How to derive general relativity from Wolfram Physics with Jonathan Gorard
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- Опубліковано 20 вер 2023
- Here’s a masterclass from Jonathan Gorard.
One of the most compelling results to come out of the Wolfram Physics is Jonathan’s derivation of the Einstein equations from the hypergraph.
Whenever I hear anyone criticize the Wolfram model for bearing no relation to reality, I tell them this: Jonathan Gorard has proved that general relativity can be derived from the hypergraph.
In this excerpt from our conversation, Jonathan describes how making just three reasonable assumptions - causal invariance, asymptotic dimension preservation and weak ergodicity - allowed him to derive the vacuum Einstein equations from the Wolfram model.
In other words, the structure of space-time in the absence of matter more or less falls out of the hypergraph.
And making one further assumption - that particles can be treated as localized topological obstructions - allowed Jonathan to derive the non-vacuum Einstein equations from the Wolfram model.
In other words, the structure of space-time in the presence of matter, too, falls out of the hypergraph.
It’s difficult to overstate the importance of this result.
At the very least, we can say that the Wolfram model is consistent with general relativity.
To state it more strongly: we no longer need to take general relativity as a given; instead, we can derive it from Wolfram Physics.
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Jonathan’s seminal paper on how to derive general relativity
• Some Relativistic and Gravitational Properties of the Wolfram Model arxiv.org/abs/2004.14810 also published in Complex Systems www.complex-systems.com/abstr...
Jonathan Gorard
• Jonathan Gorard at The Wolfram Physics Project www.wolframphysics.org/people...
• Jonathan Gorard at Cardiff University www.cardiff.ac.uk/people/view...
• Jonathan Gorard on Twitter / getjonwithit
• The Centre for Applied Compositionality www.appliedcompositionality.com/
• The Wolfram Physics Project www.wolframphysics.org/
People mentioned by Jonathan
• Alfred Gray en.wikipedia.org/wiki/Alfred_...)
Research mentioned by Jonathan
• The volume of a small geodesic ball of a Riemannian manifold by Alfred Gray projecteuclid.org/journals/mi...
• Tubes by Alfred Gray archive.org/details/springer_...
Concepts mentioned by Jonathan
• Hausdorff dimension en.wikipedia.org/wiki/Hausdor...
• Geodesic balls, tubes & cones www.wolframphysics.org/techni...
• Ricci scalar curvature en.wikipedia.org/wiki/Scalar_...
• Ricci curvature tensor en.wikipedia.org/wiki/Ricci_c...
• Einstein equations en.wikipedia.org/wiki/Einstei...
• Einstein-Hilbert action en.wikipedia.org/wiki/Einstei...
• Relativistic Lagrangian density en.wikipedia.org/wiki/Relativ...
• Causal graph writings.stephenwolfram.com/2...
• Tensor rank mathworld.wolfram.com/TensorR...
• Trace en.wikipedia.org/wiki/Trace_(...)
From A Project to find the Fundamental Theory of Physics by Stephen Wolfram:
• Dimension www.wolframphysics.org/techni...
• Curvature www.wolframphysics.org/techni...
Images
• Spinning and chargend black hole with accretion disk commons.wikimedia.org/wiki/Fi... by Simon Tyran, Vienna (Симон Тыран) commons.wikimedia.org/wiki/Us... licensed under CC BY-SA 4.0 creativecommons.org/licenses/...
• Альфред Грэй в Греции commons.wikimedia.org/wiki/Fi... by AlionaKo licensed under CC BY-SA 3.0 creativecommons.org/licenses/...
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The Last Theory lasttheory.com/ is hosted by Mark Jeffery markjeffery.com/ founder of the Open Web Mind www.openwebmind.com/
Prefer to listen to the audio? Search for The Last Theory in your podcast player, or listen at lasttheory.com/podcast/047-ho...
Kootenay Village Ventures Inc.
haha, great thumbnail. Dr. Gorard is one of those brilliant people that has a genuine humanity and respect for all kinds of people
Yes, I've met Jonathan in person, and he's a great guy for sure!
My mind is blown ! Jonathan is gifted and explain this subject clearly but the fact that this explanation can fit in a 13 minutes video shows how this is so straight forward and natural. I am hooked by this model because even if you let aside any wish to get QFT from this framework, you already have a serious candidate for a theory of quantum gravity ! And it seems to be easy and straight forward to build discrete version of a lot of the continuous mathematical tools used in GR so the theory is certainly predictive. Fascinating ! Thank you for sharing this
Yes, absolutely, I find this truly compelling. When I first heard of Jonathan's derivation of General Relativity from the Wolfram model, it was my first indication that, as well as being fascinating at first sight, Wolfram Physics stands a strong chance of being the way forward for a fundamental theory of physics. Thanks for watching!
oh my. that was a hard one. had to re-listen to some parts quite a few times, since Jonathan tends to speak relatively fast when getting passionate 🤗
P.S.: I think you should "heart" more comments. The algorithm really seems to like that 😉
After watching your most recent video (How to tell if Space is curved), this video now makes a little bit more sense to me because I was only going off of the preservation of Lorentz transformations due to graph isomorphism (Causal Invariance). Thank you!
That's good to hear. There's so much in these Jonathan Gorard conversations that I want to go back through all of them and try to explain everything he's saying step-by-step. There'll be more of these on Jonathan's derivation of General Relativity coming soon. Thanks, as ever, for watching!
The impact of Jonathan's last statement was beautiful to watch.
Yes, this part of the conversation got me really excited!
Of your videos so far, I think this one is my favorite! It really makes the case for “here is why this really does give GR, under some natural conditions”!
The part about “how do you get the stress energy tensor” seemed a bit quickly glossed-over maybe? (I’m also unclear on what constitutes a “local topological obstruction” in a hypergraph)
but, I imagine it might be difficult to go into “only a little bit more detail” on that without going into a lot of detail on it.
Very nice!
Edit: also, I am a bit unclear on what it means for the net number of causal edges through the surface, converging to zero, means.
Like, what is the index that the limit is over? And, I would think that “net number of edges” would be an integer quantity, so, it converging to zero would mean that it eventually becomes and stays zero? But the surrounding language seemed to suggest that something was merely converging to zero, not being eventually always zero? Maybe the idea is “(net number of edges)/(some increasing number)” , so maybe like, if the net number of edges becomes negligible compared to something else?
Or maybe I’m totally misunderstanding that part?
In any case, I’m sure the actual papers make it clear.
Thanks. Yes, I’ve been really looking forward to getting this one out, it’s my favourite part of my conversation with Jonathan, too.
And yes, “local topological obstruction” is a bit cryptic. The Wolfram model does have a precise definition of energy, but you’re right, it’d take a serious digression to go into it. For a future video!
By “converges to zero,” Jonathan means that _statistically_ the flux is always about zero. It’s just like the number of molecules of gas passing through a plane. Because the motion of molecules in a gas is random, the flux is sometimes positive, sometimes negative, but the distribution is centred around zero and can be explained purely by that randomness.
Thanks, as ever, for watching!
@@lasttheory Ah, thank you for that clarification!
Lovely!
A beautiful mind is a wondrous thing to behold. 😊
Yes, Jonathan's the real deal!
As I understand it, the curvature of space-time can be interpreted as a local increase in spatial dimension? For example from 3 to 3.05. A local change in quantum fields increases the dimension of space, and this effect spreads spherically, gradually decreasing, spreading over an increasingly larger volume of space, i.e. gravity.
It’s not quite the same as an increase in spatial dimension. There are two _separate_ terms: one for the spatial dimension; one for the curvature. I’ll be explaining this in more detail in a future video. I find General Relativity is always a little difficult to wrap my mind around, so I’ll try to keep it as easy-to-follow as I can! Thanks for watching and commenting; stay tuned for more!
Curvature couples to heat, while temperature is the average velocity of molecules.
@@lasttheory Intuitively, I understand space as a network of nodes connected by edges of different (lets say random) lengths. Let's say the edges always form triangular faces. Using distance based geometry and the Pythagorean theorem, we select some node as the origin of coordinates, then calculate the coordinates of the nearest nodes so that the triangles are not degenerate. If we get a degenerate triangle, we simply increase the dimension of the node coordinates. Thus, for example, most nodes will have 3 coordinates, and some will have 4 or more. And on average over a limited volume we obtain a dimension of space slightly larger than 3. I will even assume that these nodes having a coordinate dimension greater than 3 are the reason for the quantum entanglement of parts of space. It seems to me that this is similar to Erik Verlinde's emergent gravity theory where entropic gravity can emerge from quantum entanglement.
Interesting, thanks Yarov.
I’m reluctant to apply any arbitrary restrictions on the hyperedges, such as that they form triangles. Stephen Wolfram imposed a similar restriction early in his exploration of these ideas, but it proved unnecessary, and anything so arbitrary demands an explanation.
It’s interesting how often quantum entanglement comes up in these theories. Instead of thinking of quantum entanglement as something weird, it seems we’d do well to think of it as emerging naturally from the hypergraph.
Yep. Nods head
I'm sure that was a neat explanation for someone who has a graduate degree in relativity, but could get it one aimed at a high school student or an undergrad?
Yes, absolutely. This is a difficult topic, so I’m planning to do several videos on this channel to explain it step-by-step. It’s going to take a little while, but look out for them over the next few months! And thanks for the feedback.
I think this should be fine for a physics undergrad?
Like, we teach undergrads who might not even be in STEM at all, *some* concept of curvature in the like, third calculus class? Granted, that’s about the curvature of a 1D curve in 2D or 3D space,
but, still.
Actually, when we are talking about a function of 2 variables, when doing the second derivative test, that is kinda also about curvature? Though we don’t really frame it that way I guess.
Anyway: consider a sphere of radius R (this isn’t the same R as in the video, I just wanted to use ‘r’/‘R’ because of the word “radius”), and take a point on it (e.g., the North Pole of the sphere. Doesn’t matter where.)
If we consider all points on the sphere surface that are at most r away, meaning, distance traveled on the surface of the sphere, not tunneling through it)
we can look at how the area of that region depends on r and on R.
When we take the radius of the sphere to be very big, the curvature becomes approximately zero (approximating it locally as flat becomes a better and better approximation, as we take R bigger and bigger, if we hold r constant)
and in this case, the area ends up being about pi r^2 ,
but, when R isn’t too big, then the area is somewhat smaller.
We can ask the same kinds of questions on other shapes, like the surface of a doughnut.
Here, what we do is we consider when r is very small, and within that context, look at how the area of the region within distance r (traveling along the surface of the doughnut) of a starting position, depends on r, and also how that in turn depends on what starting position we chose.
If we look at the innermost edge of the doughnut, we will actually find that the area of these regions is *more* than pi r^2 . This is because it has negative curvature there.
Now, that’s just the part about the scalar curvature,
the curvature tensor is, yeah, more complicated.
Still. There’s no need to give up and say “this video is just too hard for me”. One could write down the first part that is talking about something non-Wolfram-physics-project-specific and which one finds confusing or unfamiliar, and look it up. Maybe try Wikipedia or simple.wikipedia ?
General Relatively can describe space-time curvature but offers no explanation. The hypergraph describes the operation of how space-time construcs itself and GR equations turn out to be a natural consequence of space itself.
Yes, exactly. I’ll try to illustrate precisely what you’re saying @drdca8263 in a future video. Pictures can really help people understand!
Yes, that’s a great overview, @l3lixx!
I understand Einstein's version of GR and how we can come to the EFEs. I cannot say the same about this version of derivation.
Yes, thanks Cesar. Jonathan's derivation isn't easy to grasp! I need to dig into it further myself, but understanding the evolution of the hypergraph and the assumptions Jonathan makes helps. I'll be publishing more videos on this soon!
If I understand right, nothing here says that the graphs should have three space like dimensions and one time like dimension, right? Can you make graph rewrite rules that give the right number of dimensions?
Great question, thanks Tim. You’re right, there’s nothing here to say there are three dimensions. So that’s a fascinating open question: how do we get three dimensions? Should we simply reject any rules that _don’t_ give us three dimensions? Or is there something about applying a particular subset of rules that _does_ give us three dimensions? Or are we, as three-dimensional thinkers, somehow reducing the higher-dimensional hypergraph to three dimensions because perceiving it in all it’s complexity would just be too much for us?
@@Igdrazil Take a few pencils. You can put three all at right angles to each other, but not four. So we have three spatial dimensions. Vague deepities aren't going to explain that. The fact that you can do it with three pencils and not four is a fundamental property of the space we live in, not an obscure consequence of human psychology.
@@Igdrazil I hope Poincaré wasn't hopelessly vague. You are not inspiring me to find out.
He knows what he is talking about! But how on earth would anyone who isn't fully versed in GR have any clue? So what's the bloody point???
Proving that the Wolfram model can lead to General Relativity is just a start. Ultimately, it’ll need to make testable predictions that depart from the Einstein equations. If it does, that’ll be the point: the Wolfram model a better theory of _our_ universe than we’ve had before.
amazing stuff ..
Yes, this one really gets me going! Thanks Dan!
What the heck does a topological obstruction look like in a discrete (hyper-) graph? I mean there are just rules for updating discrete values at nodes. I can't imagine what he's talking about.
Think knots. Imagine a fishing net with knots in it, except that the updating rules cause the knots to propagate across the net.
@@lasttheory Is there a formal definition that does not use 3D metaphors?
Perhaps the updating rules give conflicting values for some nodes?
@@lasttheory or maybe instead of having a discrete set of values at each node, we can have real numbers that could go infinite? Or real component vectors that can go to 0? I'm just not sure what knots are in a discrete graph theory.
@@iuvalclejan Good question. Wolfram Physics doesn't yet have a well-defined idea of exactly what a particle would look like. Jonathan Gorard _does_ have an idea for toy particles, i.e. a formal definition of topological obstructions that can be mathematically proved to be persistent. But he doesn't suggest that these toy particles actually maps on to any real particles in our universe: he presents them instead merely as an illustration of how particles might work in the model.
The persistent topological obstructions Jonathan talks about are non-planar tangles in an otherwise planar hypergraph, specifically K₅ and K₃,₃. For Jonathan's explanation of this, take a look at our 7-minute conversation _A toy model of particles_ ua-cam.com/video/RcZbK9BmXa0/v-deo.html
Thanks for digging deeper, Iuval.
The notion that particles are regions of locally higher dimensionality makes a heck of a lot of sense. It also accords with string theory.
Yes, absolutely. I'm beginning to rethink my ideas of dimensionality. When _we_ large-scale creatures think of dimensions, we think of the flat(ish), three-dimensional space we live in. But really dimensionality is just a measure of connectedness of nodes by edges. You're right, it's not suprising that particles, if they're persistent tangles of nodes and edges in the hypergraph, would have higher connectedness.
Sounds cool, except that particles don't exist in reality. They are a failed mental model from 1905.
@@lasttheory I am an experimental physicist. Where do I see nodes and edges in my experiments???? ;-)
@@schmetterling4477 I agree, particles are a model, but I wouldn't say it's a "failed" model, any more than waves are a "failed" model. They're both simplifications of an underlying reality. After all, those tracks in cloud chambers, they're _real,_ right?
@@schmetterling4477 Well, nodes and edges are very _(very)_ small, so they're going to be hard to see. But if I know experimental physicists at all, I know that a phenomena's being hard to see won't stop you! Take a look at my video _Where's the evidence for Wolfram Physics?_ with Jonathan Gorard ua-cam.com/video/XLtxXkugd5w/v-deo.html for ideas about where to start looking.
Causal graphs appear to be so underconstrained, that it's hard to believe 3 simple assumptions can lead to Einstein equations. It's clear they *should* lead, when properly constrained, though.
"Dimension of spacetime" correspond to temperature 🌡️
Yes, it’s a surprising result, isn’t it? But that, to me, is what makes it so compelling. I have yet to dig into the details of the three assumptions or how they lead to the Einstein equations, but I don’t doubt that Jonathan has this right! Thanks for the comment!
@@lasttheory Sure, it is. It is probably best thought in thermodynamic language. Curvature couples to temperature 🌡️ (your so called 'dimension of spacetime). It makes a volume of a geodesic ball 🏀 deviate .
Would A. Einstein understand what he's saying?
I think Einstein would love this! Obviously, he was not exposed to computation from an early age, so he might have found that a bit difficult to take, but I think he’d be fully accepting of the idea that space might be quantized, and that the discrete structure of the hypergraph might underlie his continuous equations of relativity. Just a guess! Thanks for the question, Michael.
I think I understood less than 5% of this, but it was still awesome. I can't wait until AI takes our jobs so I can spend 10 hours a day learning math.
Yes, I'm still around the 5% level, too, but I'm working on it! I need to spend more time studying Jonathan's papers.
Loved it however too much mathematical jargon. Please explain some of your comments in the video as not only are you using unfamiliar terms, they aren't explained fully. Imagine an Arvin Ash video where he uses nothing but high graduate level theoretical physics terms and doesn't explain anything. Loved the video! Thank you!
Yes, absolutely. My plan is to do several videos explaining this step-by-step with illustrations. Thanks for the feedback and thanks for watching!
Please show some equations. Otherwise this is pure handwaving.
It's not just handwaving, I assure you! I try to avoid too many equations in my video, because many people are able to understand things better visually. But take a look at my series of videos on dimensionality ua-cam.com/play/PLVwcxwu8hWKlSYJ6iwzquLm5rOrykyg8c.html and my videos on the curvature of space ua-cam.com/video/r8vOj4QKUJE/v-deo.html and ua-cam.com/video/Bbuvfh6qTsU/v-deo.html for a more mathematical approach.
And if you _really_ want equations, check out Jonathan's paper _Some Relativistic and Gravitational Properties of the Wolfram Model_ arxiv.org/abs/2004.14810 It's not an easy read but there _are_ 147 equations in there!
Then do some more formal science, please
@@lasttheory How about a paper? I see a lot of handwaving here but no actual connection to MEASURED physics. A hypothesis has to be testable. What's the test for yours?
@@schmetterling4477 Here's Jonathan's paper on this derivation: _Some Relativistic and Gravitational Properties of the Wolfram Model_ arxiv.org/abs/2004.14810
This sounds really great... the only problem with it is that most of nature is not causal. ;-)
As if "Wolfram Physics" was even a thing..
It’s a thing all right! Whether it’s an accurate model of our universe remains to be seen, but it’s going to be fun finding out!
@@lasttheorybut Kopenhagen-Quantum-Physics is *a thing* 😆 /s
@@lasttheorythere might be some sweet sweet engagement opprotunity here 😉
What was said here seems to deserve a closer (slower) look maybe with some visuals and more time. @lasttheory how do you feel about learning 𝕄anim? 🤗
Thanks, as ever, for the comments… and the heart tip!
Yes, this is a tough one, and yes, I hope to break it down step-by-step in future videos.
I think I’ll be sticking to my own SVG animations for now, though! I’m always a bit reluctant to learn any new frameworks: HTML, CSS, SVG and Javascript have served me so well for so long!