brains, which are very specialized physical structures. They ask which substructures of the brain are required for consciousness. If physics cannot explain consciousness then why is consciousness a property of such a small subset of physical structures?

One abstract view is that maybe it isn't. Perhaps a city has consciousness that implemented by its networks of streets, cables, etc. Perhaps a galaxy or a coffee cup is also conscious. If we reject those possibilities then one way "physics cannot explain the reality of our dynamical experience" could work out is for any physical theory that explains the consciousness of brains to also imply the consciouness of objects we consider inanimate. (i.e. Perhaps no physical theory of consciousness can separate conscious physical structures from unconscious ones in a way that we find satisfactory.)Returning to assertion 1), it's worth mentioning the alternative. The article refers to something (time) not being an illusion, so I'll permit myself to talk about illusion versus reality without delving into the metaphysical complexities of those concepts.

The obvious scenario: The thing that is me at time t= 0 and the thing that is me at time t = 1 are both real. The version of me at time t = 1 has the illusion that t = 1 is the only "now" and that the version of me at time t = 0 is no longer real. The version of me at time t = 0 has the illusion that t = 0 is the only "now" and the version of me at time t = 1 is not yet real.

(By the way, what exactly would it mean to say that time is an illusion? Is the above scenario an example of time being an illusion?)

 

[RUTA]

The Insight shows that one can subscribe coherently to both the block universe model of physics and the reality of our dynamical experience of time. The ontology I shared (per neutral monism, see Ref 6) to make that point does entail that physics cannot account for subjective experience (such as the dynamical experience of time). That ontology is not necessitated by physics and I linked to the Smolin paper as an alternative :-)

 

[Dataatq]

Check out Existics by Gavin Wince.

 

[RUTA]

Check out Existics by Gavin Wince.

According to this article, it’s nonsense: https://goodmath.scientopia.org/2013/03/21/genius-continuum-crackpottery/ . I hope you’re not comparing the published works I cited with this.

I want to point out again that the idea my colleagues and I are trying to sell the foundations community is that the physics we have now is actually right. I so often read that QM is “incomplete” or “wrong” or that QM is ”incompatible” with SR, etc. Our publications and book are attempts to show how physics is right and beautifully self-consistent, if you don’t insist that objective reality be fundamentally understood via causal mechanisms.

In general, I’m always suspicious of people who claim that established ideas are “wrong.”

 

[.Scott]

I do not view consciousness or the illusions as discussed here.
The illusions we have are instrumental in our survival as a social animal.

When I call something an illusion, I do not imply that it is false - only that it is apparent - and more importantly, pragmatic.

It may help you to know that I have been designing and coding software for half a century now - so how should I view the human nervous system as other than an information processing system.

Our first illusion is that we, personally, exist. Of course we do exist, but we generally view ourselves as a being in a body bounded mostly by our sensor-ladened skin acting as a unified self in a social environment. There are conditions where this illusion fails (ex, anosognosia or asomatognosia) but we can always expect that the rest of society will treat us as a single being. And as a matter of survival, when part of us fails, it threatens our whole being. So having a built-in model of ourselves as human with a single agency is very pragmatic.

Which brings us to the second illusion - that we can affect our condition and our surroundings. We have a built-in notion that we can make choices and act on them. This is not to say we can't affect things - but it is significant that we hold a view that we can. Your iRobot may plan out a strategy to clean your floors in a short time while keeping itself charged. But it has no sense that it can affect its environment.

Which brings us to time. Not only do we have a spatial (skin-bound) view of ourselves, we also have the notion that what we remember from the past represents a real human being - the same one that exists now - and that will continue into the future. And we are built to record to memory pleasant and unpleasant events - and to view those memories as clues to how to make our world "better" for us.

So Physics gives us a 4D world and we evolve a built-in story about how we work within that world. Our illusions are personally pragmatic models of the key concepts important for our survival.

Now let's talk about consciousness...
Everything I just described above could be programmed into a machine - resulting in a robot that in principle could operate within human society as a human. And it would not be conscious. If you think otherwise, you have to explain to me the fundamental difference between an abacus, Babbage's Analytic engine, and contemporary computers. And if you think its complexity, how large does a Rube Goldberg machine have to be before it becomes conscious? Computers are not conscious; large networks of computers are not conscious; and if all of a sudden something fundamentally changes when you interconnect processor number 1 billion, what you have is a network problem - not consciousness.

So why are we conscious? Apparently, there are basic information-processing components useful to our survival that are easier to implement in our biology in a fundamentally different manner than the way it is done in a digital computer. And it is this different component that caries with it the foundation of consciousness.

What is important for this discussion is that the illusion of time is simply a possible content of consciousness, not a direct consequence of it.

 

[intervoxel]

A problem with cellular automata models of the universe is how to explain the cell processor (unless we live in a simulation). The block universe model can solve this puzzle by stating that time "slices" are connected blocks of information. See https://zenodo.org/record/3818303#.X3ilCMJKiM9 for further details.

 

[PeterDonis]

Check out Existics by Gavin Wince.

This is not a valid reference for PF discussion.

 

[Stephen Tashi]

The Insight shows that one can subscribe coherently to both the block universe model of physics and the reality of our dynamical experience of time.

I'm trying to understand whether the article has a deductive structure or whether it only intends to be persuasive in a subjective manner.

The ontology I shared (per neutral monism, see Ref 6) to make that point does entail that physics cannot account for subjective experience (such as the dynamical experience of time). That ontology is not necessitated by physics and I linked to the Smolin paper as an alternative :-)

After glancing at Ref 6) and the Smolin paper, the interpretation I get is summarized by:

Fact (or Assumption) 1: Current physics does not provide a way to distinguish conscious physical phenomena from phenomena that are not conscious. Furthermore, since current physics is based on concepts that do not entail consciousness, it will be unable to distinguish consciousness as an emergent phenomenon in a complicated physical model.

For example, Ref 6 says:

We agree with the panpsychists and the like that Galileo made an error and that one can’t subtract out conscious experience from what we call the “physical” universe.

and

Relatedly, there are various information theoretic models of conscious experience such as Integrated Information Theory (IIT) that purport to explain if not the very existence, at least the content, unity, degree, or types of conscious experience under certain conditions such as waking state, dreaming, psychedelics, anesthesia, etc. [4]. Pragmatically speaking, we do not doubt that there are many such valuable projects going forward, and we say let a thousand flowers bloom. However, suppose one hopes not to merely formally model conscious experience or seek the neural, dynamical, graphical, information-theoretic, or computational correlates of conscious experience. As Christof Koch puts it ([4], p. 71):
Once science sees the neural correlate of conscious experience face to face, what then? . . . But we would still not understand at a conceptual level why this mechanism but not that one
constitutes a particular experience. How can the mental be squeezed out of the physical?

Fact (or Assumption) 2: Our conscious experience defines what is real.

Conclusions:

1) To create a physics that describes consciousness requires starting with concepts that model conscious experience. Mathematical concepts such as space and time do not, in themselves, entail the presence of consciousness.

2)One may "coherently" agree with a physical model of the Block Universe (or presumably any other successful physical model) and also agree that time ( and presumably any other conscious experience) is real. Such coherence is possible because current physical models say nothing about consciousness versus unconsciousness.

 

[RUTA]

I'm trying to understand whether the article has a deductive structure or whether it only intends to be persuasive in a subjective manner.

The Insight provides a counterexample to the claim that block universe physics is incompatible with our dynamical experience of time. In that sense the counterclaim that block universe physics is not incompatible with our dynamical experience of time is deductively valid.

The statement “our dynamical experience of time is an illusion” probably has many meanings. I take it to be a derisive dismissal of our dynamical experience of time as unworth of explanation. For example, suppose you and a friend watch a magician put his assistant in a box and saw it in half. Your friend exclaims, “OMG, how can she still be alive after being sawed in half?!?” You would not feel compelled to explain why the assistant is alive after being sawed in half because in fact the assistant was not sawed in half. That perception is an illusion, it’s not true and therefore in no need of explanation.

 

[Jimster41]

I am kind of hoping that in the book you speculate on just what might cause such a difference of quality... why would flour and yeast molecules in a loaf of bread think they were in toast-slice movie? What could cause, for lack of a better description, such vertigo of perception - when the actors in that movie suddenly picture the loaf - which they hadn't pictured up until some specific flour and yeast actor/guy named Einstein came a long and said, "picture this".

 

[Jimster41]

This reminds me of that old chestnut (Douglas Adams was the person who introduced my to the idea btw) that we are part of some optimizer, 3D annealing to characterize the 4D object (which is maybe mysterious to some 11D perspective). It's mostly good for a laugh but then I'm like... it does look sort of like that.

 

[madness]

Would neuroscientists agree with assertion 2)? They study brains, which are very specialized physical structures. They ask which substructures of the brain are required for consciousness. If physics cannot explain consciousness then why is consciousness a property of such a small subset of physical structures?

There are some tentative answers to this question, but they are still controversial. Based on some of your replies in this thread you may find the following theory interesting:

https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1003588
https://www.nature.com/articles/nrn-2016-44
http://var.scholarpedia.org/article/Information_Integration_Theory

And a rebuttal:
https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004286

Assertion 1) is self evident if we assume our conscious experience is , on the whole, not an illusion.

Also controversial unfortunately https://www.ingentaconnect.com/content/imp/jcs/2016/00000023/F0020011/art00002, https://www.amazon.com/dp/1845409574/?tag=pfamazon01-20

 

[paradisePhysicist]

Continue reading...

my 2 cent reply:

time is not an illusion, consciousness creates time. time is simply adding data to the universe, its not an illusion it is an enhancement, changing 3d to 4d.

 

[bhobba]

The block universe is just one way of looking at relativity that has its ahrendts. It is not something all physicists take to. What I hope to show, while these ideas can be interesting, when doing physics, there is a simple operational idea of time that is all that is required. It bypasses all these 'deep' questions of what it 'actually' is.

It does not matter what you think time is, an illusion, really psychological, philosophical, or anything you can think of, we can go ahead and do physics by looking at it in a 'simple' way. While not usually emphasized, here is the full definition of an inertial frame. An inertial frame is a conventional standard of rest such that all points, instants of time, and directions are equivalent as far as the laws of physics go. Note I have not tried to describe what time is - I leave that up in the air so to speak. In an inertial frame it doesn't matter when you perform something that takes some time, it will take exactly the same amount of time. So to assign a number to time, which is all that is required to do physics, you simply repeat the same process over and over counting the number of times it occurs. This is a more carefull explanation of what many physicists say when asked about time - time is what a clock measures. The process I described is really a simple clock.

This is why questions like what is time, do not need to be answered to understand or make progress in physics. It's OK to discuss these issues, but mentors will keep an eye on the thread to ensure it does not slip into personal theories, or pure philosophy. While philosophy is off topic on this forum because we do not have moderators qualified well enough to moderate them, it is accepted some legitimate discussions will touch on philosophical issues, but the moderators will keep a tight rein on it. As a moderator I can assure everyone this is quite hard, and many people will likely be unsatisfied with decisions made. All I can do is assure everyone such decisions are not made lightly, and usually made after discussion with other mentors.

Thanks
Bil

 

[bhobba]

Check out Existics by Gavin Wince.

As Peter said, that would not generally be considered a valid reference on this forum. Please be carefull to stick to peer reviewed literature, textbooks etc. It can be hard sometimes to decide - if in doubt contact a mentor.

Thanks
Bill

 

[MacCrimmon]

Sorry. Still new to the forums. I don't know how to use the quotes yet. Please explain.
Also I have read (sorry I don't know source now and it may not be acceptable to the Forums but please don't point me this time.) that scientists are using superposition to get atomic clocks to measure time more accurately. You said "time is what a clock measures." And as the measurements are compared for greater and greater accuracy without accepting an averaged time between devices, doesn't it imply that the time a clock measures is effected by more than GR and SR, mass, acceleration, gravity waves. Those lead to one comparison of measured time to different frames of reference. But if measured clock time is also influenced by another measurable factor, wouldn't that imply another possible dimension measuring time from one inertial frame to another?

 

[bhobba]

doesn't it imply that the time a clock measures is effected by more than GR and SR, mass, acceleration, gravity waves. Those lead to one comparison of measured time to different frames of reference. But if measured clock time is also influenced by another measurable factor, wouldn't that imply another possible dimension measuring time from one inertial frame to another?

Indeed measured time is affected by all those things you mention. Here on Earth we usually think of it as inertial - but it isn't really. We now have clocks so sensitive that even lifting them one foot shows time is different than one not lifted. That's why I talked about inertial frames. You can view an inertial frame as a standard of rest in interstellar space. Even that is not strictly inertial because we have the Cosmic Microwave Background Radiation (CMBR) - but it is good enough to get the general gist. Inertial frames are abstractions useful in discussing theoretical ideas. It is like a point in geometry is defined as having position but no size. Such of course do not exist, still in proving theorems we often appeal to diagrams with points that have a small, but actual size. It is part of the reasoning toolkit we have as humans to analyse things, often called abstracting away inessentials. It is a matter of experience and tact on what to abstract away.

Thanks
Bill

 

[PeterDonis]

I have read (sorry I don't know source now and it may not be acceptable to the Forums but please don't point me this time.) that scientists are using superposition to get atomic clocks to measure time more accurately.

First, you evidently know you're supposed to give a specific reference, so asking not to get a warning for not doing so is not going to work. You have just gotten a warning.

Second, I have no idea what "scientists are using superposition to get atomic clocks to measure time more accurately" even means, so without a reference we can't say anything useful about it. That is why we need references.

wouldn't that imply another possible dimension measuring time from one inertial frame to another?

No, since this is a meaningless phrase.

 

[berkeman]

I don't know how to use the quotes yet. Please explain.

If you want to quote a whole post, click "Reply" in the lower right of the post window. If you want to quote a portion of a post, use click-drag to highlight the part that you want to quote, and you should see a pop-up that says "Reply" that you can click.

 

[MacCrimmon]

Oct. 23 (UPI) -- Not even the most precise atomic clocks are immune to the quantum phenomenon known as superposition, according to a new theory developed by a team of physicists from Dartmouth College, Saint Anselm College and Santa Clara University.

Superposition describes the ability of an atom to simultaneously exist in multiple states. In a new study, published Friday in the journal Nature, scientists theorize that superposition leads a correction in atomic clocks -- an effect the study's authors call "quantum time dilation."
I hope Nature and the physicists of those colleges qualify.

 

[MacCrimmon]

I think I looked it up in the rules. I think Nature magazine is there.
NATURE
Publisher:
NATURE RESEARCH , HEIDELBERGER PLATZ 3, BERLIN, GERMANY, 14197
ISSN / eISSN:
0028-0836 / 1476-4687
And in the above those physicists claim a "quantum time dilation"..
And it is pertinent to the discussion of block universe and time as illusion or just illusive.

 

[Sunil]

In a new study, published Friday in the journal Nature, scientists theorize that superposition leads a correction in atomic clocks -- an effect the study's authors call "quantum time dilation."

Thanks. No problem with the time dilation effect itself, but with the philosophy behind this:

What allowed Einstein to transcend Newton’s absolute time was his insistence that time is what is shown by a clock
...
In this regard, time observables are identified with positive-operator valued measures (POVMs) that transform covariantly with respect to the group of time translations acting on the employed clock system8,9. This covariance property ensures that these time observables give the optimal estimate of the time experienced by the clock; that is, they saturate the Cramer–Rao bound10. Furthermore, covariant time observables allow for a rigorous formulation of the time-energy uncertainty relation4,5,6,7, circumvent Pauli’s infamous objection to the construction of a time operator11,12, and play an important role in relational quantum dynamics13,14,15,16,17.

"Transcend" sounds nice but is misleading in this case. Don't forget that Newton has already defined and distinguished two notions:

Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration. Relative, apparent, and common time is any sensible and external measure “(precise or imprecise)“ of duration by means of motion; such a measure -- for example, an hour, a day, a month, a year -- is commonly used instead of true time

That means, Einstein simply uses Newtonian apparent time as time and ignores Newtonian absolute time.
This decision is nothing worth to be named "transcending".

Then, "covariance property" is misleading, given that what is claimed is about the "covariance" with respect to time translations, which, first of all, assumes that such a group of time translations exists, thus, presupposes an absolute, preferred time which defines the meaning of "time translation". This absolute time is present in quantum theory, it is the parameter of the Schrödinger equation. Newtonian "apparent time" is something observable, and therefore has to be described differently. (Of course, following Kretschmann's objection, one can present every theory, inclusive Newtonian theory, in a covariant form. Nonetheless, this does not seem to be in agreement with how "covariant" is used by physicists, which usually suggests some form of relativistic symmetry.)

Then, Pauli's objection, which is not infamous at all, is not really circumvented if one uses a POVM instead of an operator. There is, last but not least, an important conceptual difference between operators and POVMs: Operators can be used to prepare eigenstates which have well-defined results for a repetition of the measurement. POVMs don't have such eigenstates and cannot be used for preparations.

 

[MacCrimmon]

I liked what Sunil wrote but:
Question: Is the parameter of the Schrödinger equation a useful temporal Inertial frame that is an abstraction useful in discussing theoretical ideas? Is it like a point in geometry defined as having position but no size which of course does not exist. Still in proving theorems we often appeal to diagrams with points that have a small, but actual size?

 

[PeterDonis]

Is the parameter of the Schrödinger equation a useful temporal Inertial frame

What does "temporal Inertial frame" mean?

 

[MacCrimmon]

What does "temporal Inertial frame" mean?

I am referring to the use of an absolute time used by Schrödinger as being used in the manner of an inertial frame being the absolute beginning position.
But my comparison or conceptualisation of these two uses may be incorrect hence the question.

 

[PeterDonis]

I am referring to the use of an absolute time used by Schrödinger

Which means he is using non-relativistic mechanics, so "inertial frame" means the Newtonian concept, not the relativistic concept. I don't know if that helps.

an inertial frame being the absolute beginning position.

I don't know what this means.

 

[MacCrimmon]

So if Schrödinger is using non-relativistic mechanics, meaning the Newtonian absolute time concept, not the relativistic concept, then doesn't there develop a contradiction between understanding time as relativistic with covarient properties and Schrödinger's equation using Newtonian absolute time just for the convenience of studying quantum mechanisms? Unless these two starting points are resolved with an explanation, doesn't it threaten to lead to the wrong conclusions in quantum mechanics and ignore possibilities in observable POVMs?

 

[PeterDonis]

doesn't there develop a contradiction between understanding time as relativistic with covarient properties and Schrödinger's equation using Newtonian absolute time just for the convenience of studying quantum mechanisms?

No, because if you're using Schrodinger's Equation, as I said, you're using non-relativistic mechanics, so you aren't "understanding time as relativistic". If you want to do relativistic quantum mechanics, you don't use Schrodinger's Equation; you use quantum field theory.

 

[MacCrimmon]

I liked your reply. Is there a reciprocal theory to understand POVM using superposition? ( If I am correct in thinking what Schrödinger's equation is referring to?)

 

[PeterDonis]

Is there a reciprocal theory to understand POVM using superposition? ( If I am correct in thinking what Schrödinger's equation is referring to?)

I don't know what this means.

 

[MacCrimmon]

Well... Could it be (and I am asking not speculating) that entangled particles seem to be influencing each other nonlocally because they are still sharing an absolute time reference. This would in no way be a contradiction or violation of GR or SR. And may not be limited to quantum particles.?

 

[PeterDonis]

Could it be (and I am asking not speculating) that entangled particles seem to be influencing each other nonlocally because they are still sharing an absolute time reference.

I'm not aware of any proposed model that claims this.

 

[MacCrimmon]

Cool

 

[Sunil]

If you want to do relativistic quantum mechanics, you don't use Schrodinger's Equation; you use quantum field theory.

The Schrödinger equation works fine in the relativistic field theory too. There are, of course, the field-theoretic problems with handling an infinite number of degrees of freedom, but this is not different from non-relativistic field theory. Despite being relativistic, the momentum of, say, a scalar field follows the same formula $\pi(x) = \dot{\phi}(x)$ and the energy is quadratic in this momentum. The spatial partial derivatives are functions on the configuration space, which is the space of functions $\phi(x)$ (but not of t).

One can get rid of these problems with an infinite number of degrees of freedom using a regularization with a finite lattice on a big cube with periodic boundary conditions, then one can apply standard the Schrödinger picture with the Schrödinger equation for $\vec{q} = (\phi(n_i,n_j,n_k)),\vec{p} = (\pi(n_i,n_j,n_k)), H=\frac12 |\vec{p}|^2 + V(\vec{q})$ where the potential V contains also dependencies on the finite differences approximating the spatial partial derivatives.

Is there a reciprocal theory to understand POVM using superposition?

The best way to understand POVMs is to see how they describe approximate measurements of position and momentum together. To measure $\hat{p}$ and $\hat{q}$ at the same time approximately, prepare another test particle in the ground state of some harmonic oscillator and measure $\hat{p}+\hat{p}_{test}$ and $\hat{q}-\hat{q}_{test}$. They commute, so that you can measure them at the same time, in the same measurement. But the result is some two-particle state, which does not define any pure one-particle wave function. So you cannot use it for preparation of some one-particle state which would give the same measurement result again. (The mathematics of this representation on the p-q plane is very beautiful, the wave functions are not only continuous but, modulo a simple weight factor $e^{-\frac12 z\bar{z}}$, holomorph (complex analytic) in $z=p+iq$. The analog of the basis are known as coherent states.)

The situation with time is similar. You have no operator for time measurement, the POVM does not give you as the result a state where the same measurement immediately after this would give the same result. There are no clocks giving accurate time, all clocks with some nonzero probability go sometimes even backward in Schrödinger time.

Well... Could it be (and I am asking not speculating) that entangled particles seem to be influencing each other nonlocally because they are still sharing an absolute time reference. This would in no way be a contradiction or violation of GR or SR. And may not be limited to quantum particles.?

This is essentially how realistic interpretations explain the violations of the Bell inequalities. They use a hidden preferred frame. The key is that from the Schrödinger equation follows a continuity equation for the probability in configuration space $\rho(q)=|\psi{q}|^2$:
$$ \partial_t \rho + \nabla (\rho \vec{v}(q)) = 0. $$
with the Bohmian velocity v. That means, one can define a Bohmian trajectory by $\dot{q}(t) = \vec{v}(q)$. The velocity $\vec{v}(\vec{q})$ depends on the whole configuration $\vec{q}$ at that moment of absolute time. So, absolute time is indeed essential to explanation of the non-local quantum effects.

This is obviously not in contradiction with SR, but simply the Lorentz ether interpretation of SR which has a preferred frame. It is in some weak conflict with GR, given that GR has solutions which don't allow to define a global time-like time coordinate, like wormholes or Goedel's universe with causal loops. So, this requires a minor modification of GR which excludes such things. Such a generalization has been proposed in Schmelzer, I. (2012). A Generalization of the Lorentz Ether to Gravity with General-Relativistic Limit. Advances in Applied Clifford Algebras 22(1), 203-242, resp. arxiv:gr-qc/0205035. It is, roughly, a Lorentz ether interpretation of the field-theoretic version of GR on a flat background $\mathbb{R}^4$ in harmonic coordinates.