2.6.0 Here I'll briefly review some other aspects of physics that seem possibly important in supporting life and complexity, but which aren't quite as central to the main story I'm telling. This subsection is a bit of a grab bag.
2.6.1 Time, Space, Locality
2.6.1.1 Some rather obvious features of our world: we live in a universe with time, space, and local interactions, i.e. when evolving in time, the state of a given region is determined by some local neighbourhood of the region.
2.6.1.2 These are features we share with the Game of Life and other cellular automata, so they don't explain why we don't live there, although they do differentiate us from other mathematical structures.
2.6.1.3 Time seems necessary for something like life to exist, period. After all, what is life if it doesn't try to change or optimize its surroundings, which definitionally implies some sort of ability to affect other things, i.e. they are a function of the life-form's decisions, i.e. they are in its future. Whereas things that a life-form perceives are in its past.
2.6.1.3.1 There are subtleties here related to the low-entropy initial conditions and the retroactive nature of quantum branching. More on this later.
2.6.1.4 As for space, we're very used to it because we live here, but it's worth appreciating how much complexity and ability to create useful structures comes "for free" from living in 3D space. Padlocks and proteins are good examples (but really, you can probably look at anything around you and see this).
2.6.1.4.1 Sokoban puzzles are a fun way to see how even the most primitive notions of space (pushing blocks on a 2D grid) can give rise to extremely intricate and complex interactions. The Game of Life is another example, of course.
Figure: Locks and proteins are examples of the complex structures that can be created in 3D space. From here and here.
2.6.1.5 Locality means that the time evolution of some bounded region is only affected by some bounded neighbouring region. This is important because it would otherwise be impossible for finite beings to predict their surroundings without knowing the state of the entire universe, since any part of the universe could interact with any other on the next time step.
2.6.1.5.1 Locality is implemented in our universe by the speed of light, an absolute limit on the rate at which information can propagate. Our universe also has apparent Galilean invariance, i.e. the laws of physics look the same to observers moving at different velocities. This generates an apparent contradiction, since this seemingly would make it possible to increase one's velocity past the speed limit. This is resolved by special relativity, which states that the appearance of space and time transforms for observers depending on their velocity (thus the laws are really invariant under these Lorentz transforms, not Galilean transforms).
2.6.1.6 That said, locality on its own is not enough to imply that local conditions can be predicted from a finite amount of information. For it could be the case that a local region contains an infinite density of information.
2.6.1.6.1 You might imagine a version of physics with an infinite hierarchy of ever-smaller entities. Conformal field theory is somewhat like this.
2.6.1.6.2 More mundanely, even ordinary classical physics can have this property! For if the position of a particle is a precise real number, it encodes an infinite amount of information.
2.6.1.6.3 In reality, particles are quantum mechanical, so they're spread out: their "resolution" depends on their energy. However, nothing in quantum mechanics prevents an arbitrary amount of energy in a small space, so information density can still be arbitrarily high.
2.6.1.7 What would prevent this would be a regional entropy bound of some kind, meaning a bound on the information encoded in a given region in terms of the region's size. For instance, the Game of Life has such a bound: A region with N tiles has 2^N possible states, so its entropy is bounded by its area.
2.6.1.8 Interestingly enough, despite the examples in 2.6.1.6, our universe has a regional entropy bound! Combining the theories of quantum mechanics and general relativity leads to such a bound, although either on its own does not have one. Let's turn to this next.
2.6.2 Entropy Bounds
2.6.2.1 The Holographic Entropy Bound
2.6.2.1.1 The ideas leading to the holographic entropy bound were originally developed by Jacob Bekenstein. Surprisingly enough, the holographic bound controls the entropy of a region (with certain caveats) in terms of the region's surface area (thus the name "holographic").
2.6.2.1.2 This line of thought has its origins in attempts to quantify the entropy of a black hole. Certain paradoxical results could be obtained by considering experiments in which objects were dropped into black holes. Since black holes are characterizable in terms of a handful of parameters, dropping high-entropy matter into them can seemingly lead to a loss of information (entropy) for external observers, in contradiction to the laws of thermodynamics.
2.6.2.1.3 To resolve this, Bekenstein proposed that black holes carry an intrinsic entropy. Based on analyzing thought experiments where matter with a given entropy is dropped into a black hole in a maximally efficient manner -- i.e. extracting as much "work" as possible so that the black hole grows as little as possible -- it turns out that the second law is preserved overall if we associate an entropy with black holes proportional to their area.
2.6.2.1.4 Bekenstein went on to consider another thought experiment where a system fitting inside a sphere of radius R, with energy E, was lowered into a black hole. Again by analyzing the process of slowly lowering the system, the black hole will gain entropy proportional to E*R. Thus the second law requires the system's initial entropy (which is lost) to be less than this. This is known as the Bekenstein entropy bound.
2.6.2.1.4.1 This can then be turned into a bound in terms of area by noting that the energy of a system fitting inside a sphere of radius R must be less than (a constant multiplied by) R in order for a black hole to not have already formed. This form of the bound was proposed by Leonard Susskind. This is the holographic bound (also known as the spherical entropy bound).
2.6.2.1.5 On an intuitive level, you could say the idea is that entropy is bounded by surface area because information density requires energy density, and energy density in a sphere of given surface area can only be so high before a black hole is formed[1].
2.6.2.1.6 However, the holographic bound does not apply to arbitrary regions of spacetime.
2.6.2.1.6.1 An important counterexample: it seems that the universe is spatially homogeneous and infinite in extent. Consider a sequence of spheres centered at the Earth and growing ever larger. At large enough scales, the entropy density in the interior should be approximately uniform, so the overall entropy must grow with the volume of the sphere.
2.6.2.1.6.2 Raphael Bousso proposed a modified bound which fixed this issue, and can be applied to arbitrary surfaces in spacetime.
2.6.2.2 The Covariant Bound
2.6.2.2.1 Bousso's bound is formulated in terms of light-sheets, surfaces traced out by light orthogonal to a spacelike (n-1) surface S, n being the dimension of space.
2.6.2.2.2 For instance, the causal diamonds from the last section are composed of two lightsheets meeting at a surface S. Generally in our spacetime an example to keep in mind is light emitted from the surface of a sphere, moving either inwards or outwards and either towards the past or the future.
2.6.2.2.3 For a surface S, there will be exactly 4 such lightsheets: one oriented towards the interior in the past, one towards the interior in the future, one towards the exterior in the past, and one towards the exterior in the future.
Figure: Four null hypersurfaces orthogonal to a spherical surface. Note we use the term lightsheet more broadly than Bousso (see footnote 2). From here, page 19.
2.6.2.2.4 A lightsheet in spacetime will trace over matter and radiation. We want to bound the entropy of this matter in terms of the area of the surface S.
2.6.2.2.5 Bousso's bound applies to lightsheets with non-positive expansion. What does this mean? The idea is to consider how the volume of slices of the lightsheet changes as we move along it starting from the surface S. For instance, if we consider a causal diamond, the interior past and future lightsheets have negative expansion, because the area of cross sections decreases as we move towards the tips from the surface S.
2.6.2.2.5.1 We can pair up the lightsheets: continuing the interior future sheet past the surface S leads to the exterior past sheet, and vice versa. If one member of a pair initially has positive expansion, the other will have negative expansion, so at least 2 of the 4 lightsheets will be non-positively-expanding (possibly more if some sheets have zero expansion).
2.6.2.2.5.2 Once we have a lightsheet with non-positive expansion, we continue tracing it, locally stopping when part of it begins to have positive expansion. This is the portion of the lightsheet that the bound will apply to.[2]
2.6.2.2.6 Now, the statement of the bound is: for lightsheets of non-positive expansion, the entropy of matter and radiation on the sheet is bounded by (a constant times) the area of the surface S.
2.6.2.2.7 Consider the case of a sphere enclosing some matter on a relatively small scale (e.g. a planet, a solar system). In this case, the interior light sheets will be non-expanding, and they will trace over the degrees of freedom inside, thus recovering the holographic bound.
2.6.2.2.8 How does the bound evade the problem with the very large spheres in our cosmology? The idea is that since the universe is expanding, both the interior and exterior future lightsheets in fact have positive expansion in the future direction once the sphere is large enough. In such cases the bound instead applies to both past lightsheets, which will trace out a shell moving backwards into the past and terminating at the Big Bang. The total volume covered will be proportional to the surface area of the sphere, removing the problematic scaling with the sphere's volume.
2.6.2.2.9 So those are entropy bounds. Of course, since we don't have a full theory of quantum gravity, the exact cause in the underlying theory is not clear, but everything we do know about QG indicates that the bounds hold. I find this counterintuitive way our universe implements the principle of limited information per spatial region fascinating. It also has the interesting implication that everything around us, even seemingly continuous systems and space itself, ultimately consists of a finite number of degrees of freedom, quantum-mechanically speaking.[3]
2.6.3 Heat Death. Symmetries
2.6.3.1 Cosmology and heat death
2.6.3.1.1 While we're talking about cosmology, I'd also like to say a bit about the distant future and "heat death".
2.6.3.1.2 As you're likely aware, heat death refers to the approach of the universe to a state of maximum entropy. As we approach heat death, it becomes impossible for systems to maintain their structure by dumping entropy elsewhere. So it seems that all structure will be destroyed by this point.
2.6.3.1.3 However, the increase of entropy to a maximum applies most strongly to closed systems with a finite number of degrees of freedom. Our universe is plausibly infinite, so the approach to a single equilibrium state is not so clear. The universe is going to expand and gradually cool, and under some models of cosmology (asymptotically flat ones) its temperature will approach zero.
2.6.3.1.4 Dyson pointed out that indefinitely decreasing temperature could in theory allow a system to induce infinitely many irreversible transitions (e.g. erasing bits) with a finite amount of energy, by waiting longer and longer between each action so that less energy is needed (the energy needed to erase a bit decreases linearly with temperature). So systems that rely on bit erasure to maintain their structure could in theory last forever.
2.6.3.1.5 Even in such a scenario, for a truly infinite civilization one would like to avoid loops in state space. For this to occur, it is additionally necessary to expand outwards and acquire more degrees of freedom to compute with.
2.6.3.1.6 Doing this forever seems difficult, but not obviously impossible. So in such a universe, life continuing forever, in an ever colder and sparser universe, doesn't seem out of the question[4]. Meeting aliens who want to do the same thing could be a help or hindrance.
2.6.3.1.7 Our current best guess at cosmology has a positive cosmological constant and is thus not asymptotically flat, which would rule out both the indefinite decrease in temperature and the acquisition of an indefinite number of degrees of freedom. However, there are plausible explanations of cosmological data that do not rely on a cosmological constant, such as quintessence or the Timescape theory. So life persisting forever doesn't seem impossible.
2.6.3.2 Symmetries
2.6.3.2.1 One more interesting feature of our physics is the presence of many types of symmetry relating different states.
2.6.3.2.2 As mentioned previously, the laws of physics have many symmetries such as time, translation, rotation, Lorentz boost. Noether's theorem guarantees that each continuous symmetry has an associated conserved quantity. As we've seen, conserved quantities are important in making thermodynamics work, and therefore in maintaining pretty much all structure in the universe.
2.6.3.2.3 There's also a somewhat different class of symmetries known as gauge symmetries. These have to do with the description of the fundamental forces and gravity. These forces are described by fields filling space. Roughly speaking, the fields have the property that they can be transformed by an entire (group-valued) function across all of space. Think of choosing a gauge in electromagnetism if you're familiar. So this is a much broader class of symmetries than the ones mentioned above.
2.6.3.2.4 However, gauge symmetries are in a sense not "real" symmetries because, technically speaking, they don't transform state space. Rather, state space is only defined up to the action of gauge symmetries. Two states that are related by gauge symmetries are in fact the same state. So gauge symmetries could be thought of as a redundancy in our description of physics.
2.6.3.2.5 So what's up with that? Why do the laws of nature have such a symmetric character? Honestly, I'm not really sure. There are of course the conserved quantities which are important, but there doesn't seem to be an obvious reason that those couldn't arise in some other manner. And it doesn't explain the gauge symmetries. Nevertheless I find the abundance of symmetries quite striking so I thought I'd mention it.
2.6.3.2.5.1 Symmetries relate different perspectives on the world. So perhaps part of the explanation is that symmetric laws make it easier for different vantage points on the world to be related to each other.
Again speaking intuitively, you can see that entropy density should be proportional to energy by considering that energy in quantum mechanics roughly corresponds to the frequency of the waves (small complex spirals) defining particles. Only so many distinguishable bits of information can be fit into waves of low frequency per unit area, so higher energy is required to pack in more.
Of course, technically any quantum system has a continuum of possible states, as they are complex vector spaces. However, as observers we can only probe them by taking a set of measurements whose discrete outcomes have a high degree of correlation, and for finite-dimensional systems we can only take finitely many measurements with independent outcomes. So "morally speaking" finite-dimensional quantum systems are closer to finite systems than continuous classical systems.
Bousso and Susskind gave their proposal for a branching definition in an asymptotically flat universe for a related reason. They wanted hypothetical observers to be able to make infinitely many observations so as to obtain a limiting precise value for quantum probabilities.
2.6.0 Here I'll briefly review some other aspects of physics that seem possibly important in supporting life and complexity, but which aren't quite as central to the main story I'm telling. This subsection is a bit of a grab bag.
2.6.1 Time, Space, Locality
2.6.1.1 Some rather obvious features of our world: we live in a universe with time, space, and local interactions, i.e. when evolving in time, the state of a given region is determined by some local neighbourhood of the region.
2.6.1.2 These are features we share with the Game of Life and other cellular automata, so they don't explain why we don't live there, although they do differentiate us from other mathematical structures.
2.6.1.3 Time seems necessary for something like life to exist, period. After all, what is life if it doesn't try to change or optimize its surroundings, which definitionally implies some sort of ability to affect other things, i.e. they are a function of the life-form's decisions, i.e. they are in its future. Whereas things that a life-form perceives are in its past.
2.6.1.4 As for space, we're very used to it because we live here, but it's worth appreciating how much complexity and ability to create useful structures comes "for free" from living in 3D space. Padlocks and proteins are good examples (but really, you can probably look at anything around you and see this).
Figure: Locks and proteins are examples of the complex structures that can be created in 3D space. From here and here.
2.6.1.5 Locality means that the time evolution of some bounded region is only affected by some bounded neighbouring region. This is important because it would otherwise be impossible for finite beings to predict their surroundings without knowing the state of the entire universe, since any part of the universe could interact with any other on the next time step.
2.6.1.6 That said, locality on its own is not enough to imply that local conditions can be predicted from a finite amount of information. For it could be the case that a local region contains an infinite density of information.
2.6.1.7 What would prevent this would be a regional entropy bound of some kind, meaning a bound on the information encoded in a given region in terms of the region's size. For instance, the Game of Life has such a bound: A region with N tiles has 2^N possible states, so its entropy is bounded by its area.
2.6.1.8 Interestingly enough, despite the examples in 2.6.1.6, our universe has a regional entropy bound! Combining the theories of quantum mechanics and general relativity leads to such a bound, although either on its own does not have one. Let's turn to this next.
2.6.2 Entropy Bounds
2.6.2.1 The Holographic Entropy Bound
2.6.2.1.1 The ideas leading to the holographic entropy bound were originally developed by Jacob Bekenstein. Surprisingly enough, the holographic bound controls the entropy of a region (with certain caveats) in terms of the region's surface area (thus the name "holographic").
2.6.2.1.2 This line of thought has its origins in attempts to quantify the entropy of a black hole. Certain paradoxical results could be obtained by considering experiments in which objects were dropped into black holes. Since black holes are characterizable in terms of a handful of parameters, dropping high-entropy matter into them can seemingly lead to a loss of information (entropy) for external observers, in contradiction to the laws of thermodynamics.
2.6.2.1.3 To resolve this, Bekenstein proposed that black holes carry an intrinsic entropy. Based on analyzing thought experiments where matter with a given entropy is dropped into a black hole in a maximally efficient manner -- i.e. extracting as much "work" as possible so that the black hole grows as little as possible -- it turns out that the second law is preserved overall if we associate an entropy with black holes proportional to their area.
2.6.2.1.4 Bekenstein went on to consider another thought experiment where a system fitting inside a sphere of radius R, with energy E, was lowered into a black hole. Again by analyzing the process of slowly lowering the system, the black hole will gain entropy proportional to E*R. Thus the second law requires the system's initial entropy (which is lost) to be less than this. This is known as the Bekenstein entropy bound.
2.6.2.1.5 On an intuitive level, you could say the idea is that entropy is bounded by surface area because information density requires energy density, and energy density in a sphere of given surface area can only be so high before a black hole is formed[1].
2.6.2.1.6 However, the holographic bound does not apply to arbitrary regions of spacetime.
2.6.2.2 The Covariant Bound
2.6.2.2.1 Bousso's bound is formulated in terms of light-sheets, surfaces traced out by light orthogonal to a spacelike (n-1) surface S, n being the dimension of space.
2.6.2.2.2 For instance, the causal diamonds from the last section are composed of two lightsheets meeting at a surface S. Generally in our spacetime an example to keep in mind is light emitted from the surface of a sphere, moving either inwards or outwards and either towards the past or the future.
2.6.2.2.3 For a surface S, there will be exactly 4 such lightsheets: one oriented towards the interior in the past, one towards the interior in the future, one towards the exterior in the past, and one towards the exterior in the future.
Figure: Four null hypersurfaces orthogonal to a spherical surface. Note we use the term lightsheet more broadly than Bousso (see footnote 2). From here, page 19.
2.6.2.2.4 A lightsheet in spacetime will trace over matter and radiation. We want to bound the entropy of this matter in terms of the area of the surface S.
2.6.2.2.5 Bousso's bound applies to lightsheets with non-positive expansion. What does this mean? The idea is to consider how the volume of slices of the lightsheet changes as we move along it starting from the surface S. For instance, if we consider a causal diamond, the interior past and future lightsheets have negative expansion, because the area of cross sections decreases as we move towards the tips from the surface S.
2.6.2.2.6 Now, the statement of the bound is: for lightsheets of non-positive expansion, the entropy of matter and radiation on the sheet is bounded by (a constant times) the area of the surface S.
2.6.2.2.7 Consider the case of a sphere enclosing some matter on a relatively small scale (e.g. a planet, a solar system). In this case, the interior light sheets will be non-expanding, and they will trace over the degrees of freedom inside, thus recovering the holographic bound.
2.6.2.2.8 How does the bound evade the problem with the very large spheres in our cosmology? The idea is that since the universe is expanding, both the interior and exterior future lightsheets in fact have positive expansion in the future direction once the sphere is large enough. In such cases the bound instead applies to both past lightsheets, which will trace out a shell moving backwards into the past and terminating at the Big Bang. The total volume covered will be proportional to the surface area of the sphere, removing the problematic scaling with the sphere's volume.
2.6.2.2.9 So those are entropy bounds. Of course, since we don't have a full theory of quantum gravity, the exact cause in the underlying theory is not clear, but everything we do know about QG indicates that the bounds hold. I find this counterintuitive way our universe implements the principle of limited information per spatial region fascinating. It also has the interesting implication that everything around us, even seemingly continuous systems and space itself, ultimately consists of a finite number of degrees of freedom, quantum-mechanically speaking.[3]
2.6.3 Heat Death. Symmetries
2.6.3.1 Cosmology and heat death
2.6.3.1.1 While we're talking about cosmology, I'd also like to say a bit about the distant future and "heat death".
2.6.3.1.2 As you're likely aware, heat death refers to the approach of the universe to a state of maximum entropy. As we approach heat death, it becomes impossible for systems to maintain their structure by dumping entropy elsewhere. So it seems that all structure will be destroyed by this point.
2.6.3.1.3 However, the increase of entropy to a maximum applies most strongly to closed systems with a finite number of degrees of freedom. Our universe is plausibly infinite, so the approach to a single equilibrium state is not so clear. The universe is going to expand and gradually cool, and under some models of cosmology (asymptotically flat ones) its temperature will approach zero.
2.6.3.1.4 Dyson pointed out that indefinitely decreasing temperature could in theory allow a system to induce infinitely many irreversible transitions (e.g. erasing bits) with a finite amount of energy, by waiting longer and longer between each action so that less energy is needed (the energy needed to erase a bit decreases linearly with temperature). So systems that rely on bit erasure to maintain their structure could in theory last forever.
2.6.3.1.5 Even in such a scenario, for a truly infinite civilization one would like to avoid loops in state space. For this to occur, it is additionally necessary to expand outwards and acquire more degrees of freedom to compute with.
2.6.3.1.6 Doing this forever seems difficult, but not obviously impossible. So in such a universe, life continuing forever, in an ever colder and sparser universe, doesn't seem out of the question[4]. Meeting aliens who want to do the same thing could be a help or hindrance.
2.6.3.1.7 Our current best guess at cosmology has a positive cosmological constant and is thus not asymptotically flat, which would rule out both the indefinite decrease in temperature and the acquisition of an indefinite number of degrees of freedom. However, there are plausible explanations of cosmological data that do not rely on a cosmological constant, such as quintessence or the Timescape theory. So life persisting forever doesn't seem impossible.
2.6.3.2 Symmetries
2.6.3.2.1 One more interesting feature of our physics is the presence of many types of symmetry relating different states.
2.6.3.2.2 As mentioned previously, the laws of physics have many symmetries such as time, translation, rotation, Lorentz boost. Noether's theorem guarantees that each continuous symmetry has an associated conserved quantity. As we've seen, conserved quantities are important in making thermodynamics work, and therefore in maintaining pretty much all structure in the universe.
2.6.3.2.3 There's also a somewhat different class of symmetries known as gauge symmetries. These have to do with the description of the fundamental forces and gravity. These forces are described by fields filling space. Roughly speaking, the fields have the property that they can be transformed by an entire (group-valued) function across all of space. Think of choosing a gauge in electromagnetism if you're familiar. So this is a much broader class of symmetries than the ones mentioned above.
2.6.3.2.4 However, gauge symmetries are in a sense not "real" symmetries because, technically speaking, they don't transform state space. Rather, state space is only defined up to the action of gauge symmetries. Two states that are related by gauge symmetries are in fact the same state. So gauge symmetries could be thought of as a redundancy in our description of physics.
2.6.3.2.5 So what's up with that? Why do the laws of nature have such a symmetric character? Honestly, I'm not really sure. There are of course the conserved quantities which are important, but there doesn't seem to be an obvious reason that those couldn't arise in some other manner. And it doesn't explain the gauge symmetries. Nevertheless I find the abundance of symmetries quite striking so I thought I'd mention it.
Again speaking intuitively, you can see that entropy density should be proportional to energy by considering that energy in quantum mechanics roughly corresponds to the frequency of the waves (small complex spirals) defining particles. Only so many distinguishable bits of information can be fit into waves of low frequency per unit area, so higher energy is required to pack in more.
Note that Bousso uses the term "lightsheet" only for this portion of what we are calling lightsheets.
Of course, technically any quantum system has a continuum of possible states, as they are complex vector spaces. However, as observers we can only probe them by taking a set of measurements whose discrete outcomes have a high degree of correlation, and for finite-dimensional systems we can only take finitely many measurements with independent outcomes. So "morally speaking" finite-dimensional quantum systems are closer to finite systems than continuous classical systems.
Bousso and Susskind gave their proposal for a branching definition in an asymptotically flat universe for a related reason. They wanted hypothetical observers to be able to make infinitely many observations so as to obtain a limiting precise value for quantum probabilities.