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What is the shape of spacetime?
I want to read this question in a different way: what is a faithful characterization of space?
Let's start by saying that right or wrong are too simplistic reductions to describe nature. Reality is an hallucination that we build to make sense of what is around us, nothing is true or false, rather we have shades of correctness, or credence in Bayesian terms. My credence lies in 3 facts: causality, refusing infinities and the strong effects that black holes have on us.
Without taking out my favourite book https://de.wik https://de.wikipedia.org/wiki/G%C3%B6del on the topic we can simplify this by saying: all models (statements) are wrong, some are useful. In particular those who can help us predict the future are most useful.
Said that, to answer the original question we need to find a topological https://en.wikipedia.org/wiki/Topological_space characterization of space that allow us to describe the hallucination we perceive as reality and predict its evolution. Such characterization is formed by a set of elements, described in terms of group theory, and a topology h https://en.wikipedia.org/wiki/Topological_space#Definition_via_open_sets possibly of the metric kind so we can have a concept of causality.
The first wrong model we have is the Newtonian model. The elements are dimensionless points, the fundamental group is `R4`, or the product of 4 infinite lines, with metric signature `(4,0)`. This means we have 3 infinite dimensions of space and one infinite dimension of time. But then we notice that this would violate causality, as it would allow information to travel infinitely fast, which Newton called action at a distance 1 h https://en.wikipedia.org/wiki/Action_at_a_distance#Newtonian_gravity and so we have to move on.
The second wrong model we consider is the Einstein model, again with dimensionless points and a similar fundamental group `(R3,R)`, or the semidirect product https://en.wikipedia.org/wiki/Semidirect_product between 3 infinite lines and a infinite line, this time with metric signature of `(3,-1)`. This still breaks causality at large scales (general relativity), but we can save it for small scales (special relativity). The problem is that then we get very strange results: infinite densities in black holes, local violations of causality around black holes, and we still have the pesky problem of action at a distance.
The third wrong model we have is the semiclassical quantum model. Here the elements are again points, the fundamental group is again `R4` with signature `(4,0)` locally, like in Newtonian mechanics, but at large scales the signature becomes `(3,0)`. It means that time is separable from the equations and hence disappear from our equations, and it's called the problem of time https://en.wikipedia.org/wiki/Problem_of_time#The_frozen_formalism_problem Causality is recovered via loss of locality, also called entanglement, but when we try to go at human scales very strange things happen, like time freezes and we get infinite energies https://en.wikipedia.org/wiki/Cosmological_constant_problem.
But we notice some very cool things: first is the kaluza miracle h https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory#Group_theory_interpretation a real marvel of mathematics, which tells us that we can use extra dimensions to model physics and hence abandon the concept of dimensionless points, we also notice the hawking radiation h https://en.wikipedia.org/wiki/Hawking_radiation#Large_extra_dimensions which tells us about important properties of the topological space around black holes, and reconnect mechanics with thermodinamics https://en.wikipedia.org/wiki/Black_hole_thermodynamics#Statement_of_the_laws. Also we observe the AdS/CFT correspondence h https://en.wikipedia.org/wiki/AdS/CFT_correspondence#Geometry_of_anti-de_Sitter_space which allow us to scale quantum physics to macroscopic scales.
The result is the holographic principle https://en.wikipedia.org/wiki/Holographic_principle: locally, at low energies, space is `(R2,Sn,R)` with metric `(4+n,0)`, or a cylinder, which means we have 2 large dimensions for space, many small dimensions for fundamental forces, and no locality. Time becomes an emergent property, like gravity or thermodynamics, and not a fundamental trait of nature, like angular motion or field theory.
At large scale the situation becomes even stranger, because the metric becomes `(2,-2)`, and the large spatial dimensions gets compactified through a mechanism called Alexandroff extension h https://en.wikipedia.org/wiki/Alexandroff_extension#Example:_inverse_stereographic_projection and we end up in Anti de Sitter https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#Artistic_realizations hyperbolic space.
This means that local properties are described as angular motion along a small dimension of a small string: if you rotate clockwise your charge is positive, counterclockwise for negative charge. The speed of rotation is the intensity of the charge. Same for spin, color charge, and weak charge. These strings exist on a plane and as humans we perceive a third spatial dimension which is not really there, but is how our brain perceive the pauli's exclusion principle: like electron do not sit in increasingly larger orbits around the atoms but rather simply try to avoid being in the same space at the same time, we perceive energy levels as the spatial dimension perpendicular to the plane of gravity.
We then look at the stars and we see infinity, but is actually a finite volume. It's like we are sitting at the center of a black hole https://en.wikipedia.org/wiki/Black_hole_cosmology#Evidence: the universe is not expanding but the measure of the distance between us and the cosmological horizon grows by the minute. It behaves like the event horizon of a black hole, the universe is stationary but what is moving is the concept of distance itself, what yesterday was 1 meter tomorrow will be 2.
This is the most fucked up model, but also the best model we currently have. Do you understand now why I call reality an hallucination?
**Note:** I hope my physicist friends will forgive the extreme simplifications and romanticizations I used for the sake of entertaining the reader, very much like as a mathematician I forgive their liberal use of mathematics lol
#science
source
I want to read this question in a different way: what is a faithful characterization of space?
Let's start by saying that right or wrong are too simplistic reductions to describe nature. Reality is an hallucination that we build to make sense of what is around us, nothing is true or false, rather we have shades of correctness, or credence in Bayesian terms. My credence lies in 3 facts: causality, refusing infinities and the strong effects that black holes have on us.
Without taking out my favourite book https://de.wik https://de.wikipedia.org/wiki/G%C3%B6del on the topic we can simplify this by saying: all models (statements) are wrong, some are useful. In particular those who can help us predict the future are most useful.
Said that, to answer the original question we need to find a topological https://en.wikipedia.org/wiki/Topological_space characterization of space that allow us to describe the hallucination we perceive as reality and predict its evolution. Such characterization is formed by a set of elements, described in terms of group theory, and a topology h https://en.wikipedia.org/wiki/Topological_space#Definition_via_open_sets possibly of the metric kind so we can have a concept of causality.
The first wrong model we have is the Newtonian model. The elements are dimensionless points, the fundamental group is `R4`, or the product of 4 infinite lines, with metric signature `(4,0)`. This means we have 3 infinite dimensions of space and one infinite dimension of time. But then we notice that this would violate causality, as it would allow information to travel infinitely fast, which Newton called action at a distance 1 h https://en.wikipedia.org/wiki/Action_at_a_distance#Newtonian_gravity and so we have to move on.
The second wrong model we consider is the Einstein model, again with dimensionless points and a similar fundamental group `(R3,R)`, or the semidirect product https://en.wikipedia.org/wiki/Semidirect_product between 3 infinite lines and a infinite line, this time with metric signature of `(3,-1)`. This still breaks causality at large scales (general relativity), but we can save it for small scales (special relativity). The problem is that then we get very strange results: infinite densities in black holes, local violations of causality around black holes, and we still have the pesky problem of action at a distance.
The third wrong model we have is the semiclassical quantum model. Here the elements are again points, the fundamental group is again `R4` with signature `(4,0)` locally, like in Newtonian mechanics, but at large scales the signature becomes `(3,0)`. It means that time is separable from the equations and hence disappear from our equations, and it's called the problem of time https://en.wikipedia.org/wiki/Problem_of_time#The_frozen_formalism_problem Causality is recovered via loss of locality, also called entanglement, but when we try to go at human scales very strange things happen, like time freezes and we get infinite energies https://en.wikipedia.org/wiki/Cosmological_constant_problem.
But we notice some very cool things: first is the kaluza miracle h https://en.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory#Group_theory_interpretation a real marvel of mathematics, which tells us that we can use extra dimensions to model physics and hence abandon the concept of dimensionless points, we also notice the hawking radiation h https://en.wikipedia.org/wiki/Hawking_radiation#Large_extra_dimensions which tells us about important properties of the topological space around black holes, and reconnect mechanics with thermodinamics https://en.wikipedia.org/wiki/Black_hole_thermodynamics#Statement_of_the_laws. Also we observe the AdS/CFT correspondence h https://en.wikipedia.org/wiki/AdS/CFT_correspondence#Geometry_of_anti-de_Sitter_space which allow us to scale quantum physics to macroscopic scales.
The result is the holographic principle https://en.wikipedia.org/wiki/Holographic_principle: locally, at low energies, space is `(R2,Sn,R)` with metric `(4+n,0)`, or a cylinder, which means we have 2 large dimensions for space, many small dimensions for fundamental forces, and no locality. Time becomes an emergent property, like gravity or thermodynamics, and not a fundamental trait of nature, like angular motion or field theory.
At large scale the situation becomes even stranger, because the metric becomes `(2,-2)`, and the large spatial dimensions gets compactified through a mechanism called Alexandroff extension h https://en.wikipedia.org/wiki/Alexandroff_extension#Example:_inverse_stereographic_projection and we end up in Anti de Sitter https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model#Artistic_realizations hyperbolic space.
This means that local properties are described as angular motion along a small dimension of a small string: if you rotate clockwise your charge is positive, counterclockwise for negative charge. The speed of rotation is the intensity of the charge. Same for spin, color charge, and weak charge. These strings exist on a plane and as humans we perceive a third spatial dimension which is not really there, but is how our brain perceive the pauli's exclusion principle: like electron do not sit in increasingly larger orbits around the atoms but rather simply try to avoid being in the same space at the same time, we perceive energy levels as the spatial dimension perpendicular to the plane of gravity.
We then look at the stars and we see infinity, but is actually a finite volume. It's like we are sitting at the center of a black hole https://en.wikipedia.org/wiki/Black_hole_cosmology#Evidence: the universe is not expanding but the measure of the distance between us and the cosmological horizon grows by the minute. It behaves like the event horizon of a black hole, the universe is stationary but what is moving is the concept of distance itself, what yesterday was 1 meter tomorrow will be 2.
This is the most fucked up model, but also the best model we currently have. Do you understand now why I call reality an hallucination?
**Note:** I hope my physicist friends will forgive the extreme simplifications and romanticizations I used for the sake of entertaining the reader, very much like as a mathematician I forgive their liberal use of mathematics lol
#science
source
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