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Wednesday, 5 May 2021

The Dark is Already There

Terry Pratchett- The Light Fantastic
A little bit of knowledge can be a dangerous thing.

A bit of physics fun today, methinks...

While trying to find my way back to writing regularly, I've been trawling some of the things I've talked about over the years elsewhere, and I came across something interesting and fun. I was involved in a discussion about some research being carried out by Eagleworks, NASA's advanced propulsion physics laboratory at Johnson Space Centre.

The research was focused on whether warp technology was feasible. The discussion was started by a person who was incredulous NASA would engage with this sci-fi pseudoscience and take it seriously. So, is it pseudoscience? There's a lot of ground to cover in answering this question, so we'd better hit the warp button and get on with it...

Ensign Crusher...

So, where to begin? Let's start with where the accusation of pseudoscience comes from and what motivates it.

We've given a fair bit of ground over the history of this tiny corner of the web to a particular scientific paper published in 1905 Annalen der Physik, a German physics journal and one of the oldest physics journals in the world. This paper shook the world of physics, overturning one of the most successful scientific theories in history, and giving us an entirely new way of looking at reality.

I won't delve into the Special Theory of Relativity (SR) in any depth here, as I've covered it in quite a bit of detail elsewhere. However, a quick précis of the relevant hit points is in order.

The central feature of SR is the unification of space and time into a single, dynamic framework; spacetime. This has some critical implications for, among other things, motion. In particular, it imposes a de facto speed limit on 'travel' through space. The scare quotes there are important, for reasons we'll come back to, as they'll have implications in terms of what it means to go from one place to another in the context of physics.

Let's revisit the analogy we saw in The Idiot's Guide to Relativity:


Here we see two cars on a racetrack. If we imagine this as a graph, with the direction from left to right as one dimension, and the direction from bottom to top as another dimension, and we further picture both of these cars hitting the start line at the same time and speed, it's intuitively obvious car 1 will get to the finish first, because it's only travelling through one dimension, from left to right. Car 2, because its travel is through two dimensions simultaneously, will get there slightly later. In other words, his motion in the vertical direction reduces his motion in the horizontal direction. The more of his motion in the vertical, the less motion in the horizontal. They're counter-correlated. As one increases, the other is reduced. It's fairly easy to work out exactly what the difference in time will be just based on the speed and the distance between start and finish. 

In SR, because spacetime is a single unified entity, we can begin, with a little careful thought, to tease out the notion any travel through one dimension reduces travel in all other dimensions. In particular, if we treat all spatial dimensions as one unified dimension*, we can treat any direction and speed with a single vector**. From there, we can revisit the diagram, and substitute our horizontal (\(x\)) direction (axis) and our vertical (\(y\)) axis, and label them \(s\) and \(t\) respectively, denoting space and time.

Now, with the addition of a single principle, we're set to see how SR works in a really straightforward way. Here it is:
The speed of light (\(c\)) is the same for all observers, regardless of how the observer or the source of the light are moving.
As we've discussed previously, this was the principle underpinning the formulation of SR. The term for the speed of light appeared in Maxwell's field equations for electromagnetism, with no specification for motion other than this speed. In the early part of the 20th century, while others were trying to measure differences in the speed of light and, on the back of the Michelson-Morley experiment (MME) in 1887, trying to work out what was wrong with the experiment, Einstein ran with the notion \(c\) was constant in all inertial frames, and determined, in order for this to be true, space and time would have to be merged into a single, unified entity, through which all observers (conscious or otherwise) were travelling at a single fixed speed, a speed corresponding to the speed light travels through space. This speed, and this speed only, is the core of the unification of spacetime. In fact, although we talk about light in this context, light isn't at all special. Photons, the quanta - constituent 'packets' - of light, are merely the things we can most readily point to which travel at this speed. There are other principles, especially relating to mass and momentum, but their explication takes us too far afield for today's purpose. Again, this is covered in the larger article on SR, for which a link will be found at the bottom.

This brings us nicely back to where we started, and we can consider out diagram again, now with space on the \(x\) axis and time on the \(y\) axis, which we're now representing with \(s\) and \(t\) respectively. It's worth noting \(s\) is also the term we use for this speed, the combined speed through spacetime at which everything travels.

Let's picture our cars now, both of them travelling at the same fixed speed as before, but now we're talking about a fixed rate \(s\) through spacetime. 

Car 1 is only travelling along the \(s\) axis, meaning it isn't travelling at all along the \(t\) axis. Car 2, on the other hand, is travelling partially along the \(s\) axis and partially along the \(t\) axis.

For car 1, time has stopped. All its motion is through space, and none of its motion is through time. For car 2, time has slowed down, as some of its motion is through space. If we were to add a third car, travelling orthogonally to the \(s\) axis, all its motion would be through time, which corresponds to standing still in space.

There are some subtleties beyond this, especially concerning the notion mentioned in the first footnote below, namely each observer has equal claim to being at rest, and also concerning mass and momentum. These are dealt with in the main SR piece. Those caveats aside, this is how we arrive at a speed limit; the absolute limit of travel through space.

Is it correct? Are any of our theories? We've learned repeatedly throughout these pages one of the core foundations of modern scientific epistemology (the study of knowledge); it must be possible, at least in principle, to prove a theory or other scientific statement to be false. This is known as falsifiability. If an idea cannot be falsified in principle, it isn't possible even to test it, thus it can't even be considered a scientific idea. To hold up any scientific idea or principle as unassailable is to misunderstand, at a foundational level, what science is and how it works. Even physical laws cannot be said to be inviolable, as we're going to rediscover shortly.

So now the groundwork is laid, let's move on. Time to generalise.

There was an obvious problem with Einstein's work on SR. It disagreed with Newton in fundamental ways not pertaining directly. Particularly, Newton's theory was predicated on certain assumptions; First, time moved at the same rate for everybody everywhere meaning, if it was 12:00pm UTC (the standard term for Greenwich Mean Time), this was just as true for an observer on the other side of the known universe as it was for an observer in London. Second, gravity propagated instantaneously. Both of these assumptions were further rooted in the assumption time and space were independent, fixed and immutable. SR shows us this cannot be the case. SR tells us spacetime has to stretch and squeeze just to accommodate the relative motions of observers. 

But Einstein's theory wasn't about gravity. In fact, for reasons we'll discover, SR isn't even capable of incorporating gravity, (it's what, in fact, makes it 'special'). However, for the above reasons, Newton's picture of gravity was simply and fundamentally incompatible with SR. A further incompatibility arises when considering the nature of light. Newton's explanation for light was rooted in Boyle's idea of a 'luminiferous aether', the detection of which was the object of the MME, which we met earlier. It's unclear whether Einstein's thought was driven at all by this, possibly the most famous and far-reaching null result in the history of science since the death of phlogiston theory, though he certainly knew of it. MME falsified the aether hypothesis, which cast serious doubt on some of Newton's core ideas about the nature of light. Newton's core ideas about light were already in serious jeopardy anyway, after Thomas Young's famous 'double-slit' experiment, which seemed to rule out the corpuscular theory††.

Putting all this together, it was clear we needed a new model for gravity and, in the ten years following the publication of SR, Einstein turned his thought to resolving this incompatibility.

Einstein is on record as having the breakthrough or, as he called it 'my happiest thought' when thinking about somebody in an elevator. He thought about what experiments a person in an elevator with no information about the outside of the elevator could do to distinguish between different circumstances. One could not, he reasoned, even in principle, conduct an experiment capable of distinguishing between being stationary on the ground and being accelerated in space at a rate of 1G. He further reasoned one could not, even in principle, conduct an experiment capable of distinguishing between being adrift in space (i.e., with no forces acting upon the elevator) and the elevator falling in its shaft. 

Thus, he concluded, being accelerated and standing in a gravitational well are equivalent. As a corollary, drifting in space free of any forces and freefall are equivalent.

This is the famous Equivalence Principle, and it is, to my mind, the beating heart of General Relativity, at least from the perspective of gaining the right perspective. It meant, among other things, gravity isn't a force pulling you down, it's a force pushing you up (to the extent it's a force at all; this is a topic of discussion in itself). Those with strictly Newtonian views of the world may be suffering from some cranial dissolution at this point, because I seem to be saying a 
body in freefall experiences no forces; a falling body is not subject to gravity.

Wait, what? Surely not! 

Let's see if we can tease out what's going on. We need a question; the right kind of question. Luckily, I have one to hand:

What shape is a raindrop?

There's a trap for the unwary in the question, so it's worth considering carefully. A Newtonian intuition might tell you its shape should conform to something like the conceptual ideal of a teardrop, pinched at the top, with a larger bulk at the bottom. This is reasonable, of course, given the well-grasped inverse-square law. We know gravitational attraction decreases as the square of the distance, so the force has to be greater at the bottom. 

Is this really what happens, though? It's not easy to see raindrops properly without modern technology, so is there some process we can look at for a clue? 

There is indeed. To get to it, we need to shoot a plumber, err... checks notes... oops, I mean we need to talk about shot and Plumbum. Since the late 18th century, lead shot has been made by a process of dropping from a high tower while molten into a vat of water. These shot, to a first approximation, come out spherical every time. They usually need some attention, but they're not weighted to one area, they have uniform mass distribution. This tells you they were at their lowest energy state while falling, because the lowest energy state for a liquid is a sphere. This in turn tells you no forces were acting on it while falling, because any forces would change the distribution as our Newtonian intuition tells us. 

Shot towers are all over the world and were used well into the 20th century. The Wiki tells me the last one was built in the year I was born. Physicists use something very closely related even today, a drop tube, and it's precisely to study objects in weightless conditions. Recall from other discussions weight is the effect of gravity on mass, and it should hopefully join all the dots together. Freefall, weightlessness, and not being subject to gravity are the same thing. In freefall, there is no down.

Nowadays, we have technology, and we can look at raindrops in slow motion. I'll pop a video in at the bottom. As you'll easily see, to a first approximation, they're all pretty much perfectly spherical.

So how would it work, then? Well, we have the beginnings of an answer, because we have SR. SR is a theory rooted in geometry, so how could geometry be applied here?

The answer, it turns out (I'm skipping an awful lot here, it should be noted; I have an idea for a more complete post on GR, but it would make this post colossal), is curvature. SR isn't actually capable of handling curves or, more accurately, it is, but only in a very specific sense; it can accommodate curves in flat space, as the sums of vectors known as geodesics. What if it's spacetime curving in the presence of mass. What if spacetime is non-Euclidean?

Einstein reasoned that spacetime curves in the presence of mass.

To try to develop a visual intuition, let's imagine imposing a grid on spacetime. This is, in essence, what SR does for spacetime. It gives us a metric. A metric is something giving us a notion of distance between events‡‡, which in turn means we can make measurements. It's not an overstatement to say loudly here and now: this grid is entirely arbitrary. What we're doing is choosing a coordinate system; a preferred frame of reference. There is no preferred frame of reference but, as long as we're careful, we can pop a grid over spacetime in order to make some measurements.

You're free to imagine this as an infinite three-dimensional lattice moving freely (freefalling) in space, with the lines representing rulers and the junctions representing clocks. 

Now let's look at a gravity well. There are many ways to visualise this, and I'm sure we've all seen the classic illustrations of funnelled grids and bowling balls on rubber sheets, but I want to try a slightly different approach, as I feel like it develops a slightly better intuition, and certainly in the context of where I'm going with this post. This is just like the classic view of a gravity well, but we're going to look at it from 'above'.

This should give a clearer intuition than the popular view of a gravity well, not least because this is considerably closer to how you'd see your grid behaved if you imposed it over this portion of spacetime regardless of where you viewed it from (this pinching of space, in other words, is the same from every angle), with only the caveat all observers must agree on: units are measured consistently by all observers, which is to say a second and a metre, for example, are each the same length in every frame.

As we should be able to see, space is being 'pinched' by the mass of the planet, in four dimensions. The notion of 'down' doesn't make any sense at all except in the context of the gravity well. Down is specifically the centre of the pinching. To Mr Crusher in the elevator car, all his measurements will tell him there are no forces acting on him (although they're about to be in a massive way§; lol), and he's absolutely correct. Time moves for him at the usual rate, and all his length measurements are as expected.

To an outside observer in orbit, however, things will look different. In particular, suppose he could compare Wesley's watch with his own? He'd observe Wesley's watch progressively slowing down, compared to his own, a function of both an SR component and a GR component. Time runs slower for observers in motion, recall. Time also runs slower in a gravity well. Both these factors will increase, gradually slowing time down for Wes as measured by our outside observer. He'll also be slightly truncated in the direction of travel. These will be incredibly difficult to measure, but this is an in practice challenge rather than an in principle one. Bear in mind, however, his measurements are, other than employing the agreed upon definitions for length, just as arbitrary as Wes's. Neither is objectively correct, but neither is either one incorrect. We can, in fact, take all their respective measurements, apply a Lorentz transformation, and calculate 'proper time', time as measured by an observer moving between the frames.

The point to take away here is how mass (energy) warps spacetime. We can see, because of the curvature of spacetime, things will fall into gravity wells.

So how does this help? Well, we need one more piece to the puzzle, and it's an observational one. As we know from previous posts, the bits of the universe furthest away from us are receding from us at greater than \(c\). In the discussion which motivated this, it was asserted this was nonsense because we can't look at things outside our light horizon. The latter is true, but it's not the full picture. Objects we can still observe are moving away from us faster than light can travel. It might seem odd we can, but what we can do is measure changes in redshift (\(z\), the fractional change in wavelength) over time, and calculate their recessional velocity. Any object with a redshift greater than \(z=1.48\) is receding from us at greater than \(c\), observably. This tells us two points in space can 'move' apart from each other at velocities exceeding the purported speed limit of the universe It isn't the galaxies 'travelling'. Space itself is somehow immune to the speed limit imposed by SR. This could have interesting implications. Could space move through space at faster than \(c\)?

OK, so there are some deep caveats here before we get carried away. It does at least seem to provide a back door to... something. In particular, it can, with a little squinting, look like there's an escape to faster than light travel. Is there?

It's unclear, but some interesting results have turned up over the years of playing with GR. Exact solutions are difficult to find and horrendously difficult to calculate for most things (which is why we still use the incorrect Newtonian model; we didn't know it was incorrect until we had a more correct model showing the errors to be off down the decimal expansion a way). The first exact solution of note is the Schwarzschild Metric, the solution giving us the black hole. The Schwarzschild radius is the radius a body's mass must be entirely contained within in order for escape velocity to exceed \(c\), the definition of a black hole. This is a useful guide to an important notion; energy density. Recall from SR mass and energy are equivalent. We can say the positive energy density of a given amount of 'stuff' will pinch space to such a degree. 

So, suppose space really is immune to the lightspeed limitation? Could we travel faster than light?

And this is where we really need to look at this word; travel. It's problematic. In general, in this context, travel is taken to mean something occurring through space. I'm going to make a distinction here, and define travel as any journey, by any means, from one spacetime event to another. I'm going to use a different term, traverse, to talk about moving through space and having momentum (the momentum is important, because momentum plays a critical role in the speed limit, as this momentum \(p\) is the product of mass \(m\) and velocity \(v\); \(p=mv\)). In short, to treat travelling and traversing, as I've defined them here, as if they're the same, is like treating plesiosaurs as if they were mammals.

So now let's ask the question again. Could we traverse faster than light? No, almost certainly not. I say 'almost' for very good reasons, not the least of which is this speed limit, while a perfectly valid conclusion to draw from a theory we have good confidence in owing to voluminous observational data entirely in accord with it, can never circumvent the problem of induction. It is, however, a mathematical extrapolation of the theory, thus it can never be regarded as sacrosanct. Indeed, the notion of any idea ever being be regarded as sacrosanct is the absolute antithesis of science. It's as unscientific as it's possible to get, for reasons exposed in this very thread. The epistemic status of Newtonian gravity was unassailable for centuries, but it turned out to be only an approximation.

Could we travel? Is there some way to get from a to b quicker than light could get there through space? This is a different question, and it's where we're headed now.

There's another solution to Einstein's field equations worth looking at. Proposed in 1994 by a Mexican Ph.D. student studying for his doctorate at Cardiff University, the Alcubierre Metric is a speculative solution to the GR equations. Specifically, it takes the notion of warping space and runs with it.

He envisaged his grid, with positive energy density at some location (in fact, his representation was very different, but it's describing the same thing, so we'll stick with this for consistency).

This is all well and good. It's a gravity well. Nothing to see here at all. We know the solution for a gravity well is perfectly physical. All we need is a positive energy density (\(\Omega+\)) and we have a gravity well. Everything else is a matter of magnitude.

But what would happen if we could find a way to decrease the energy density to below the density of the vacuum..?

Hmm... this sounds odd. All else aside, we define the vacuum loosely as zero energy density (in fact, this is a massive simplification; for various reasons, the energy-density of spacetime cannot be zero, but it's a complication not really relevant to this discussion). Can there be such a thing as negative energy? There's the first of our problems. While negative energy density (\(\Omega-\)) is not a problem from a theoretical perspective (we just plug in the numbers and the theory spits out solutions, in essence), it's a huge problem from a practical perspective, because it's never been observed, and presents hurdles to our intuition making it seem insurmountable. Our intuitions are rarely the whole story, however. 

Let's pretend, for a second, negative energy density isn't an issue, and consider what it might look like. So let's get ourselves some spacetime and stick a ship in it:


Yes, I know. I had wanted to avoid all the Star Trek references. When I was making this diagram, I'd wanted to try to design it based on the earliest entry in fiction of the notion of warp, and I tracked it down on the Gutenberg project. Unfortunately, the design of the craft as described therein was a vague cylinder made of some exotic composite of condensed photons such you couldn't really make out the outline. 

In any event, let's follow Jean-Luc's instruction, and engage the warp drive, just as soon as we can persuade the ever-uncouth Commander Riker to get his feet off the fucking bridge furniture (it's not a rock you've tied your horse to, pardner).


Enterprise Engaged

This is what it might look like. As I've drawn it here, I've envisaged our craft inside a bubble of spacetime, what Trekkies might refer to as a 'warp shell'. There are theoretical reasons for this, mostly to do with those theoretical limitations imposed by SR. By having a bubble of space, we can construct (not suggesting this is physically feasible) in such a way the craft isn't traversing space at all, merely sitting still in a pocket of moving space, free of the theoretical constraints of SR. The challenges of grabbing onto space so such a bubble could be constructed are another matter, and it may be that they're not necessary (Alcubierre's metric included no such bubble). I've included it purely to address a potential theoretical complaint, by isolating the craft from the space outside the bubble.

Having a \(\Omega+\) region in front and a \(\Omega-\) region behind, a differential is created. As we know from all our lessons concerning energy distribution, this differential needs to equalise. The most direct route to equalisation is for the \(\Omega-\) region to flow into the \(\Omega+\) region, which in turn creates a moving wave of spacetime upon which we might theoretically surf. 

There's the theory. 

Does any of this look unscientific? Some of it may look odd, but it's well within the confines of current theory. There are some problems, such as the seemingly fatal requirement of already having an Alcubierre drive to be able to construct an Alcubierre drive, but this may not be the whole story. As with any good physical theory, there are implications to relativity, special and general, yet to be discovered. We also know, just as Newton's theory provided an incomplete picture, so does Einstein's, not least because Einstein's model can't deal with events at quantum scales, which may provide further solutions. One proposal for negative energy densities leans on the energy density differential manifest in the Casimir effect, for example. There's a lot of discussion about whether this is a real negative energy density or just a lower-value positive density than exists outside the plates. In future posts, we'll have a brief look at some recent papers delivering some hopeful news, such as an update to Alcubierre's model which, although it looks like it probably rules out warp for FTL, suggests warp may still be a workable means of subluminal propulsion, and has reduced the required negative energy density by orders of magnitude. It may even be, once we have a working quantum model of gravity, which would of necessity be a more complete picture than either SR or GR, all these considerations vanish in a puff of future research. The last is the killer answer to our question, however.

Is it pseudoscience? Most certainly not.

Conscious this is already considerably longer even than I'm usually given to, I'm going to leave this here for now. I haven't remotely covered all the ground I want to, so I'll use future posts to explore this in a little more depth and to look at other potential ways of travelling (rather than traversing) between two spacetime events at apparently superluminal speeds without running hard up against the constraints of relativity.

Nits, crits and comments welcome, as always.

Further reading:

The Idiot's Guide to Relativity - SR in a nutshell.
The Certainty of Uncertainty Quantum Mechanics, the Casimir effect and a potted history of our ideas about light.
Gravity and Freefall An experiment you can do to demonstrate the equivalence principle.
Deduction, Induction, Abduction and Fallacy How logic is used in the sciences and the demarcation between scientific, unscientific and pseudoscientific statements.
It Wasn't Big and it Didn't Bang. Pre-Planck cosmology, how the energy density relates to the geometry of spacetime
You Must Be Off Your Brane. Pre-Planck cosmology, M-Theory, brane-worlds and more history.
In On The Secret Cartesian coordinates and jargon demystified.

* There's good reason to do this, and it's all about another of the central lessons of Special Relativity, which is there are no preferred inertial frames, meaning any choice of coordinate systems is arbitrary. In the vernacular, this would best be expressed in the statement 'every observer can state - correctly - they are at rest; their position is the origin for any Cartesian representation of space'. For a deeper explanation of Cartesian coordinates, see In on the Secret.

** A vector is a quantity combining speed and direction, as opposed to a scalar, which is a quantity containing only speed. The statement 'he was travelling at 100 mph' is a scalar statement, which can be made into a vector by the addition of a direction such as 'going East'. 

† In fact, photons travel at this speed regardless of their energy. That might seem a slightly odd thing to say, but it's pointing to something important, which is that what we generally refer to as 'light' is a tiny, tiny band of the electromagnetic spectrum. The entire spectrum is made up of photons, and all travel at this speed. Of course, it would be easy enough to argue that there's a case for describing the whole electromagnetic spectrum as light. This does not mirror convention, however, so I'll avoid it, even if it does lead to the lovely conclusion that we are beings of light, because it would also describe the photons that bind our atoms together and give us a degree of permanence. They're also not alone, as all 'particles' (see ††) that have zero rest mass must travel at this speed in a vacuum. The point is that it's the speed that matters here, not the light. Indeed, light isn't actually playing any role in Einstein's equation.

†† The corpuscular theory was the idea that light came in corpuscles, or what we'd now call 'quanta'. In general, the term we use is 'particle', but this is misleading, for reasons we've talked about in posts about Quantum Mechanics. Only slightly less misleading is a face-value assessment of what replaced it, the so-called 'wave-particle' duality, which suggests that photons are both particle and wave, or 'wavicles'. This is wrong. A photon is neither a particle nor a wave, it's a photon!

‡ Plumbum is the Latin name for lead, and the reason that the chemical symbol for lead is Pb. It's also why people who fix pipes are called plumbers, because pipes were typically made of lead for most of the last couple of millennia. Plumb lines - devices used to ensure perpendicularity in architecture fir millennia, are named for the lead weight at the end that pulls them taut.

‡‡ It's worth noting what we mean by 'event' here. In the vernacular, we think of an event as something that happens. In this scheme, an event is not the thing it happens, but the precise spacetime coordinates at which it happens (indeed, there needn't be anything of note happening at the coordinates). It might seem like a trivial distinction, but it's critical, because these are the things we're measuring. An event in SR and GR is very specifically a geometric object whose existence is independent of any reference or label, including whatever may or may not occur at the event.

§ No Wesley Crushers were harmed in the making of these diagrams. Some physics dogmatists had their egos severely bruised, though.

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