Is gravity a force?

 Believe it or not, this is a question that even physicists don’t seem to agree on. Viktor T Toth, whom I’ve referenced in previous posts and whom I follow on Quora, is almost dismissive of the question. Paraphrasing, he said something like, ‘If you don’t think gravity is a force, just drop a rock onto your foot.’ Ouch! I’d say it’s one of those paradoxes that seem to pop up everywhere in our comprehension of the Universe. 

This question takes me back to my teenage years when I first encountered Newton’s universal theory of gravitation. I struggled to understand how all bodies from the most massive to the smallest could all fall at the same rate in the same gravitational field – it made no sense to me.

 

The equation F = ma is one of the most basic in physics and embodies Newton’s second law in a succinct formula that is universal – it applies everywhere in the Universe. 

 

Leaving gravity aside for the moment, if you apply a uniform force to different masses they will accelerate at different rates. But in gravity, we get a converse relationship. We have different forces dependent on the mass, so we always get the same acceleration. It’s like gravity adjusts its ‘force’ according to the mass it effects. This was the dilemma I couldn’t resolve as a high school physics student.

 

Around the same time, I remember watching a documentary on Einstein (when TV was still in black and white), which was inspiring and thought-provoking in equal measure. One of the more mind-bending ideas I remember was someone explaining how Einstein had changed the concept of gravity as a force to gravity as a curve. This made absolutely no sense to me and it says a lot that I still remember it after more than 5 decades. Of course, I didn’t realise at the time, that these 2 conundrums I was contemplating were related.

 

I’ve recently been reading a book called Emmy Noether’s Wonderful Theorem by Dwight E. Neuenschwander, which is really a university-level text book, not a book for laypeople like me. Emmy Noether famously developed a mathematical formulation to show the relationship between conservation laws and symmetry. Basically, energy is conserved with transformations in time, momentum is conserved with transformations in space and angular momentum is conserved with transformations in rotation. She originally developed this for relativity theory but it equally applies to quantum mechanics. As someone who is not known outside of the physics community, she’s had an enduring and significant input into that field. In her own time, she was not even paid for giving lectures to students, such was the level of prejudice towards women in the sciences in her time (pre WW2).

 

Reading Neuenschwander’s book, I was surprised to learn how much a role the Lagrangian played in her work. Relevant to this topic, Noether was the first to apply the Lagrangian to general relativity, which is actually the easiest way to understand it.

 

To quote from Neuenschwander:

 

The general theory of relativity came along in 1915, and by 1918 the equation of motion of a particle falling in a gravitational field was shown, especially by Emmy Noether and David Hilbert, to be derivable from a variational principle: The world line of a freely falling particle would be that for which the elapsed time between two events was maximised, making the world line a geodesic in spacetime.

 

David Hilbert was Noether’s mentor, and also her greatest champion it has to be said. Hilbert was arguably the greatest living mathematician of his day. The last part of that quote is effectively a description of the Lagrangian, which the author compares to Fermat’s principle. In other words, Fermat’s principle gives the path of least time for light being refracted, and the Lagrangian in a gravitational field gives the geodesic, which is the path of maximum elapsed time for a particle.

 

Notice that in that description of a particle following a geodesic in a gravitational field there is no mention of its mass. This means that its path and its elapsed time is independent of its mass. In fact, we even know it applies to photons, which are, to all intents and purposes, massless.

 

And this revelation finally resolves the conundrum I wrestled with in high school. If you are in free fall, say in a falling lift, or in an orbiting space station; in both cases, you will experience weightlessness.

 

So where is the force? The force is experienced when an object is stopped from following the geodesic, which is the normal everyday experience we all have and don’t even think about. We call it the force of gravity, because gravity is the underlying cause, but the force is actually created by whatever it is that is stopping you from falling, and not the other way round. And, of course, that force is proportional to your mass otherwise you wouldn’t be stationary, relative to whatever’s holding you up. And if you drop a rock on your foot (not recommended) you’ll experience a force that is directly proportional to its mass; no surprise there either.

This is a video, by someone who knows more about this than me, which is even more mind-bending. He argues, quite convincingly, that we are all accelerating by just standing still - on Earth.

Reality, nature, the Universe and everything abhors a contradiction

 People used to say that ‘nature abhors a vacuum’, which is more often than not used metaphorically. Arguably, contradiction avoidance is more fundamental in that you don’t even need a universe for it to be requisite. Many mathematical proofs are premised on the unstated axiom that you can’t have a contradiction - reductio ad absurdum.

However, the study of physics has revealed that nature seems to love paradoxes and the difference between a paradox and a contradiction is often subtle and sometimes inexplicable. So, following that criterion, I believe that reality exists in that sliver of possibility between paradox and contradiction.

 

Clifford A Pickover, who normally writes about mathematics and physics, wrote an entertaining and provocative book, The Paradox of God and the Science of Omniscience. One of the conclusions that I took from that book is that there are rules of logic that even God can’t break. And one of those rules is the rule of contradiction that something can’t ‘be’ and ‘not be’ at the same time. It turns the argument on its head that God created logic. 

 

It’s obvious to anyone who reads this blog that I’m unceasingly fascinated by science and philosophy, and, in particular, where they meet and possibly crossover. So an unerring criterion for me is that you can’t have a contradiction. My recent exposition on the famous twin paradox demonstrates this. It can only be resolved if one accepts that only one twin experiences time dilation. If they both did, you would arrive at a contradiction, both mathematically and conceptually. Specifically, each twin would perceive the other one as younger, which is impossible. 

 

The opposite to contradiction is consistency, and if you look at the mathematical analysis of the twin paradox, you’ll see it’s consistent throughout for both twins.

 

To give another example, physicists tell us that time does not flow (as Paul Davies points out in this presentation, 48.30min mark), yet we all experience time ‘flowing’. Davies and I agree that the sensation we have of time ‘flowing’ is a psychological experience; we disagree on how or why that happens. 

 

Time is a dimension determined by the speed of light. Everything is separated, not only spatially, but also in time, because it takes a finite time for light to travel between events separated in space. This leads to the concept of spacetime, which is invariant for different observers, while space and time (independently) can be ‘measured’ to be different for different observers. Spacetime is effectively an extension of Pythagoras’s theorem into 4 dimensions, only it involves negatives.

Δs² = c²Δt² - Δx² - Δy² - Δz²

So Δs is the ‘interval’ that remains invariant, and cΔt is the time dimension converted into a spatial dimension, otherwise the equation wouldn’t work. But note that it’s c (the constant speed of light) that makes time a dimension, and it’s one of the dimensions of the Universe as a whole. Einstein’s equations work for the entire Universe, which is his greatest legacy.

 

I’ve been reading Brian Greene’s The Fabric of the Cosmos (2004), which I came across while browsing in a bookshop and bought it on impulse – I’m glad I did. It’s a 500 page book, covering all the relevant topics on cosmology, so it’s very ambitious, but also very readable.

 

Theories of gravity started with Newton, and he came up with a thought experiment that was still unresolved when Einstein revolutionised his theory. Greene discusses it in some detail. If you take a bucket of water and hang it from a rope, then turn the bucket many times so that the rope is twisted. When you release the rope the bucket spins and the water surface becomes concave in the bucket due to the centrifugal force. The point is, what is the reference frame that the bucket is spinning to? Is it the Universe as a whole? Newton would have argued that it was spinning relative to absolute space.

 

Now imagine that you could do this experiment in space; only, instead of a bucket of water, you have 2 rocks tied together. You could do it on the space station. As you spin them, you’d expect the rope or cord between them to tension. So again, what are they spinning in reference to? Greene gives a detailed historical account because it involves Mach’s principle. But in the end, when Einstein applied his theories of relativity, he came to the conclusion that they spin relative to spacetime. While we don’t have absolute space and absolute time, we have absolute spacetime, according to Einstein.

 

Why have I taken so much trouble to describe this? Because there is a frame of reference that is used to determine what direction and what speed our solar system travels in the context of the overall Universe. And it’s the Cosmic Microwave Background Radiation. Paul Davies described this in his book, About Time (1995). By measuring the difference in the Doppler effect (for CMBR) in different regions of the sky, we can deduce we are travelling in the direction of the Pisces constellation. Greene also points this out in his book.

 

All physicists that I’ve read, or listened to, argue that ‘now’ is totally dependent on the observer. They will tell you that if you move backwards and forwards here on Earth, then the time changes on some far off constellation will be in the order of hundreds of years because of the change in angle of the time slice across that part of the Universe. Greene himself explains this in a video, as well as in his book. But there is an ‘age’ for the Universe, so you have an implicit contradiction. Greene is the only person I’ve read who actually attempts to address this, though not very satisfactorily (for me).

 

But one doesn’t need to look at far off galaxies, one can look at Einstein’s original thought experiment involving 2 observers: one on a train and one on a platform. The reason physicists argue that there is no objective now is because simultaneity is different according to different observers, and Einstein uses a train as a thought experiment to demonstrate this.

 

If you have a light source in the centre of a moving carriage then a person in the carriage will observe that it gets to both ends at the same time. But a person on the platform will see that the back of the train will receive the light before the front of the train. However, the light source is moving relative to the observer on the platform, so they will see a Doppler shift in the light showing that it is moving relative to them. I contend that only the observer in the same frame of reference as the light source sees the true simultaneity. In other words, I argue that you can have 'true simultaneity' in the same way as you can have ‘true time’. Also, what many people don’t realise, that different observers not only see simultaneity happening at different times but different locations, as this video demonstrates.

 

I’m a subscriber to Quora, mainly because I get to read posts by lots of people in various fields, most of whom are more knowledgeable than me. In fact, I claim my only credential is that I read a lot of books by people much smarter than me. One of the regular contributors to Quora is Viktor T Toth, whom I’ve referenced before, and who calls himself a ‘part-time physicist’. Toth knows a lot about cosmology, QFT (quantum field theory) and black holes. He occasionally shows his considerable mathematical abilities in dealing with a question, but most of the time keeps them in reserve. What I like about Toth, is not just his considerable knowledge, but his no-nonsense approach. He doesn’t pretend or bluff; he has no problem admitting what he doesn’t know and is very respectful to his peers, while not tolerating fools.

 

Toth points out that while 2 observers can experience different durations in time (like the twins in the twin paradox) they agree on the time when they meet again. In other words, there is a time reference (like a space reference) that’s independent of the path they took to get there. As I pointed out in my post explaining relativity based on waves, it’s not only the time duration that 2 observers disagree on, but also the space duration. If someone was able to travel so fast that they could cross the galaxy in years instead of thousands of years, then they would also traverse a much shorter distance (according to them). In other words, not only does time shrink, but so does distance. We don’t tend to think that space can be just as rubbery as time.

 

And this brings me to a point that Toth and Greene seem to disagree on. Toth is adamant that space is not an entity but just the ‘distance’ between physical objects. Regarding the expanding universe, he says ‘space’ does not ‘stretch’ but it’s just that the ‘distance’ between ‘objects’ increases. Greene would probably disagree. Greene argues that wavelengths of light lengthen as space ‘stretches’.

 

What do I think? I tend to side with Greene. I think space is an entity because it has dimensions. John Barrow, in his book, The Constants of Nature, gives an excellent account of how a universe that didn’t exist in 3 dimensions would be virtually unworkable. In particular, the inverse square law for gravity, that keeps planets in stable orbits for hundreds of millions of years, would not work in any other dimensional universe.

 

But also, space can be curved by gravity, or more specifically, spacetime, as Toth readily acknowledges. I’ll return to Toth’s specific commentary on gravity and black holes later.

 

I’ve already mentioned that Greene discussed the age of the Universe. I recently did an online course provided by New Scientist on The Cosmos, and one of the lecturers was Chris Impey, Distinguished Professor, Department of Astronomy, University of Arizona. He made the point that the Universe has an ‘edge in time’, but not an edge in space. Greene expanded on this, by pointing out that everywhere in the Universe all clocks are ‘in synchronicity’, which contradicts the notion that there is no universal now. I’ll quote Greene directly on this, because it’s an important point. According to Greene (but not only Greene) whether an observer is moving away from a distant stellar object, or towards it, determines whether they would see into that object’s distant past or distant future. Mind you, because they are so far away, the object’s future is still in our past.

 

Each angled slice intersects the Universe in a range of different epochs and so the slices are far from uniform. This significantly complicates the description of cosmic history, which is why physicists and astronomers generally don’t contemplate such perspectives. Instead, they usually consider only the perspective of observers moving solely with the cosmic flow...

 

I don’t know the mathematics behind this, but I can think of an analogy that we all observe every day. You know when you walk along a street with the Sun low in the sky, so it seems to be moving with you. It will disappear behind a building then appear on the other side. And of course, another observer in another town will see the Sun in a completely different location with respect to their horizon. Does this mean that the Sun moved thousands of miles while you were walking along? No, of course not. It’s all to do with the angle of projection. In other words, the movement in the sky is an illusion, and we all know this because it happens all the time and it happens in sync with our own movements. I remember once travelling in a car with a passenger and we could see a plane low in the sky through the windscreen. My passenger commented that the plane was travelling really fast, and I pointed out that if we stopped, we’d find that the plane would suddenly slow down to a speed more commensurate with our expectations.

 

I think the phenomenon that Greene describes is a similar illusion, only we conjure it up mathematically. It makes sense to ignore it, as astronomers do (as he points out) because we don’t really expect it happens in actual fact.

 

Back in 2016, I wrote a post on a lecture by Allan Adams as part of MIT Open Courseware (8.04, Spring 2013) titled Lecture 6: Time Evolution and the Schrodinger Equation. This was a lecture for physics students, not for a lay audience. I found this very edifying, not least because it became obvious to me, from Adams’ exposition, that you could have a wave function with superposition as described by Schrodinger’s equation or you could have an observed particle (like an electron) but you couldn’t have both. Then I came across the famous quote by William Lawrence Bragg:

Everything that has already happened is particles, everything in the future is waves. The advancing sieve of time coagulates waves into particles at the moment ‘now’.

 

In that post on Adams’ lecture, I said how people (like Roger Penrose, among others) explained that ‘time’ in the famous time dependent Schrodinger equation exists outside the hypothetical Hilbert space where the wave function hypothetically exists. And it occurred to me that maybe that’s because the wave function exists in the future. And then I came across Freeman Dyson’s lecture and his unorthodox claim:

 

... the “role of the observer” in quantum mechanics is solely to make the distinction between past and future...

What really happens is that the quantum-mechanical description of an event ceases to be meaningful as the observer changes the point of reference from before the event to after it. We do not need a human observer to make quantum mechanics work. All we need is a point of reference, to separate past from future, to separate what has happened from what may happen, to separate facts from probabilities.

The inherent contradictions in attempting to incorporate classical physics into quantum mechanics (QM) disappear, if one is describing all the possible future paths of an event while the other describes what actually happened.

 

Viktor Toth, whom I’ve already mentioned, once made the interesting contention that the wave function is just a ‘mathematical construct’ (his words) and he’s not alone. He also argued that the ‘decoherence’ of the wave function is never observed, which infers that it’s already happened. Toth knows a great deal about QFC (and I don’t) but, if I understand him correctly, the field exists all the time and everywhere in spacetime.

 

Another contributor on Quora, whom I follow, is Mark John Fernee (PhD in physics, University of Queensland), who obviously knows a great deal more than me. He had this to say about ‘wave function collapse’ (or decoherence) which corroborates Toth.

 

The problem is that there is no means to detect the wavefunction, and consequently no way to detect a collapse. The collapse hypothesis is just an inference that can't be experimentally tested.

 

And, in a comment, he made this point:

 

In quantum mechanics, the measurement hypothesis, which includes the collapse of the wave function, is an irreversible process. As we perceive the world through measurements, time will naturally seem irreversible to us.

 

And I’ve made this exact same point, that we also get the ‘irreversibility’ or asymmetry between past and future, from QM physics becoming classical physics.

 

Dyson is also contentious when it comes to gravity and QM, arguing that we don’t need to combine them together and that, even if the graviton existed, it’s impossible to detect. Most physicists argue that we need a quantum gravity – that it’s the ‘missing link’ in a TOE (Theory of Everything).

 

It’s occurred to me that maybe the mathematics is telling us something when it appears obstructive to marrying general relativity with QM. Maybe they don’t go together, as Dyson intimated.

 

Again, I think Toth gives the best reasoning on this, as I elaborated on in another post.

 

We can do quantum field theory just fine on the curved spacetime background of general relativity.

Then he adds this caveat:

What we have so far been unable to do in a convincing manner is turn gravity itself into a quantum field theory.

 

Toth explains how and why they are ‘incompatible’, though he loathes that term, because he doesn’t think they’re incompatible at all.

 

Basically, Einstein’s field equations are geometrical, whereby one side of the equation gives the curvature of spacetime as a consequence of the energy, expressed on the other side of the equation. The energy, of course, can be expressed in QM, but the geometry can’t. 

 

Someone else on Quora, Terry Bollinger (retired Chief Scientist), explains this better than me:

 

It all goes back to that earlier point that GR is a purely geometric theory, which in turn means that the gravity force that it describes is also specified purely in terms of geometry. There are no particles in gravity itself, and in fact nothing even slightly quantum. Instead, you assume the existence of a smooth fabric called spacetime, and then start bending it. From that bending emerges the force we know as gravity.

 

Bollinger wrote a lengthy polemic on this, but I will leave you with his conclusion, because it is remarkably similar to Toth’s.

 

Even if you finally figure out a clever way to define a gravity-like quantum force that allows objects in spacetime to attract each other, what are those force particles traveling across?

 

He provides his own answer:

 

Underneath the quantum version, since like all of the other forces in the Standard Model this swarm-like quantum version of a gravity must ride on top of spacetime. (Emphasis in the original)

 

In other words, Bollinger claims you’ll have a redundancy: a quantum field gravity on top of a spacetime gravity. Spacetime provides the ‘background’ to QFT that Toth described, which doesn’t have to be quantum.

 

So what about quantum gravity that is apparently needed to explain black holes?

 

There is an inherent contradiction in relation to a black hole, and I’m not talking about the ‘information paradox’. There is a debate about whether information (meaning quantum information) gets lost in a black hole, which contradicts the conservation of quantum information. I’m not going to get into that as I don’t know enough about it.

 

There is a more fundamental issue: according to Toth (but not only Toth) the event horizon of a black hole is always in an observer’s future. Yet for someone at the event horizon, they could cross over it without even knowing they had done so, especially if it was a really big black hole and the tidal effects at the event horizon weren’t strong enough to spaghettify them. Of course, no one really knows this, because no one has ever been anywhere near a black hole event horizon.

 

The point is that, at the event horizon of a black hole, time stops according to an external observer, which is why, theoretically it’s always in their future. Toth makes this point many times. So, basically, for someone watching something fall into a black hole, it becomes frozen in time (at the event horizon). In fact, the Russian term for a black hole is ‘frozen star’.

 

What we can say is that light from any object gets red-shifted so much that it disappears even before it reaches the event horizon. But what about an observer at or near the event horizon looking back out at the Universe. Now, I don’t see any difference in this scenario to the twin paradox, only it’s in extremis. The observer at the event horizon, or just outside it, will see the whole universe pass by in their lifetime. Because, if they could come back and meet up with their twin, their twin would be a hologram frozen in time, thousands of years old. Now, Toth makes the point, that as far as an observer at or near the event horizon, the speed of light is still constant for them, so how can that be?

 

There is another horizon in the Universe, which is the theoretical and absolute practical limit that we can see. Because the Universe is expanding, there is a part of the Universe that is expanding faster than light, relative to us. Now, you will say, how is that possible? It’s possible because space can travel faster than light. Now Toth will confirm this, even though he claims space is not an entity. Note that some other observer in a completely different location, would see a different horizon, in the same way that sailors in different locations in the same ocean see different horizons.

 

So, a hypothetical observer, at the horizon of the Universe (with respect to us) would still see the speed of light as c relative to their spacetime. Likewise, an observer at the event horizon of a black hole also sees light as c relative to their spacetime.

 

Now most black holes, we assume, are spinning black holes and they drag an accretion disk around with them. They also drag space around with them. Is it possible then that the black hole drags space along with an observer across the event horizon into the black hole? I don’t know, but it would resolve the paradox. According to someone on Quora, Leonard Susskind argues that nothing ever crosses the event horizon, which is how he resolves the quantum information paradox.

 

This is a very lengthy post, even by my standards, but I need to say something quickly about entanglement. There appears to be a contradiction between relativity and entanglement, but not in practical terms. If there is a universal 'now', implicit in the Universe having an ‘edge in time’ (but not in space) and if QM describes the future, then entanglement is not a mystery, because it’s a correlation between events separated in space, but not in time. 

 

Entanglement involves a ‘decoherence’ in the wave function that predicts the state of a decoherence in the particle it’s entangled with, because they share the same wave function. Schrodinger understood this better than anyone else, because he realised that entanglement was an intrinsic consequence of the wave function. He famously said that entanglement was the defining characteristic of quantum mechanics.

 

But there is no conflict with relativity because the entangled particles, whatever they are, can’t be separated at any greater rate than the speed of light. However, when the correlation occurs, it appears to happen instantaneously. But this is no different to a photon always being in the future of whatever it interacts with, even if it crosses the observable universe. However, for someone who detects such a photon, they instantaneously see something in the distant past as if there has been a backward-in-time connection to its source.

 

There is no reason to believe that anything I’ve said is true and correct. I’ve tried to follow a simple dictum that nature abhors a contradiction and apply it to what I know about the Universe, while acknowledging there are lots of people who know a great deal more than me, who probably disagree.

 

I see myself as an observer on the boundary line of the history of ideas. I try to make sense of the Universe by reading and listening to people much cleverer than me, including people I have philosophical differences with.

Addendum 1: I referenced Paul Davies 1995 book, About Time; Einstein's Unfinished Revolution. I mentioned that Earth is travelling relative to the CMBR towards Pisces (at 350 km/s), and according to Davies:

This is about 0.1 percent of the speed of light, and the time-dilation factor is only about one part in a million. Thus, to an excellent approximation, Earth's historical time coincides with cosmic time, so we can recount the history of the universe contemporaneously with the history of the Earth, in spite of the relativity of time. Similar hypothetical clocks could be located everywhere in the universe, in each case in a reference frame where the cosmic background heat radiation looks uniform... we can imagine the clocks out there, and legions of sentient beings dutifully inspecting them. This set of imaginary observers will agree on a common time scale and a common set of dates for major events in the universe, even though they are moving relative to each other as a result of the general expansion of the universe. They could cross-check dates and events by sending each other data by radio; everything would be consistent. So cosmic time as measured by this special set of observers constitutes a type of universal time... It is the existence of this pervasive time scale that enables cosmologists to put dates to events in cosmic history - indeed, to talk meaningfully at all about "the universe" as a single system. (my emphasis)

Addendum 2: This is a PBS video, which gives the conventional physics view on time. I don't know who the presenter is, but it would be fair to say he knows more about this topic than me. He effectively explains Einstein's 'block universe' and why 'now' is considered totally subjective. Remember Einstein's famous words in a letter to the mother of a friend who had died:

We physicists know that the past, present and future is only a stubbornly persistent illusion.

This was a consequence of simultaneity being different for different observers, as I discussed in the main text, and is described in the video. It's important to point out that this does not undermine causality, so it refers to events that are not causally related. The video presenter goes on to point out that different observers will see different pasts and different futures on worlds far far away, dependent on their motion on this world. This infers that all events are predetermined, which is what Einstein believed, and explains why so many physicists claim that the Universe is deterministic. But it contradicts the view, among cosmologists, that the Universe has an 'edge in time but not in space'.

It's certainly worth watching the video. Curiously, his logic leads him to the conclusion that we live in a quantum multiverse (the many worlds interpretation of QM). I agree with him that different observers in different parts of the Universe must have different views of 'Now'. That's just a logical consequence of the finite speed of light. Motion then distorts that further, as he demonstrates. My view is that what we perceive is not necessarily what actually 'is'. If one looks at the clock of a moving observer their time is dilated compared to ours, and likewise they see our clocks showing time dilation compared to them. But logic tells us that they both can't be right. The twin paradox is resolved only if one acknowledges that time dilation is an illusion for one observer but not the other. And that's because one of the twin travels relative to an absolute spacetime if not an absolute space or an absolute time.

Back to the video, my contention is that one observer can't see another observer's future, even though we can see another observer's 'present' in our 'past'; I don't find that contentious at all.

The Twin paradox, from both sides now (with apologies to Joni)

 I will give an exposition on the twin paradox, using an example I read in a book about 4 decades ago, so I’m relating this from memory.

Imagine that one of the twins goes to visit an extra-terrestrial world 20 light years away in a spaceship that can travel at 4/5 the speed of light. The figures are chosen because they are easy to work with and we assume that acceleration and stopping are instantaneous. We also assume that the twin starts the return journey as soon as they arrive at their destination.

 

From the perspective of the twin on Earth, the trip one-way takes 25 years because the duration is T = s/v, where s = 20 (light years) and v = 4/5c. 

So 5/4 x 20 = 25.

 

From the perspective of the twin on the spaceship, their time is determined by the Lorentz transformation (γ).

 

γ = 1/√(1 – v²/c²)

 

Note v²/c² = (4/5)² = 16/25

So √(1 – v²/c²) = √(9/25) = 3/5

 

Now true time for the space ship (τ) is given by τ = T/γ

So for the spaceship twin, the duration of the trip is 3/5 x 25 = 15

So the Earth twin has aged 25 years and the spaceship twin has aged 15 years.

 

But there is a relativistic Doppler effect, which can be worked out by considering what each twin sees when the spaceship arrives at its destination. 

 

Note that light, or any other signal, takes 20 years to come back from the destination. So the Earth twin will see the space ship arrive 25 + 20 = 45 years after it departed. But they will see that their twin is only 15 years older than when they left. So, from the Earth twin’s perspective, the Doppler effect is a factor of 3. (3 x 15 = 45). So the Doppler effect slowed time down by 3. Note: 45 years has passed but they see their twin has only aged 1/3 of that time.

 

What about the spaceship twin’s perspective? They took 15 years to get there, but the Doppler effect is a factor of 3 for them as well. They’ve been receiving signals from Earth ever since they left so they will see their twin only 5 years older because 15/3 = 5, which is consistent with what their twin saw. In other words, their Earth twin has aged 5 years in 15, or 1/3 of their travel time. 

 

If spaceship twin was to wait another 20 years for the signal to arrive then it would show Earth twin had aged 5 + 20 = 25 years at the time of their arrival. But, of course, they don’t wait, they immediately return home. Note that the twins would actually agree on each other’s age if they allowed for the time it takes light to arrive to their respective locations.

 

So what happens on the return trip? The Lorentz transformation is the same for the spaceship twin on the return trip, so they only age another 15 years, but according to the Earth twin the trip would take another 25 years, so they would have aged 50 years compared to the 30 years of their twin.

 

But what about the Doppler effect? Well, it’s still a factor of 3, only now it works in reverse, speeding time up. For the spaceship twin, their 15 years of observing their Earth twin is factored by 3 and 15 x 3 = 45. And 5 + 45 = 50, which is how much older their twin is when they arrive home.

 

For the Earth twin, their spaceship twin’s round trip is 50 years, so the return trip appears to only take 5 years. And allowing for the same Doppler effect, 3 x 5 = 15.

When the Earth twin adds 15 to the 15 years they saw after 45 years, they deduce the age of their spaceship twin is 30 years more (against their own 50). So both twins are in agreement.

 

Now, the elephant in the room is why do we only apply the Lorentz transformation to the spaceship twin? The usual answer to this question is that the spaceship twin had to accelerate and turn around to come back, so it’s obvious they did the travelling.

 

But I have another answer. The spaceship twin leaves the surface of Earth and even leaves the solar system. It’s obvious that the spaceship didn’t remain stationary while the solar system travelled through the cosmos at 4/5 the speed of light. There is an asymmetry to the scenario which is ultimately governed by the gravitational field created by everything, and dominated by the solar system in this particular case. In other words, the Lorentz transformation only applies to the spaceship twin, even when they only travel one way.

Relativity makes sense if everything is wavelike

 When I first encountered relativity theory, I took an unusual approach. The point is that c can always be constant while the wavelength (λ) and frequency (f ) can change accordingly, because c = λ x f. This is a direct consequence of v = s/t (where v is velocity, s distance and t time). We all know that velocity (or speed) is just distance divided by time. And λ represents distance while f represents 1/t. 

So, here’s the thing: it occurred to me that while wavelength and frequency would change according to the observer’s frame of reference (meaning relative velocity to the source), the number of waves over a specific distance would be the same for both, even though it’s impossible to measure the number of waves. And a logical consequence of the change in wavelength and frequency is that the observers would ‘measure’ different distances and different periods of time.

 

One of the first confirmations of relativity theory was to measure the half-lives of cosmic rays travelling through the Earth’s atmosphere to reach a detector at ground level. Measurements showed that more particles arrived than predicted by their half-life when stationary. However, allowing for relativistic effects (as the particles travelled at high fractional lightspeeds), the number of particles detected corresponded to time dilation (half-life longer, so more particles arrived). This means from the perspective of the observers on the ground, if the particles were waves, then the frequency slowed, which equates to time dilation - clocks slowing down. It also means that the wavelength was longer so the distance they travelled was further. 

 

If the particles travelled slower (or faster), then wavelength and frequency would change accordingly, but the number of waves would be the same. Of course, no one takes this approach - why would you calculate the Lorentz transformation on wavelength and frequency and multiply by the number of waves, when you could just do the same calculation on the overall distance and time.

 

Of course, when it comes to signals of communication, they all travel at c, and changes in frequency and wavelength also occur as a consequence of the Doppler effect. This can create confusion in that some people naively believe that relativity can be explained by the Doppler effect. However, the Doppler effect changes according to the direction something or someone is travelling while relativistic effects are independent of direction. If you come across a decent mathematical analysis of the famous ‘twin paradox’, you’ll find it allows for both the Doppler effect and relativistic effects, so don’t get them confused.

 

Back to the cosmic particles: from their inertial perspective, they are stationary and the Earth with its atmosphere is travelling at high fractional lightspeed relative to them. So the frequency of their internal clock would be the same as if they were stationary, which is higher than what the observers on the ground would have deduced. Using the wave analogy, higher frequency means shorter wavelength, so the particles would ‘experience’ the distance to the Earth’s surface as shorter, but again, the number of waves would be the same for all observers.

 

I’m not saying we should think of all objects as behaving like waves - despite the allusion in the title - but Einstein always referred to clocks and rulers. If one thinks of these clocks and rulers in terms of frequencies and wavelengths, then the mathematical analogy of a constant number of waves is an extension of that. It’s really just a mathematical trick, which allows one to visualise what’s happening.

Quantum mechanics, entanglement, gravity and time

I wrote a post on Louisa Gilder’s well researched book, The Age of Entanglement, 10 years ago, when I acquired it (copyright 2008). I started rereading it after someone on Quora, with more knowledge than me, challenged the veracity of Bell’s theorem, also known as Bell’s Inequality, which really changed our perception of quantum phenomena at its foundations. Gilder’s book is really all about Bell’s theorem and its consequences, whilst covering the history of quantum mechanics over most of the 20^th Century, from Bohr through to Feynman and beyond.

Gilder is not a physicist, from what I can tell, yet the book is very well researched with copious notes and references, and she garnered accolades from science publications as well as literary reviewers. Her exposition on Bell’s theorem is technically correct to the best of my knowledge, which she provides very early in the book. 

She goes to some length to explain that the resolution of Bell’s theorem is not the obvious intuitive answer that entangled particles are like a pair of shoes separated in space and time, so that if you find the right-handed shoe you automatically know that the other one must be left-handed. This is what my interlocutor on Quora was effectively claiming. No, according to Gilder, and everything else I’ve read on this subject, Bell’s theorem is akin to finding too many coincidences than one would expect to find by chance. The inequality means that if results are found on one side of the inequality then the intuitive scenario is correct, and if they are on the other side, then the QM world obeys rules not found in classical physics.

The result is called ‘non-local’, which is the opposite of ‘local’, a term with a specific meaning in QM. Local means that objects are only affected by ‘signals’ that travel at the speed of light. Non-local means that objects show a connectivity that is not dependent on lightspeed communication or linkage.

It was Schrodinger who coined the term ‘entanglement’, claiming that it was the defining characteristic of QM.

I would not call that ‘one’ but rather ‘the’ characteristic trait of quantum mechanics. The one that enforces its entire departure from classical lines of thought.

I’ve also recently read an e-book called An Intuitive Approach to Quantum Field Theory by Toni Semantana (only available in e-book, 2019), so it’s very recent. It’s very good in that Semantana obviously knows what he’s talking about, but, even though it has minimum mathematical formulae, it’s not easy to follow. Nevertheless, he covers esoteric topics like the Higgs field, gauge theories, Noether’s theorem (very erudite) and Feynman diagrams. It made me realise how little I know. It’s relevance to this topic is that he doesn’t discuss entanglement at all.

Back to Gilder, and it’s obvious that you can’t discuss entanglement and locality (or non-locality) without talking about time. If I can digress, someone else on Quora provided a link to an essay by J.C.N. Smith called Time – Illusion and Reality. Smith said you won’t find a definition of time that doesn’t include clocks or things that move. In fact, I’ve come across a few people who claim that, without motion, time has no reality. 

However, I have a definition that involves light. Basically, time is the separation between events as measured by light. This stems from the startling yet obvious fact, that if lightspeed was not finite (instantaneous) then everything would happen at once. And, because lightspeed is the same for all observers, it determines the time difference between events, even though the time measured may differ for different observers, as per Einstein’s special theory of relativity. (Spacetime between events for all observers is the same.)

When I was in primary school at the impressionable age of 10 or 11, I was introduced to relativity theory, without being told that is what it was. Nevertheless, it had such an impact on my still-developing brain that I’ve never forgotten it. I can’t remember the context, but the teacher (Mr Hinton) told us that if you travel fast enough clocks will slow down and so will your heart. I distinctly remember trying to mentally grasp the concept and I found that I could if time was a dimension and as you sped up the seconds, or whatever time was measured in, they became more frequent between each heartbeat, so, by comparison, your heart slowed down. One of the other students made the comment that ‘if a plane could fly fast enough it would come back to land before it took off’. I’m unsure if that was a product of his imagination or if he’d come across it somewhere else, which was the impression he gave at the time. Then, thinking aloud, I said, It’s impossible to go faster than time, as if time and speed were interdependent. And someone near me turned, in a light-bulb moment, and said, You’re right.

My attempt at conceptually grasping the concept was flawed but my comment was prescient. You can’t travel faster than time because you can’t travel faster than light. For a photon of light, time is zero. The link between time and light is an intrinsic feature of the Universe, and was a major revelation of Einstein’s theory of relativity.

J.C.N. Smith argues in his essay that we have the wrong definition of time by referring to local events like the rotation of the planet or its orbit about the sun, or, even more locally, the motions of a pendulum or an atomic clock. He argues that the definition of time should be the configuration of the entire universe, because at any point in time it has a unique configuration, and, even though we can’t observe it completely, it must exist. 

There is a serious problem with this because every observer of that configuration would see something completely different, even without relativistic effects. If you take the Magellanic Clouds, which you can see in the southern hemisphere with the naked eye on a cloudless, moonless night, you are looking 150,000 to 190,000 years into the past (there are 2 of them), which is roughly when homo sapiens emerged from Africa. So an observer on a world in the Magellanic Clouds, looking at the Milky Way galaxy, would see us 150,000 to 190,000 years in the past. In other words, no observer in the Universe could possibly see the same thing at the same time if they are far enough apart.

However, Smith is right in the sense that the age of the Universe infers that there is a universal ‘now’, which is the edge of the Big Bang (because it’s still in progress). The Cosmic Microwave Background Radiation is the earliest light we can see (from 380,000 years after the Big Bang) yet our observation of it is part of our ‘now’.

This has implications for entanglement if it’s non-local. If Freeman Dyson is correct that QM describes the future and classical physics describes the past, then the collapse or decoherence of the wave function represents ‘now’. So ‘now’ for an observer is when a photon hits your retina and you immediately see into the past, whether the photon is part of a reflection in a mirror or it comes from the Cosmic Background Radiation. It’s also the point when an entangled quantum particle (which could be a photon or something else) ‘fixes’ the outcome of its entangled partner wherever in the Universe it may be.

If entangled particles are in the future until one of them is observed then they infer a universal now. Or does it mean that it creates a link back in time across the Universe? 

John Wheeler believed that there was a possibility of a connection between an observer and the distant past across the Universe, but he wasn’t thinking of entanglement. He proposed a thought experiment involving the famous double-slit experiment, whereby one makes an observation after the particle (electron or photon) has passed through the slit but before it hits the target (where we observe the outcome). He predicted that this would change the pattern from a wave going through both slits to a particle going through one. He was later vindicated (after his death). Wheeler argued that this would imply that there is a ‘backwards-in-time’ signal or acausal connection to the source. He argued that this could equally apply to photons from a distant quasar, gravitationally lensed by an intervening galaxy.

Wheeler’s thought experiment makes sense if the wave function of the particle exists in the future until it is detected, meaning before it interacts with a classical physics object. Entanglement also becomes ‘known’ after one of the entangled particles interacts with a classical physics object. Signals into the so-called past are not so mysterious if everything is happening in the future of the ‘observer’. Even microwaves from the Cosmic Background Radiation exist in our future until we ‘detect’ them.

Einstein’s special theory of relativity tells us that simultaneity can’t be determined, which seems to contradict the non-locality of entanglement according to Bell’s theorem. According to Einstein, ‘now’ is subjective, dependent on the observer’s frame of reference. This implies that someone’s future could be another person’s past, but this has implications for causality. No matter where an observer is in the Universe, everywhere they look is in their past. Now, as I explained earlier, their past maybe different to your past but, because all observations are dependent on electromagnetic radiation, everything they ‘see’ has already happened.

The exception is the event horizon of a black hole. According to Viktor T Toth (a regular contributor to Quora), the event horizon is always in your future. This creates a paradox, because it is believed you could cross an event horizon without knowing it. On the other hand, an external observer would see you frozen in time. Kip Thorne argues there is no matter in a black hole, only warped spacetime. Most significantly, once you pass the event horizon, space effectively becomes uni-directional like time – you can’t go backwards the way you came.

As Toth has pointed out a number of times, Einstein’s theory of gravity (the general theory of relativity) is mathematically a geometrical theory. Toth also points out that We can do quantum field theory just fine on the curved spacetime background of general relativity. Another contributor, Terry Bollinger, explains why general relativity is not quantum:

GR is a purely geometric theory, which in turn means that the gravity force that it describes is also specified purely in terms of geometry. There are no particles in gravity itself, and in fact nothing even slightly quantum.

In effect, Bollinger argues that quantum phenomena ‘sit’ on top of general relativity. I contend that gravity ultimately determines the rate of time, and QM uses a ‘clock’ that exists outside of Hilbert space where QM ‘sits’ (according to Roger Penrose, as well as Anil Ananthaswamy, who writes for New Scientist). 

So what happens inside a black hole, which requires a theory of quantum gravity? As Freeman Dyson observed, no one can get inside a black hole to report or perform an experiment. But, if it’s always in one’s future, then maybe quantum gravity has no time. John Wheeler and Bryce de-Witt famously attempted to formulate Einstein’s theory of general relativity (gravity) in the same form as electromagnetism, and time (denoted as t) simply disappeared. And as Paul Davies pointed out in The Goldilocks Enigma, in quantum cosmology (as per the Wheeler de-Witt equation), time vanishes. But, if quantum cosmology is attempting to describe the future, then maybe one should expect time to disappear.

Another thought experiment: if you take an entangled particle to the other side of the visible universe (which would take something like the age of the Universe) and then they instantly ‘link’ or ‘connect’ non-locally, it still requires less than lightspeed to separate them. So you won’t achieve instantaneous transmission, even in principle, because you have to wait until its entangled ‘partner’ arrives at its destination. Or, as explained in the video below, the 'correlation' can only be checked in classical physics.

Addendum: This is the best explanation of QM entanglement and Bell's Theorem (for laypeople) that I've seen:

The Lagrangian – possibly the most fundamental mathematical principle in physics

This is something I wrote on Quora, which was ‘upvoted’ by a physics tutor (Mike Milner) and someone with an MSc (Dimitrios Kalemis), which gives it some credence.

I’ve written about all of this before in previous posts, but probably not as succinctly, which hopefully makes it easier to follow.


How does an electron know beforehand that it's a single slit or double slit so it decides whether to create an interference pattern or not?

Obviously it doesn’t. It’s like asking how does a ball thrown in the air know what path to follow? These 2 questions have more in common than you might think.

There is a fundamental principle in physics called the principle of least action, and Richard Feynman used it to describe the trajectory of a ball in a gravitational field and also as the basis for his path integral method of quantum mechanics (QM).

The principle of least action is that the difference between the potential energy and the kinetic energy of a particle will always be a minimum and, mathematically, this is called a Lagrangian. In his book, Six Not-So-Easy Pieces, Feynman demonstrates how this applies to a body in a gravitational field when it follows the path dictated by a geodesic, which, in Einstein’s theory of relativity, is the path of maximum relativistic time. It turns out that this is the shortest path and also the path of least action, as determined by the Lagrangian.

Feynman gives the following analogy. Imagine a lifesaver needing to run along a beach and then swim out to rescue a bather in distress in the surf. The lifesaver could run along the beach (at a diagonal) until he (or she) is perpendicular to the swimmer in the waves and swim out. Or the lifesaver could run straight into the surf and swim diagonally to the swimmer. But the optimum path is something in between these 2 and that’s the path of least action or least time. It’s also the path of light when it refracts through glass or any other medium.

It was Paul Dirac who originally wrote a Lagrangian for QM and Feynman used his result to derive Schrodinger’s equation. Feynman’s approach to the 2 slit problem or any other QM problem was to combine all the possible paths the electron (or a photon) could take. By ‘combine’ this means adding all the phases of the wave function, most of which cancel each other out. Then, using Born’s rule, he derived the probabilities of where the electron would hit the screen on the other side of the slit(s).

In his book, QED, he provides a graphic demonstration using this method to derive the path of a photon hitting a mirror. He says ‘the light goes where the time is least.’

In response to your specific question, the electron’s path is only determined retrospectively after it hits the screen on the other side of the slit(s). Freeman Dyson (who collaborated with Feynman) argues that QM cannot describe the past but only the future. So prior to the electron hitting the screen, QM describes the probabilities of where it will go, which is mathematically dependent on it being able to go everywhere at once. If there are 2 slits then this means it can go through both and if there is only one slit then it can only go through one. So the observation made retrospectively confirms this.

Addendum: Sabine Hossenfelder gives a much more erudite exposition in this video. And I agree with her - it's the closest we have to a 'theory of everything'.

Understanding Einstein’s special theory of relativity

In imagining a Sci-Fi scenario, I found a simple way of describing, if not explaining, Einstein’s special theory of relativity.

Imagine if a flight to the moon was no different to flying half way round the world in a contemporary airliner. In my scenario, the ‘shuttle’ would use an anti-gravity drive that allows high accelerations without killing its occupants with inertial forces. In other words, it would accelerate at hyper-speeds without anyone feeling it. I even imagined this when I was in high school, believe it or not.

The craft would still not be able to break the speed of light but it would travel fast enough that relativistic effects would be observable, both by the occupants and anyone remaining on the Earth or at its destination, the Moon.

So what are those relativistic effects? There is a very simple equation for velocity, and this is the only equation I will use to supplement my description.

v = s/t

Where v is the velocity, s is the distance travelled and t is the time or duration it takes. You can’t get much simpler than that. Note that s and t have an inverse relationship: if s gets larger, v increases, but if t gets larger, v decreases.

But it also means that for v to remain constant, if s gets smaller then so must t.

For the occupants of the shuttle, getting to the moon in such a short time means that, for them, the distance has shrunk. It normally takes about 3 days to get to the Moon (using current technology), so let’s say we manage it in 10 hrs instead. I haven’t done the calculations, because it depends on what speeds are attained and I’m trying to provide a qualitative, even intuitive, explanation rather than a technical one. The point is that if the occupants measured the distance using some sort of range finder, they’d find it was measurably less than if they did it using a range finder on Earth or on the Moon. It also means that whatever clocks they were carrying (including their own heartbeats) they would show that the duration was less, completely consistent with the equation above.

For the people on the Moon awaiting their arrival, or those on Earth left behind, the duration would be consistent with the distance they would measure independently of the craft, which means the distance would be whatever it was all of the time (allowing for small variances created by any elliptic eccentricity in its orbit). That means they would expect the occupants’ clocks to be the same as theirs. So when they see the discrepancy in the clocks it can only mean that time elapsed slower for the shuttle occupants compared to the moon’s inhabitants.

Now, many of you reading this will see a conundrum if not a flaw in my description. Einstein’s special theory of relativity infers that for the occupants of the shuttle, the clocks of the Moon and Earth occupants should also have slowed down, but when they disembark, they notice that they haven’t. That’s because there is an asymmetry inherent in this scenario. The shuttle occupants had to accelerate and decelerate to make the journey, whereas the so-called stationary observers didn’t. This is the same for the famous twin paradox.

Note that from the shuttle occupants’ perspective, the distance is shorter than the moon and Earth inhabitants’ measurements; therefore so is the time. But from the perspective of the moon and Earth inhabitants, the distance is unchanged but the time duration has shortened for the shuttle occupants compared to their own timekeeping. And that is special relativity theory in a nutshell.


Footnote: If you watch videos explaining the twin paradox, they emphasise that it’s not the acceleration that makes the difference (because it’s not part of the Lorentz transformation). But the acceleration and deceleration is what creates the asymmetry that one ‘moved’ respect to another that was ‘stationary’. In the scenario above, the entire solar system doesn’t accelerate and decelerate with respect to the shuttle, which would be absurd. This is my exposition on the twin paradox.

Addendum 1: Here is an attempted explanation of Einstein’s general theory of relativity, which is slightly more esoteric.

Addendum 2: I’ve done a rough calculation and the differences would be negligible, but if I changed the destination to Mars, the difference in distances would be in the order of 70,000 kilometres, but the time difference would be only in the order of 10 seconds. You could, of course, make the journey closer to lightspeed so the effects are more obvious.

Addendum 3: I’ve read the chapter on the twin paradox in Jim Al-Khalili’s book, Paradox: The Nine Greatest Enigmas in Physics. He points out that during the Apollo missions to the moon, the astronauts actually aged more (by nanoseconds) because the time increase by leaving Earth’s gravity was greater than any special relativistic effects experienced over the week-long return trip. Al-Khalili also explains that the twin who makes the journey, endures less time because the distance is shorter for them (as I expounded above). But, contrary to the YouTube lectures (that I viewed) he claims that it’s the acceleration and deceleration creating general relativistic effects that creates the asymmetry.

The Centenary of Einstein’s General Theory of Relativity

This month (November 2015) marks 100 years since Albert Einstein published his milestone paper on the General Theory of Relativity, which not only eclipsed Newton’s equally revolutionary Theory of Universal Gravitation, but is still the cornerstone of every cosmological theory that has been developed and disseminated since.

It needs to be pointed out that Einstein’s ‘annus mirabilis’ (miraculous year), as it’s been called, occurred 10 years earlier in 1905, when he published 3 groundbreaking papers that elevated him from a patent clerk in Bern to a candidate for the Nobel Prize (eventually realised of course). The 3 papers were his Special Theory of Relativity, his explanation of the photo-electric effect using the newly coined concept, photon of light, and a statistical analysis of Brownian motion, which effectively proved that molecules made of atoms really exist and were not just a convenient theoretical concept.

Given the anniversary, it seemed appropriate that I should write something on the topic, despite my limited knowledge and despite the plethora of books that have been published to recognise the feat. The best I’ve read is The Road to Relativity; The History and Meaning of Einstein’s “The Foundation of General Relativity” (the original title of his paper) by Hanoch Gutfreund and Jurgen Renn. They have managed to include an annotated copy of Einstein’s original handwritten manuscript with a page by page exposition. But more than that, they take us on Einstein’s mental journey and, in particular, how he found the mathematical language to portray the intuitive ideas in his head and yet work within the constraints he believed were necessary for it to work.

The constraints were not inconsiderable and include: the equivalence of inertial and gravitational mass; the conservation of energy and momentum under transformation between frames of reference both in rotational and linear motion; and the ability to reduce his theory mathematically to Newton’s theory when relativistic effects were negligible.

Einstein’s epiphany, that led him down the particular path he took, was the realisation that one experienced no force when one was in free fall, contrary to Newton’s theory and contrary to our belief that gravity is a force. Free fall subjectively feels no different to being in orbit around a planet. The aptly named ‘vomit comet’ is an aeroplane that goes into free fall in order to create the momentary sense of weightlessness that one would experience in space.

Einstein learnt from his study of Maxwell’s equations for electromagnetic radiation, that mathematics could sometimes provide a counter-intuitive insight, like the constant speed of light.

In fact, Einstein had to learn new mathematics (for him) and engaged the help of his close friend, Marcel Grossman, who led him through the technical travails of tensor calculus using Riemann geometry. It would seem, from what I can understand of his mental journey, that it was the mathematics, as much as any other insight, that led Einstein to realise that space-time is curved and not Euclidean as we all generally believe. To quote Gutfreund and Renn:

[Einstein] realised that the four-dimensional spacetime of general relativity no longer fitted the framework of Euclidean geometry… The geometrization of general relativity and the understanding of gravity as being due to the curvature of spacetime is a result of the further development and not a presupposition of Einstein’s formulation of the theory.

By Euclidean, one means space is flat and light travels in perfectly straight lines. One of the confirmations of Einstein’s theory was that he predicted that light passing close to the Sun would be literally bent and so a star in the background would appear to shift as the Sun approached the same line of sight for an observer on Earth as for the star. This could only be seen during an eclipse and was duly observed by Arthur Eddington in 1919 on the island of Principe near Africa.

Einstein’s formulations led him to postulate that it’s the geometry of space that gives us gravity and the geometry, which is curved, is caused by massive objects. In other words, it’s mass that curves space and it’s the curvature of space that causes mass to move, as John Wheeler famously and succinctly expounded.

It may sound back-to-front, but, for me, Einstein’s Special Theory of Relativity only makes sense in the context of his General Theory, even though they were formulated in the reverse order. To understand what I’m talking about, I need to explain geodesics.

When you fly long distance on a plane, the path projected onto a flat map looks curved. You may have noticed this when they show the path on a screen in the cabin while you’re in flight. The point is that when you fly long distance you are travelling over a curved surface, because, obviously, the Earth is a sphere, and the shortest distance between 2 points (cities) lies on what’s called a great circle. A great circle is the one circle that goes through both points that is the largest circle possible. Now, I know that sounds paradoxical, but the largest circle provides the shortest distance over the surface (we are not talking about tunnels) that one can travel and there is only one, therefore there is one shortest path. This shortest path is called the geodesic that connects those 2 points.

A geodesic in gravitation is the shortest distance in spacetime between 2 points and that is what one follows when one is in free fall. At the risk of information overload, I’m going to introduce another concept which is essential for understanding the physics of a geodesic in gravity.

One of the most fundamental principles discovered in physics is the principle of least action (formulated mathematically as a Lagrangian which is the difference between kinetic and potential  energy). The most commonly experienced example would be refraction of light through glass or water, because light travels at different velocities in air, water and glass (slower through glass or water than air). The extremely gifted 17th Century amateur mathematician, Pierre de Fermat (actually a lawyer) conjectured that the light travels the shortest path, meaning it takes the least time, and the refractive index (Snell’s law) can be deduced mathematically from this principle. In the 20th Century, Richard Feynman developed his path integral method of quantum mechanics from the least action principle, and, in effect, confirmed Fermat’s principle.

Now, when one applies the principle of least action to a projectile in a gravitational field (like a thrown ball) one finds that it too takes the shortest path, but paradoxically this is the path of longest relativistic time (not unlike the paradox of the largest circle described earlier).

Richard Feynman gives a worked example in his excellent book, Six Not-So-Easy Pieces. In relativity, time can be subjective, so that a moving clock always appears to be running slow compared to a stationary clock, but, because motion is relative, the perception is reversed for the other clock. However, as Feynman points out:

The time measured by a moving clock is called its “proper time”. In free fall, the trajectory makes the proper time of an object a maximum.

In other words, the geodesic is the trajectory or path of longest relativistic time. Any variant from the geodesic will result in the clock’s proper time being shorter, which means time literally slows down. So special relativity is not symmetrical in a gravitational field and there is a gravitational field everywhere in space. As Gutfreund and Renn point out, Einstein himself acknowledged that he had effectively replaced the fictional aether with gravity.

This is most apparent when one considers a black hole. Every massive body has an escape velocity which is the velocity a projectile must achieve to become free of a body’s gravitational field. Obviously, the escape velocity for Earth is larger than the escape velocity for the moon and considerably less than the escape velocity of the Sun. Not so obvious, although logical from what we know, the escape velocity is independent of the projectile’s mass and therefore also applies to light (photons). We know that all body’s fall at exactly the same rate in a gravitational field. In other words, a geodesic applies equally to all bodies irrespective of their mass. In the case of a black hole, the escape velocity exceeds the speed of light, and, in fact, becomes the speed of light at its event horizon. At the event horizon time stops for an external observer because the light is red-shifted to infinity. One of the consequences of Einstein’s theory is that clocks travel slower in a stronger gravitational field, and, at the event horizon, gravity is so strong the clock stops.

To appreciate why clocks slow down and rods become shorter (in the direction of motion), with respect to an observer, one must understand the consequences of the speed of light being constant. If light is a wave then the equation for a wave is very fundamental:

v = f λ , where v is velocity, f is the frequency and λ is the wavelength.

In the case of light the equation becomes c = f λ , where c is the speed of light.

One can see that if c stays constant then f and λ can change to accommodate it. Frequency measures time and wavelength measures distance. One can see how frequency can become stretched or compressed by motion if c remains constant, depending whether an observer is travelling away from a source of radiation or towards it. This is called the Doppler effect, and on a cosmic scale it tells us that the Universe is expanding, because virtually all galaxies in all directions are travelling away from us. If a geodesic is the path of maximum proper time, we have a reference for determining relativistic effects, and we can use the Doppler effect to determine if a light source is moving relative to an observer, even though the speed of light is always c.

I won’t go into it here, but the famous twin paradox can be explained by taking into account both relativistic and Doppler effects for both parties – the one travelling and the one left at home.

This is an exposition I wrote on the twin paradox.