An impossible thought experiment

I recently watched a discussion between Roger Penrose and Jordan Peterson, which was really a question and answer session, with Peterson asking the questions and Penrose providing the answers. There was a third person involved as moderator, but I’ve forgotten his name and his interaction was minimal. It was mostly about consciousness, but also touched on quantum mechanics and Godel’s theorem.

 

I can’t remember the context, but (at point 1.06) Penrose trotted out the well-worn thought experiment of 2 people crossing a street in opposite directions, and somewhere, in some far-flung part of the cosmos, an armada of spaceships is departing for a journey to Earth. Now, according to Einstein’s theory of relativity, one of these ‘observers’ will 'say' the fleet left 100s of years in the past and the other will 'say', no, they're leaving 100s of years in the future.

I’ve always had a problem with this ‘scenario’, and I’ve discussed it previously. The thing is that neither of them can ‘observe’ anything at all, because the ‘event’ (space fleet departing) is outside the light cone of influence of Earth (in either the future or the past). So neither of them receive a signal telling them that this is what they ‘observe’. In other words, it’s something they’ve worked out with equations or a space-time diagram. Brian Greene illustrates it graphically in a YouTube video.

 

Of course, my interpretation is considered ‘naïve’ and completely wrong by Penrose and every other physicist I know of.

 

Now, some thought experiments, like the famous EPR experiment, in combination with Bell’s Theorem, can be done in the real world, and was done after Einstein’s death and effectively proved Einstein wrong (on that particular point). Another example is John Wheeler’s delayed decision thought experiment for the double-slit experiment, which was also physically done after Wheeler’s death.

 

But this thought experiment is impossible to do, even in principle. My interpretation is that you have a clear contradiction, and where you have a contradiction there is usually something wrong with one or more of your premises. My proposed resolution is that what they 'perceive' is not reality, because the event is outside the cones-of-influence (past and future) of the observers.

But let’s take the thought experiment to its logical conclusion. Let’s say the observers record their deduced ‘time of departure’ with respect to their frame of reference, and it can be looked up centuries later when the fleet actually arrives on Earth. Now, when the fleet arrives, its trajectory through spacetime is within Earth’s past light cone. The fleet has its own time record of their journey and we know how far they’ve travelled. In fact, this is no different to the return leg of the famous twin paradox thought experiment. Now observers can apply relativistic corrections to the fleet’s recorded elapsed time, and deduce a time of departure based on Earth’s frame of reference. This will give a ‘time’ which I expect will fall somewhere between the 2 times recorded by the original observers.

 

Of course, this is still an impossible thought experiment because there is no way the 2 pedestrians could know when the fleet was departing. But if a fleet of spaceships did arrive on Earth from somewhere ‘far, far away’, we could calculate exactly when it left (ref. Earth time) and there would be no contradiction.

 

 

Footnote: I know that this stems from Einstein’s discovery that simultaneity varies according to an observer’s frame of reference, and there is an excellent video that explains the maths behind it. But here’s the thing: if the observer is equi-distant from the 2 signals in the same frame of reference you’ll get ‘true’ simultaneity (watch the video). On the other hand, an observer moving with respect to the sources will not see simultaneity. A little known fact is that you have to allow for length contraction as well as time dilation to get the right answer. But here’s another thing: on a cosmic scale, 2 observers can see 2 events in opposite sequence even if they’re not moving relative to each other. BUT, if the events have a causal relationship, then all observers see the same sequence, irrespective of relative motion. (Refer Addendum 2)*

 

 

Addendum 1: I’ve given this more thought, by having imaginary dialogue with a physicist, who would tell me that my ideas are inconsistent with relativity. Naturally, I would disagree with them. I would say it’s consistent with relativity, because, for the thought experiment to actually work would require instantaneous communication, which is as contradictory with relativity theory as one can get. For the 2 hypothetical observers to ‘know’ when the far-flung event took place would require them to observe it in their ‘now’, which is impossible. So my response is strictly a philosophical one: you can’t apply relativistic theory to this situation because the 'observation' would appear to 'violate' a tenet of relativity theory. And that’s because the event is outside the observers’ light cone. Am I missing something here? 

 

*Addendum 2: I watched a video by Sabine Hossenfelder, who addresses the last sentence in the footnote of this post. She says, in fact, that 2 events that have a causal relationship in our frame of reference could appear ‘independently’ simultaneous to ‘different’ observers in another part of the Universe (watch the video). This is a variation on the thought experiment that I discuss. In practice, because there’s no possible causal relationship between the events and the far-off observers, they wouldn’t observe anything. And it doesn’t change the sequence of causal events.

 

But she’s arguing that there are a multitude of ‘nows’, in accordance with one of Einstein’s premises that all observers have the same validity. That may be correct, but why does it apply to events they can’t even observe? To be fair to Sabine, she does try to address this at the end of the video. I’ve long argued that different observers see a different ‘now’, even without relativity, but if they see a different sequence of events, at least one of them has to be wrong.

 

I want to emphasise that I don’t think Einstein’s theories of relativity are wrong, as some people do. My point is purely a philosophical one: if you have a multitude of perspectives with different versions of events, they can’t all be right. I’m simply arguing that there is an objective reality. A case in point is the twin paradox, where one twin’s clock does run slower, irrespective of what each twin ‘observes’. Mark John Fernee gives a synoptic exposition here. As he says: they each have their own ‘true time’, and one is always slower.

 

If you go far enough into the future, where the events in question fall within the observer’s past light cone, then a history can be observed. We do this with the Universe itself, right back to the CMBR, 380,000 years after the Big Bang, which is 13.8 billion years ago; both of which we claim to know with some confidence.

 

I still maintain my core point, stated explicitly in the title of this post, that, as a hypothetical, the thought experiment described is impossible to do, simply because the event can’t be observed.

 

Beliefs, prejudices and theories; where is truth?

 During COVID, New Scientist started doing a lot of online events, including courses and ‘talks’ by experts in various fields. I watched one of these talks last week by Claudia De Rahm titled What We Don’t Know About Gravity, which I thought was very good. It was informative and thought-provoking and therefore deserves special mention. Claudia is a young woman, Professor of Physics at Imperial College London, with a distinct Italian accent. She gestures a lot while she’s talking and exudes passion. Sometimes her face appeared childlike, especially at the end when she conveyed her appreciation to the presenter, Martin Davies. She’s won a number of awards and she’s done research in particle physics, gravity and cosmology.

 

One of the first things she told us is that Einstein’s GR (general theory of relativity) comes with its own ‘proof’ of its limitations. She didn’t use the word proof, but she demonstrated what she meant. If one tries to apply QM to Einstein’s mathematical theory you get probabilities of over 100%. I never knew this, but I found it a remarkable revelation. From what I could gather, it happens near the Planck scale where the curvature of spacetime becomes so large the physics breaks down. She pointed out that this doesn’t occur near the event horizon of a black hole, so for everything we can observe, GR is perfectly valid. But I was astounded to learn that GR predicts its own failure at certain scales of the Universe.

 

She also questioned whether GR breaks down at the other extreme of scale, given that there is disagreement on how fast the Universe is expanding to a significant degree (in her own words, ‘the chance of it being a fluke is 1 part in 14,000’). Of course, she also explained how 95% of the Universe is ‘missing’, meaning it can’t be accounted for. Personally, I think we’re ripe for another scientific revolution comparable to the one that occurred 100 years ago, which in turn was comparable to the one created by Copernicus, Galileo, Kepler and Newton.

 

This highlights a point I’ve raised before: the significance of scale in determining which ‘natural laws’ dominate, though they all seem to obey a Lagrangian (based on my limited knowledge of physics). Roger Penrose argues that scale is dependent on mass. If the Universe was all radiation then scale becomes irrelevant. This is essential for his CCC (Conformal Cyclic Cosmology) model of the Universe to work. Penrose also argues that there is no time without mass, because time is always zero for a photon. This creates a paradox, because the photon has an energy dependent on its frequency, which has no meaning without time. I’ve no doubt Penrose can resolve that, but I don’t know how. Perhaps gravity resolves that conundrum. But, as the Universe exists in its current epoch within our range of observations, scale plays a significant, even critical role in determining which mathematical formulations we use to model it.

 

Claudia tip-toed around the argument about whether gravity is a force or not, but gave me the impression she believes it isn’t. She did point out that a gravitational wave effectively creates a force and there are tidal forces, but this is not what people mean when they argue that there is a ‘force of gravity’ in the Newtonian sense. In answer to a question at the end, she pointed out that “gravity is related to the very structure of spacetime; you can never switch it off”.

 

On the subject of GR’s inherent limitations around a singularity inside a black hole, she seemed optimistic that new physics would overcome this eventually. Along with the questions around dark energy and dark matter, that comprise 70% and 25% of the Universe respectively, I think that only a revolution in physics and cosmology will rescue it. Towards the end of the talk, she put up a slide showing all the current theories in the running, without discussing any of them or mentioning any personal favourites she might have. She literally covered the screen with balloons of speculative ideas, demonstrating the burgeoning interest in this field.

 

And this segues into something else she said in answer to a question, where someone asked if all the current theories should be ‘thrown in the rubbish bin’ and replaced with something completely different. She pointed out that the current theories work extremely well, and whatever you replace them with has to, at the very least, account for what we already ‘know’, and you can’t just ‘throw them in the rubbish bin’. This touches on the subject of my last post where people sometimes argue that we really don’t ‘know’ anything and we only have 'beliefs'. In science, all theories have limitations. Truth is cumulative in science; just because we don’t know everything, it doesn’t mean that what we do know is wrong and should be thrown out. Personally, I don’t think there will ever be a TOE (theory of everything) simply because there’s never been one in the past, and people have always ‘believed’ that we know almost everything, which history has proved, repeatedly, is untrue.

 

And this brings me to the subject of pet theories or pet prejudices. If Claudia has her own pet theories she didn’t elaborate, yet I’m sure she has. People much smarter than me have their pet prejudices, some of which differ dramatically, so they can’t all be right, and that also applies to me. But, having said that, I like to think my prejudices are well informed and I acknowledge those who share them, and sometimes those who don’t.

 

I will quickly talk about one that is relevant and that is time. I contend that consciousness exists in a constant present, while everything we observe has already happened, which is why we ‘feel’ like we’re travelling through time. According to relativity theory, we are travelling through time just by standing still. But when we move, we start travelling through space and, as a consequence, we travel through time more slowly – that is, time slows down. In fact, if we could travel through space at the speed of light, we would stop travelling through time altogether. But here’s the thing: that’s only true in our specific frame of reference. There could be another frame of reference, like the horizon of the observable universe where space itself travels at the speed of light. I discussed this in another post.

 

This infers that everything travels through time and not just consciousness. However, while our consciousness remains in a constant present, our thoughts don’t. Our thoughts become memories as soon as we think them, otherwise we wouldn’t even know we think. Consciousness exists on the edge of time and so does the universe itself. I’ve no reason to believe that the edge of time we all experience isn’t concordant with the edge of time for the whole Cosmos. This is considered naive thinking, but it’s one of my pet prejudices.

We are not just numbers, but neither is the Universe

 A few years back I caught up with someone I went to school with, whom I hadn’t seen in decades, and, as it happened, had studied civil engineering like me. I told him I had a philosophy blog where I wrote about science and mathematics, among other things. He made the observation that mathematics and philosophy surely couldn’t be further apart. I pointed out that in Western culture they had a common origin, despite a detour into Islam, where mathematics gained a healthy and pivotal influence from India. 

I was reminded of this brief exchange when I watched this Numberphile video on the subject of numbers, where Prof Edward Frankel (UC Berkeley) briefly mentions the role of free will in our interaction with mathematics.

 

But the main contention of the video is that numbers do not necessarily have the status that we give them in considering reality. In fact, this is probably the most philosophical video I’ve seen on mathematics, even though Frankel is not specifically discussing the philosophy of mathematics.

 

He starts off by addressing the question whether our brain processes are all zeros and ones like a computer, and obviously thinks not. He continues that in another video, which I might return to later. The crux of this video is an in-depth demonstration of how a vector can be represented by a pair of numbers. He points out that the numbers are dependent on the co-ordinate system one uses, which is where ‘free will’ enters the discussion, because someone ‘chooses’ the co-ordinate system. He treats the vector as if it’s an entity unto itself, which he says ‘doesn’t care what co-ordinates you choose’. Brady, who is recording the video, takes him up on this point: that he’s effectively personifying the vector. Frankel acknowledges this, saying that it’s an ‘abstraction within an abstraction.’

 

Now, Einstein used vectors in his general theory of relativity, and one of the most important points was that the vectors are independent of the co-ordinate system. So we have this relationship between a mathematical abstraction and physical reality. People often talk about mistaking the ‘map for the terrain’ and Frankel uses a different metaphor where he says, ‘don’t confuse the menu for the meal’. I agree with all this to a point.

 

My own view is that there are 2 aspects of mathematics that are conflated. There is the language of mathematics, which includes the numbers and the operations we use, and which are ‘invented’ by humans. Then there are the relationships, which this language describes, but which are not prescribed by us. There is a sense that mathematics takes on a life of its own, which is why Frankel can talk about a vector as if it has an independent existence to him. Then there is Einstein who incorporated vectors into his mathematical formulation to describe how gravity is related to spacetime. 

 

Now here’s the thing: the relationship between gravity and spacetime still exists without humans to discover it or describe it. Spacetime is the 3 dimensions of space and 1 of time that, along with gravity, allows planets to maintain orbits over millions of years. But here’s the other thing: without mathematics, we would never know that or be able to describe it. It’s why some claim that mathematics is the language of nature. Whether Frankel agrees or not, I don’t know.

 

In the second video, Brady asks Frankel if he thinks he’s above mathematics, which makes him laugh. What Frankel argues is that there are inner emotional states, like ‘falling in love’, which can’t be described by mathematics. I know that some people would argue that falling in love is a result of biochemical algorithms, nevertheless I agree with Frankel. You can construct a computer model of a hurricane but it doesn’t mean that it becomes one. And it’s the same with the brain. You might, as someone aspired to do, create a computer model of a human brain, but that doesn’t mean it would think like one.

 

This all brings me back to Penrose’s 3 worlds philosophy of the mathematical, the mental and the physical and their intrinsic relationships. In a very real way, numbers allow us to comprehend the physical world, but it is not made of numbers as such. Numbers are the basis of the language we use to access mathematics, because I believe that’s what we do. I’ve pointed out before, that equations that describe the physical world (like Einstein’s) have no meaning outside the Universe, because they talk about physical entities like space and time and energy – things we can measure, in effect.

 

On the other hand, there are mathematical relationships, like Riemann’s hypothesis, for example, that deals with an infinity of primes, which literally can’t be contained by the Universe, by definition. At the end of the 2^ndvideo, Frankel quickly mentions Godel’s Incompleteness Theorem, which he describes in a nutshell by saying that there are truths in mathematics that can’t be formally proven. So there is a limit on what the human mind can know, given a finite universe, yet the human mind is 'not a mathematical machine’, as he so strongly argues.

 

He discusses more than I’ve covered, like his contention that our fixation with the rational is ‘irrational’, and there is no proof for the existence or non-existence of God. So, truly philosophical.

Space and time: still a mystery after all this (time?)

How’s that for a self-referential title, hence the question mark and parentheses. It highlights the fact that time is an everyday phenomenon that literally runs our lives and yet it remains one of the great mysteries of the Universe, still debated among philosophers and scientists. You may think that space is less of a mystery, yet it sparks debate as well, even without Einstein’s revelation that they are cosmologically entwined thanks to the constant speed of light, c.

The problem is with how do we categorise space and time. Are they entities, parameters, dimensions, metrics, mathematical constructions? Perhaps all of the above. I think we can safely say they are not physical objects, yet they determine the relationships between objects everywhere in the Universe, including those that we can’t perceive. In fact, some scientists would argue that time and space are all about relationships and nothing else, which I’ll return to later.

 

But let’s start with one obvious question, which was raised by Kant and still persists today, thanks to Donald Hoffman (refer my last post), and that is: are time and space simply constructs of the mind? To quote Kant from Critique of Pure Reason:

 

But this space and this time, and with them all appearances, are not in themselves things; they are nothing but representations and cannot exist outside our minds.

 

The problem with this viewpoint is that it’s readily believed by almost everyone that space and time existed for billions of years before any ‘mind’ arose in the Universe.

 

Another contentious point is to whether space is an ‘entity’ that ‘expands’ and ‘stretches’ as the Universes itself expands (which is not disputed). Viktor T Toth, a renowned expert on physics on Quora, argues very strongly that it doesn’t and what we witness is the ‘distance’ actually increasing between objects. Proponents against space expanding (like Toth) argue that the space within atoms doesn’t expand. My response is that the size of atoms is determined almost solely by Planck’s constant (h), for which there is no evidence that it changes with the universe’s expansion.

 

However, space can travel faster than light, which suggests it is an entity. This is not disputable, and it’s why there is a horizon to the observable universe (refer my post on the End of the Universe). It’s also why we can incorporate ‘inflation’ into the birth of the Universe. It also has ramifications for black holes, which I’ll come to later. According to Einstein’s theories of relativity, both space and time can change according to the observer and these changes are measurable. In other words, space and time are not ‘fixed’ and they are affected by gravity. In fact, Einstein’s famous formula for his general theory has the curvature of spacetime on one side and the momentum-energy tensor on the other side. In other words, spacetime is curved by energy/matter. To quote John Wheeler: “Spacetime tells matter how to move; matter tells spacetime how to curve.”

 

During this discussion, I’ll cite people who know a lot more than me, like Viktor T Toth and John Wheeler (already cited), even if I disagree with them. But I’m going to attempt the impossible: I’m going to argue ideas that I consider obvious, though not incontrovertible, and I will probably fail, since they will include black holes, quantum mechanics and relativity, all of which I don’t have as much knowledge as I would like. But bear with me, because it’s mostly just logic.

 

I want to point out, right at the start, that I’m not one of those people who think Einstein got it wrong, quite the contrary, but I will point out the limitations of his theory based on what we can actually observe. And that’s a good place to start. A common diagram used to visualise Einstein’s formulation of spacetime is the light cone going both forwards and backwards in time. If you are an observer at the centre of this cone you can only be affected by events from the past within the past light cone, and you can only affect events in the future within the future light cone. Everything else outside these cones can’t be observed or have a causal relationship with you, and this is what I mean when I say relativity has limitations because they are real limitations. Sometimes people will tilt the cones over, indicating movement on your part and the horizontal plane, called the 'hypersurface present', also tilts over. However, there is no causal connection along that 'hypersurface' (through spacetime), according to what I’ve just described.

But this brings one to the subject of simultaneity, because Einstein showed with his famous train and platform thought experiment that 2 observers in different frames of reference could observe different sequences of the same event or perceive a difference in what occurs simultaneously.

 

This is a video that explains this better than I can, including the mathematics involved. Two things worth mentioning: the lecturer includes the spatial Lorenz contraction as well as the time dilation in his calculations; and the observer in the same frame of reference as the source of light sees zero difference and therefore observes a ‘true simultaneity’, though no one calls it that. I’ve long argued that the ‘other observer’ who doesn’t see the simultaneity, observes a difference in the Doppler effect caused by the ‘moving’ frame of reference with the moving light source, which should tell that observer that their observation is incorrect. The Doppler effect tells the observer if the light source is in their frame of reference or a frame of reference moving relative to them. It’s the Doppler effect that tells us that the Universe is expanding uniformly in all directions – it has no centre. It also tells us that we’re moving relative to the CMBR (cosmic microwave background radiation). In other words, we can measure our ‘velocity’ relative to the whole of spacetime, which, of course, is the Universe.

 

I’ve explained elsewhere how different observers in different parts of the Universe literally see different ‘now(s)’. They can literally see events occurring in opposite sequences, as a consequence of the finite speed of light, even without relativistic effects. However, if the events have a causal relationship, then all observers will see them in the same sequence. But this also means that my present will be seen in another observer’s past in their future, but it doesn’t mean the converse: that their future can be seen in my present. In fact, the relationship is reciprocal because I will see their past in my present. Observers can only see another observer’s past, no matter where they are. No observer can see another observer’s future. 

 

To give an example, a hypothetical observer in the Small Magellanic Cloud would see us 210,000 years ago when we were just emerging from Africa. Likewise, we would observe them 210,000 years ago (relative to us) if that was physically possible. Therefore, I don’t hold to the widely held view that we can theoretically see another observer’s future (due to the tilting 'hypersurface' plane in the light cone graphic), which infers that the future must already exist for everyone.

 

We know from the twin paradox thought experiment, as well as data from orbiting satellites, that clocks do literally run at different rates due to gravity as well as motion (your satnav depends on making corrections). Also, the famous muon observations arriving on the Earth’s surface. So both special and general theories of relativity change the rate of time, yet when the clocks are back in the same reference frame, they will show a different time duration while agreeing on where they are in the spacetime co-ordinates of the solar system. In other words, they don’t exist in different ‘now(s)’ just because they measured different durations to arrive at the same destination.

 

We know that different animals see time ‘flow’ at different rates. Many birds and insects see the world in slow-motion compared to us. This means they will see the hands of a clock literally moving slower while telling the same time. As Paul Davies has pointed out, if time was to slow down or speed up, you wouldn’t notice. But you can notice if you compare clocks in relativity. My point is that ‘now’ doesn’t change for these creatures even though they perceive time flowing at a different rate to us.

 

Well, I contend the same is true on a cosmic scale. If you were to go near the event horizon of a black hole, like in the movie, Interstellar, time would slow down for you compared to everyone back on Earth, even though you wouldn’t notice it. My argument is that this is no different, perceptually, to the bird observing time going slower. If you were to use the Doppler effect of receding galaxies as a clock, they would actually appear to be going faster (assuming you could take accurate enough measurements) compared to what Earthlings observed, and when you returned, you would agree on what ‘now’ is, compared to these distant cosmic clocks, though you would be considerably younger than your counterparts, if they were still alive, but more likely you would be meeting their subsequent generations.

 

And this is true even on Earth, where clocks run infinitesimally faster on mountaintops compared to sea level. But you don’t see an accumulated difference in ‘now’ over millions of years of the Earth’s rotation. All the while, the clocks are in the same ‘present’ while they are measuring different rates of time passing.

 

Carlo Rovelli gave a talk at the Royal Institute on ‘time’, where he argues that there is no ‘universal time’. But during the 15min question time (shown in another video), he contends that we arrive at a cosmic time for the Universe by taking an ‘average’. Brian Greene, in his book, The Fabric of the Universe, said something similar. However, if you lived on a planet orbiting near a black hole, surely the age of the Universe would be much less than what we observe, because any clock would be measuring time passing at a much slower rate than what we measure on Earth. Like the clocks on top of the mountains on Earth, I don’t believe hypothetical observers orbiting close to a black hole, perceive a ‘now’ that progressively gets out of step with the ‘now’ Earthlings observe over the course of their lives in the Universe, even if they measured a different age. In other words, I contend that you can have a universal now for the whole universe even if different clocks measure different rates of time dependent on where they are located.

 

Another video, which is an interview with loop quantum gravity theorist, Lee Smolin, describes time and space as being separate, which is both heretical and interesting. I think he has a point when you consider that, on a cosmic scale, time is finite and space is possibly infinite. Space could also be finite but perceptually infinite, like a hyperbolic universe, but, as Marcus du Sautoy pointed out in his book, What We Cannot Know, if the Universe is truly spatially infinite, we might never know. Smolin conjectures that space could be a consequence of ‘causal relationships’ between physical objects, which he doesn’t elaborate on, but which I find difficult to conceptualise. Causation is determined by the speed of light, otherwise everything would happen at once (Caspar Henderson, A New Map of Wonders). Smolin also contends that time might be an ‘emergent’ property (also without elaborating). The point is that causality requires time axiomatically. The thing about both space and time is that they are dimensions and if you add light (c) into the mix, you get a 4-dimensional universe that is fundamental for it to function in the way it does. With more than 3 spatial dimensions, planets would not have stable orbits, and if there was more than 1 dimension of time you would get time loops. If you have 2 spatial dimensions you would literally fall apart. Also, more than 3 spatial dimensions causes light waves to travel inconsistently. Our universe has the ideal time-space dimensional combination for its goldilocks existence.

 

In the same video, Smolin explains how the event horizon of a black hole breaks causality. This can be seen mathematically by Schwarzchild’s equation for a static black hole, which is described in this video. As the presenter explains, the +ve and -ve signs of the equation change when you cross the event horizon, which breaks causality. Causality is caused by the space dimension being less than the (negative) time dimension, and they are reversed on the other side of the event horizon (watch the video). It should be pointed out that Einstein was initially sceptical about the existence of black holes, even though Schwarzchild derived his equation from Einstein’s tensor.

 

There is a paradox inherent in a black hole (more than one, actually) but the most fundamental one is that time theoretically stops at the event horizon because time is related to light, and light can’t escape a black hole by definition. Viktor T Toth says that ‘the event horizon is always in an observer’s future’, so how can anyone (or anything) fall into a black hole? In a previous post, I speculated that maybe ‘space’ itself ‘falls’ into the black hole and that’s exactly what the guy in the video says. This is only possible because space can travel faster than light, as I described earlier.

 

This is already a lengthy post but I can’t talk about time without mentioning quantum mechanics. The same guy (who talks about black holes), gives a very good summary explanation of Richard Feynman’s path integral formulation of QED (quantum electrodynamics) in this video. It should be pointed out that Julian Schwinger’s ‘field’ interpretation called QFT (quantum field theory) is now more popular, if that’s the right word. In QFT, particles are seen as ‘excitations’ of a quantum field which is everywhere in the Universe. Someone on Quora even suggested that the word ‘particle’ should be erased from every physics text book, because they just don’t exist. Curiously, Feynman, in his book, QED, argued that everything is ‘particles’, but that was in the context of whether quantum phenomena are ‘waves’ or ‘particles’ in the Bohr tradition. I like Freeman Dyson’s view that it depends on whether an event is in an observer’s future or past, but I’m getting ahead of myself.

 

A good place to start with QM is Schrodinger’s equation. Carlo Rovelli, whom I cited earlier, in one of his books, is almost dismissive of Schrodinger’s equation and argues that the wave function (ψ) has misled us in our understanding of QM. But Schrodinger’s wave function is the basis of Feynman’s QED, so that’s where I’ll start.

 

Schrodinger’s equation encapsulates all the characteristics of QM which make it weird: superposition, entanglement and the uncertainty principle. The wave function also incorporates time-reversal symmetry, which is an inherent feature of QM. It doesn’t incorporate relativity, but I’ll come to that later.

 

The thing about Schrodinger’s equation, which is rarely mentioned, is that it describes the future – it makes predictions about where something will be in time. It was Dirac who derived the Lagrangian for QM, and Feynman adopted that for his ‘sum over histories’ or ‘path integral’ formulation, because it calculates the path of ‘least action’, which dictates what something does. (This also applies in a gravitational field, by the way, but I don’t want to confuse you.) Feynman used the proper time (τ) in place of t (that Schrodinger used) which automatically allows for special relativity (as explained in the video).

 

As someone on Quora once explained (David Moore, who is a Sydney based GP), a probability of ONE exists in the past, after the event. In the future, the probability is always less than one. This is what happens when the wave function ‘collapses’, for want of a better word, and neatly incorporates Freeman Dyson’s view that QM describes the future while classical physics describes the past. Feynman’s formulation has an infinity of possible future paths, that he integrates (hence the ‘integral’ in path integral) and also gives the path of least action. There is an element of teleology in this, but I don’t believe it makes the universe deterministic, though others disagree. On a large enough scale, as Schrodinger himself pointed out, you get a statistical deterministic effect, which he coined ‘statistico-deterministic’. But it can’t predict individual events, like when a radioactive isotope will decay, which is the crucial component in his eponymous cat thought experiment.

 

In regard to photons being the ‘particle’ nature of light, Mark John Fernee (physicist at Queensland University and regular Quora contributor) made the point in one of his posts, that if we didn’t observe light as photons, we would not be able to see many of the distant stars that we do. If light was purely a wave, then it would be so dispersed over the massive sphere of its influence it would be too faint to see. But, as a photon, it can arrive in just one point in space, where we happen to observe it.

 

I will leave the last word to Paul Davies. Even though he’s talking about QM in reference to black holes and Hawking radiation, the principle he describes is universal.

 

The very act of measurement breaks the time symmetry of quantum mechanics in a process sometimes described as the collapse of the wave function... the rewind button is destroyed as soon as that measurement is made.

Addendum: This video gives a more detailed and accurate explanation of black holes. It's more complex than my exposition would suggest.

To the End of the Universe

I like to remind myself and others how little I know. It’s one of the reasons I like Quora, where I get to occasionally interact with people who know considerably more than me. One such person is Mark John Fernee, a physicist at the University of Queensland. I’ve learned a lot of science from an approach based on scepticism. For example, I was sceptical about relativity theory: that clocks could really slowly down and why did they slow down for one observer but not another, as demonstrated in the famous twin paradox. In fact, it’s nature’s paradoxes that provide the incentive to try and understand it to the extent that one can. 

 

Another example is quantum mechanics. For a long time, I followed David Bohm’s approach, which was really an attempt to bring QM back down to Earth so-to-speak. I believe that both Schrodinger and Einstein also believed in a ‘hidden-variables’ approach.

 

I finally gave this up when I concluded that QM and classical physics obey different rules: superposition and entanglement are not part of classical physics, either experimentally or mathematically. And I found that special relativity only made sense in the context of general relativity (which I discuss in more detail below).

 

And then you have the combination of special relativity with QM, which, from a mathematical perspective, allows anti-particles to exist. As Fernee points out, because an anti-particle can be represented mathematically by a particle going backwards in time, it ensures that charge is conserved by time’s arrow. In other words, you can turn an electron into a positron, or vice versa, by reversing time, which is why it’s never observed.

 

One of the paradoxes I now struggle with is that, according to special relativity, you can have different ‘nows’ in different parts of the universe. This is why most, if not all physicists, argue that the universe is completely deterministic, if someone’s future can be hypothetically observed by someone else’s motion. I confess I’m very sceptical about this. What they're saying is that the ‘now’ in some other part of the Universe is changed by an observer’s motion locally. Fernee quotes Roger Penrose in response to a question: can we theoretically teleport to some other location in the Universe instantaneously, like we see in science-fiction movies? According to Fernee (quoting Penrose), if you could and then teleport back, you might arrive before you left, because a random movement by you could change the ‘now’ in that distant part of the universe into your past. I’m assuming this can be demonstrated mathematically; it’s a consequence of simultaneity changing depending on the observer, according to special relativity. 

 

I’ve discussed this in other posts. I like to point out that, where there’s a causal relationship, the sequence of events can’t be changed, dependent on an observer’s perspective. Which makes me wonder: does a sequence change, dependent on an observer’s perspective, when they’re not causal? Is it possible that there is a sequence of events independent of any observer?

 

And this leads to another paradox that is hardly ever addressed which is that, despite this proliferation of ‘nows’, dependent on observers’ perspectives, we have an ‘age of the Universe’. I actually raised this with Fernee in a dialogue I had with him, and he referenced a paper by Tamara M. Davis and Charles H. Lineweaver at the University of New South Wales, titled, Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe. I’ve lost the link, and I can no longer even find the post on Quora, but I downloaded the paper, which is 24 pages long, not including the references.

 

Of course, it’s an academic paper, yet I found it easier to follow and understand than I might have expected. Which is not to say I have a full grasp of it, but I feel I can relay some of its most pertinent points. The paper is dated 13 November 2013, so it seems apt I’m writing about it on 13 Nov, 2021. Firstly, the cosmological model of the Universe the authors discuss, is referred to as ΛCDM cosmology (Lambda-CDM cosmology), where CDM is an acronym for Cold Dark Matter. Lambda (Λ) is the cosmological constant that gives us ‘dark energy’, so the model includes both dark energy and dark matter.

 

As the title suggests, the authors discuss misconceptions found in the literature concerning the horizon problem, and at the end they provide a list of examples, including one by Richard Feynman (1995), 

 

“It makes no sense to worry about the possibility of galaxies receding from us faster than light, whatever that means, since they would never be observable by hypothesis.” 

 

And this one by Paul Davies (1978): 

 

“. . . galaxies several billion light years away seem to be increasing their separation from us at nearly the speed of light. As we probe still farther into space the redshift grows without limit, and the galaxies seem to fade out and become black. When the speed of recession reaches the speed of light we cannot see them at all, for no light can reach us from the region beyond which the expansion is faster than light itself. This limit is called our horizon in space, and separates the regions of the universe of which we can know from the regions beyond about which no information is available, however powerful the instruments we use.” 

 

What the authors expound upon in the main body of their text is that there are, in effect, a number of horizons, which makes these statements erroneous at best. To be fair to both Feynman and Davies, the ΛCDM model of the Universe wasn’t known at the time. Dark energy wasn’t officially ‘discovered’ until 1998. Davis and Lineweaver provide diagrams to show these various horizons, which I can’t duplicate here, and if I did, I’d have trouble explicating them. But basically, there is a particle horizon, which is the limit of the observable universe, the Hubble sphere, which is the boundary of the expanding universe (where it equals c) and the event horizon. (To quote the authors: Our event horizon is our past light cone at the end of time, t = ∞ in this case.) There is a logical tendency to think they should all be the same thing, but they’re not, as the authors spend a good portion of their 24 pages expounding upon. To quote again:

 

The particle horizon at any particular time is a sphere around us whose radius equals the distance to the most distant object we can see... Our effective particle horizon is the cosmic microwave background (CMB).

 

Whereas:

 

Hubble sphere is defined to be the distance beyond which the recession velocity exceeds the speed of light, DHS = c/H. As we will see, the Hubble sphere is not an horizon. Redshift does not go to infinity for objects on our Hubble sphere (in general) and for many cosmological models we can see beyond it... The ratio of ∼ 3/1 is the ratio between the radius of the observable universe and the age of the universe, 46 Glyr/13.5 Gyr.

 

What you have to get your head around is that the universe is dynamic, and given the time it takes for light to reach us from the edge of the Universe, both the edge and the objects (we’re observing) have moved on, quite literally. This means we can observe objects over the horizon so-to-speak. But it’s even more complex than that, because the Hubble sphere, which is expanding, can overtake photons that were emitted beyond the horizon but are travelling towards us. According to the authors, we can observe objects that are ‘now’ travelling at superluminal speeds relative to us. 

 

This is how the authors explain it:

 

Light that superluminally receding objects emit propagates towards us with a local peculiar velocity of c, but since the recession velocity at that distance is greater than c, the total velocity of the light is away from us. However, since the radius of the Hubble sphere increases with time, some photons that were initially in a superluminally receding region later find themselves in a subluminally receding region. They can therefore approach us and eventually reach us. The objects that emitted the photons however, have moved to larger distances and so are still receding superluminally. Thus we can observe objects that are receding faster than the speed of light. 

 

One of the most illuminating aspects of their dissertation, for me, was that one needs to use a general relativistic (GR) derivation of the Doppler redshift and not a special relativistic (SR) derivation, which is usually used. They show graphically that the SR and GR derivations diverge, especially for further distances. On the same graph, they show how a non-relativistic Doppler shift, which would be ‘tired light’ (authors’ term) is actually a horizonal line, so nowhere near. The graph, of course, shows these curves against observations of super novae. As they explain it:

 

The general relativistic interpretation of the expansion interprets cosmological redshifts as an indication of velocity since the proper distance between comoving objects increases. However, the velocity is due to the rate of expansion of space, not movement through space, and therefore cannot be calculated with the special relativistic Doppler shift formula. 

 

What they are saying is that there is a distinction between the movement of the objects in space and the movement of space itself. For me, this ends the debate about whether ‘space’ is an entity or just the distance between objects. As much as I admire and respect Viktor T Toth, I’ve always had a problem with his argument that space ‘doesn’t expand’, but only the objects ‘move’ thus creating more space between them. The Hubble sphere, as I understand it, is where space equals c.

 

Later in their paper, Davis and Lineweaver describe how they derived their equation for the GR redshift.

 

For the observed time dilation of supernovae we have to take into account an extra time dilation factor that occurs because the distance to the emitter (and thus the distance light has to propagate to reach us) is increasing.

 

In other words, in calculating the redshift of a ‘comoving galaxy’, they also have to take into account the constant expansion of space in the photon’s journey to the observer. 

 

....the peculiar velocity of a photon, Rχ ̇, is c. Since the velocity of light through comoving coordinates is not constant (χ ̇ = c/R), to calculate comoving distance we cannot simply multiply the speed of light through comoving space by time. We have to integrate over this changing comoving speed of light for the duration of propagation. Thus, the comoving coordinate of a comoving object that emitted the light we now see at time t is attained by integrating.  (χ ̇is the time dependent expansion of space and R is the radial distance). 

 

Notice that in contrast to special relativity, the redshift does not indicate the velocity, it indicates the distance. That is, the redshift tells us not the velocity of the emitter, but where the emitter sits (at rest locally) in the coordinates of the universe. 

 

In other words, when we integrate χ ̇, we get χ, which is distance. The authors provide another equation for determining the velocity.

 

Now, one of the obvious aspects of this whole exercise is that they are calculating a redshift across space that changes over time, so what does time mean in this context?

 

This is how the authors explain it, just before their conclusion:

 

Throughout this paper we have used proper time, t, as the temporal measure. This is the time that appears in the RW metric and the Friedmann equations. This is a convenient time measure because it is the proper time of comoving observers. Moreover, the homogeneity of the universe is dependent on this choice of time coordinate — if any other time coordinate were chosen (that is not a trivial multiple of t) the density of the universe would be distance dependent. Time can be defined differently, for example to make the SR Doppler shift formula correctly calculate recession velocities from observed redshifts (Page, 1993). However, to do this we would have to sacrifice the homogeneity of the universe and the synchronous proper time of comoving objects.

 

I find it interesting that they adopt a ‘proper time’ for the whole universe. It makes one wonder what ‘now’ really means.

 

Footnote 1: I want to point out that in their acknowledgements, Davis and Lineweaver reference Brian Schmidt, who received a joint Nobel Prize for his work in empirically confirming dark energy, or the cosmological constant (Λ).

Footnote 2: You can download the paper here.

Addendum: This is a video by someone (who knows more than me) and doesn’t give his name. I posted a video by him before, regarding the question: Is gravity a force? His videos on Penrose tiling and the Feigenbaum constant are among the best.

 

In this video, he refutes my claim, arguing that space doesn’t expand. He makes one very compelling point that if space expanded so would atoms and so would we. Victor T Toth makes the exact same point, and I’d have to agree. The size of all atoms is determined by h (Planck's constant), which doesn't change with the expansion of the Universe. I might add that this presenter and Toth disagree on whether gravity is a force or not, so physicists don’t always agree, even in the same field, like cosmology.

 

In the video, he argues that there are 3 types of Doppler shift and contends that they are actually all the same. Most intriguing was the thought experiment that someone in ‘free fall’ wouldn’t see the Doppler shift that another observer would. In other words, it’s observer dependent.

 

But there is a spacetime metric or manifold, which forms the basis of general relativity theory (GR) and this can warp and curve (according to said theory). In fact, there is a phenomenon called ‘frame dragging’, where spacetime is dragged around by a spinning black hole. Light is always c in reference to this spacetime manifold. So when ‘space’ reaches the speed of light at the horizon relative to us, light is still c in that reference frame, even though it is expanding away from us at c or more. Space can travel faster than light, even though massive particles can’t, which is why ‘inflation’, proposed at the birth of the Universe, is possible.

 

Getting back to the Doppler shift the authors cite in their paper, they use a GR Doppler shift, which I believe isn’t covered in the video.

There are 2 paradoxes in relativity

 This post is based on an interaction I had on Quora. When people discuss the twin paradox (like I did), they always discuss the difference in time, each twin experiences relative to the other; they don’t discuss the difference in distance they experience, which is a logical consequence of the same Lorenz transformation. 

However, someone on Quora conjectured that in reality, the clocks actually measure time at the exact same rate and it’s the distance that changes. Mathematically, there is nothing wrong with this argument, so why do I disagree with it?

 

The twin paradox, of course, is a thought experiment, but there is a real-life experiment where this happens, which I’ve also discussed in another post, and that is the half-life of muons in the Earth’s atmosphere. According to the scientists observing them on the Earth’s surface, there are too many of them, which means they took longer to decay (the half-life got longer). But, from the muon’s perspective, their half-life hasn’t changed, it’s just the distance to the Earth was shorter. Now, this creates a paradox, which is equally manifest in the twin paradox thought experiment. 

 

How can the distance change depending on whose perspective you take? I don’t believe it does. I contend that the moving observer ‘measures’ a shorter distance based on their clock slowing down. But, you’ll say, the whole point of relativity is that no one knows who is moving and who is stationary. Well, there is a reference point, which, in the case of the muons, is the gravitational field of Earth. A body in free fall experiences maximum ‘proper’ time and any deviation from that causes a clock to slow down. The fact is, because the muons are travelling at high fractional light speeds, their deviation from maximum proper time is greater than the observers on the ground. I believe this logic also applies to the travelling twin in the thought experiment.

 

If you take this to extremis, someone could hypothetically travel across the entire galaxy in their lifetime if they travelled fast enough (without exceeding the speed of light). From their perspective, the distance would shrink astronomically, but if they returned to Earth, eons would have passed in their absence. I should point out that science fiction writers (like myself) routinely ignore this fundamental consequence of relativity. We ‘imagine’ that humanity has discovered ‘new physics’ to overcome this ‘problem’

 

I’m not sure how heretical it is to argue this: that time changes but the distance doesn’t. if one goes back to the argument that started this rumination: when the twins reunite their respective clocks disagree and the space-faring twin is noticeably younger than their counterpart. However, they not only disagree on the elapsed time, since the space-faring twin’s departure, they also disagree on how far that particular twin travelled. BUT, now that the space-faring twin is back in the same frame of reference as their Earthbound twin, they concur on the distance, yet still disagree on the time elapsed, evidenced by their clocks and age difference.

 

 

Addendum: Ian Miller (PhD in Chemistry and a self-described 'theoretician' on Quora) argues that it's the ruler that changes relativistically and not the distance travelled. And I'd contend that the clock is effectively the ruler.

Does QM and classical physics create the irreversibility of time?

 At long last I’ve found a YouTube video that pretty much describes quantum mechanics (QM) as I would. In particular, the narrator (Arvin Ash) expresses the possibility that the transition from QM to classical physics provides the irreversibility of time that we all experience in everyday life. In other words, QM describes the future and classical physics describes the past. The narrator cites Lee Smolin, who actually says that QM describes the ‘present’ and classical physics describes the past. Now, I’ve read Lee Smolin’s book, The Trouble with Physics, and, from memory, he made no mention of this, so maybe this is a new idea from him (I don’t know).

My knowledge of QM is rudimentary at best, so I’m hardly one who can judge, but I’ve been thinking this way since I wrote a post called What is now? in 2015. Back then, I didn’t know that Freeman Dyson had similar ideas. A contributor to Quora, Mark John Fernee, who clearly knows a lot more than me, made a similar point about QM to classical physics being irreversible in time, and whom I quoted in a not-so-recent post.

 

Ash also explains entanglement and decoherence without getting too esoteric about it, and seems to promote the view that entanglement, in principle, could involve the whole universe. Decoherence is often explained as the ‘leaking’ of information. The important point is that decoherence (or the wavefunction collapse) comes from the quantum phenomenon interacting with other particles, that one assumes already exist.

 

The narrator conjectures at the end that the multiverse interpretation is still possible, but I’m not so sure. The whole point of MWI (multiple worlds interpretation) is supposedly that decoherence never happens, but this variation means that it still would happen, only in other universes. Sabine Hossenfelder makes a similar point in a YouTube video of her own. 

 

The other problem with MWI, as I see it, is that entanglement would necessarily incorporate a multiverse. I suspect adherents to MWI (and there are a lot of them) wouldn’t have a problem with that, but I don’t really know. Some highly respected physicists, like Sean Carroll, are advocates of MWI. I really admire Sean Carroll and he readily admits that MWI is one of his personal prejudices. I recently saw a talk he gave on ‘time’ for New Scientist, but he didn’t mention any of this. Instead, he talked about the role of entropy, including its ramifications for the evolvement of the entire universe. I’m a heretic on entropy in that I think it’s a consequence of the arrow of time, not its cause. Having said that, the low entropy state of the Universe in the beginning is still a conundrum, though gravity plays a role in increasing complexity in the Universe, in spite of entropy.

 

In another video (by Closer to Truth), Lee Smolin articulates the possibility that time and 

space may be separate after all, which I’m beginning to wonder myself. Besides, if the Universe has a boundary (or edge) in time, but not in space, that would infer that they are separate. We know that time on a cosmic scale is finite, because we can estimate the age of the Universe.

 

I believe we all live on the edge of time (all of the time), which is contentious. All physicists, that I read and listen to, argue that there is no universal now, and I’m told that to think otherwise is naive, 19^th Century thinking. They argue that Einstein’s theories of relativity rule it out, because clocks run at different rates depending on where they are in the Universe and how fast they are travelling relative to other observers. Actually, it’s dependent on how fast they are travelling relative to an observer following a geodesic in a gravitational field (in free fall or in orbit).

 

I’m well aware that different observers, in different parts of the Universe, see a different ‘now’, because they all see stars 100s or 1000s of light years from them, which means they see them at different ages. And, of course, some of those observers could hypothetically see some of the same stars at different ages, which means they all see a different ‘now’. And then, if they are in motion with respect to each other, that distorts the differences even further. For example, if you have 2 super novae occurring in the field of view of 2 spatially separated observers, they may well see them happening in opposite sequences. Notice that this is true even without relativistic effects.

 

So one shouldn’t be surprised if Einstein’s special theory of relativity tells us that simultaneity appears subjective, both in space and time, because it can be, even without relative motion. But obviously, this isn’t the case where causality is involved. Causality insists on a sequence in time by definition, and it has to be objective, irrespective of what observers see.

 

I like to look at the famous twin paradox, because I think it contains almost everything we need to know about the special theory of relativity. I know it’s a thought experiment, but real experiments done with atomic clocks and aeroplanes and satellites tell us it’s true. The important point is the end result – when the spacefaring twin returns, they are younger than their Earthbound twin. The same effect could be made by the twin travelling near the event horizon of a black hole, which was the premise for Chris Nolan’s movie, Interstellar (which had Kip Thorne as a consultant).

 

But here’s the thing: both twins agree on what time it is in the 4-dimensional spacetime of the Universe. So, 2 observers travelling along different paths can measure different durations of time by whatever means they have, but when they reunite, they agree on where they are in time, in the same way they agree on where they are in space. I can’t see how this is possible if there isn’t a universal ‘clock’, which is arguably the edge of time for the whole universe.

Addendum 1: Mark John Fernee, whom I reference in the main text, and has a PhD in physics, proposes a similar, if not better, argument than I do. He also gives the same explanation of entropy as an 'emergent' property through probabilities that I do in other posts. However, I expect he may not agree with me that there is a 'universal now'. This is his answer to, How did entropy become associated with time?

 

 Addendum 2: Another post by Mark John Fernee (8Aug22) gives an excellent synoptic description of the relationship between classical physics and QM, which reflects my own point-of-view.