Posts Tagged ‘special relativity’
gravitational mysteries 2 – it’s not a force, but…
this has something to do with it all…
So I mentioned in my previous post that the Moon is tidally locked to the Earth, keeping the same face to us, presumably for eons. I left it there, without an explanation from Dr Google or anyone else. Does it have anything to do with that gravity thing?
There’s an answer on Eos, an Earth Sciences magazine I’ve shamefully never heard of, so thanks to Caroline Hasler:
A tidally locked object rotates around its axis exactly once during its orbit around a host planet or star. This physical quirk affects many planets and moons, including Earth’s Moon…
This tidally locked state is a consequence of gravity. As the Moon orbits Earth, Earth’s gravity tugs at it. This force deforms the Moon, reshaping it from a perfect sphere into something a little more akin to an American football: slightly squashed at the poles, with a bulge at its equator facing Earth and another on its far side. The same sort of deformation manifests itself in Earth’s oceans, where the Moon’s tidal forces produce watery bulges that travel around Earth as it rotates, leading to alternating high and low tides.
So gravity, this curvature of space-time, deforms the moon, and presumably the oblate spheroid that is the Earth, and billions of other bits of flotsam and jetsam spewed out by our messy universe. Interestingly, though, Hasler describes gravity as ‘this force’, while so many others, such as Veritasium and the PBS SpaceTime presenter, insist that it’s not a force… the point being that it’s hard, it seems, even for those who understand the physics of gravity (though I sometimes wonder if anybody really does), not to describe it in forceful terms. Anyway it’s this gravity spacetime curvature that ‘forces’ the Moon to be locked into facing the Earth without ‘turning away’. And yet the Earth isn’t tidally locked to the Sun. This is partly because the Earth is too far away, and partly because it’s already tidally locked to the Moon.
Exoplanets have been found that, due to their closeness to their stars, are tidally locked to them. Mercury is apparently ‘semi-tidally locked’ to the Sun at present (it has what they call a 3:2 spin-orbit resonance, rotating 1.5 times for every orbit) but presumably this tidal locking will gradually unlock as Mercury spirals away from the Sun over time – an awful lot of time. Which suggests that planets like Earth are getting further from their stars, very very gradually. And so eventually the Moon will gradually unlock itself from the Earth. As to why this is happening, I don’t know – as yet. In the planets’ case, it’s probably because the Sun is gradually losing mass. You can’t get energy out of nothing.
I’m watching Leonard Susskind’s online lectures on special relativity – or rather, I’m watching the first lecture, and I’m already lost. I’ve also bought and had a go at Susskind and Friedman’s book, Special Relativity and Classical Field Theory: The Theoretical Minimum, but haven’t got very far. I’ll keep trying, I think, and then I’ll die. It’s the maths that tends to trip me up.
So let’s go with the book, which is easier to continually refer to. It starts by telling me that special relativity is all about reference frames.
In general, reference frames are about perspectives. Everybody’s perspective is different, due to the time and place of their birth, their upbringing, which has decisively affected their neural development and so forth. All very complex, so we’re narrowing the term to refer to location in time and space. In the Cartesian sense, we have spacial co-ordinates on three axes, x, y and z, as well as an origin from which we can measure distances. And then there’s a t axis for time. So at this stage we have to imagine that time is synchronised for everyone – same starting point and same rate.
So these reference frames, in terms of space, vary individually (we can shift them around) and from other reference frames. They can be moving or (relatively!) stationary. Time, though, seems a bit trickier:
The assumption that all clocks in all frames of reference can be synchronised seems intuitively obvious, but it conflicts with Einstein’s assumption of relative motion and the universality of the speed of light.
Susskind & Friedman, Special Relativity and Classical Field Theory, p5
So we have coordinates to pin down events. The laws governing those events are apparently the same in all inertial reference frames (IRFs) – i.e in which a body, subject to no forces, moves in a straight line with uniform velocity. So, in a fast moving plane, you will be subject to the same laws as you’re subject to on the (rotating) ground, as long as your velocity is uniform. Everything’s in movement, one might say, but if your movement is uniform, then it’s as if you’re at rest. You’re in an IRF.
Now I want to jump, if it’s a jump, to Lorentz transformations, which I’ve been trying unsuccessfully to understand. Here’s how Wikipedia clarifies the matter:
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity.
I’ve taken out all the links, which, if followed, might enlighten me further, but clearly this is v complicated stuff, which I need to understand for an understanding of special relativity. I expect to fail, valiantly, so first I will say that Hendrik Lorentz was an intellectual giant in what became the transformative physics of the early 20th century, making vital contributions to the understanding of electromagnetism, electrons, the aberration of light and much much else.
Lorentz transformations are transformations within inertial reference frames. What about non-inertial reference frames? They would include accelerating and decelerating motion (obviously non-constant velocity), and, apparently, a rotating reference frame. But isn’t the Earth’s rotation something we don’t feel because of the constant velocity of that rotation? But then, doesn’t the Earth, or a ball, rotate at different rates around the axis of rotation? Isn’t it obvious that a person on the equator is moving at a faster pace than someone close to the rotational axis? Apparently, the cognoscenti define this as ‘a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame’.
Actually, there are times when I really do wish I was a bonobo.
I’ll stop here for now, having so far avoided the mathematics. Maybe next time, or maybe not, but I must add here something consoling I read today in Richard Dawkins’ most recent book, Book do furnish a life:
… practising scientists do need years of training. But you can enjoy music, appreciate music even at quite a sophisticated level, without being able to play a note. Similarly, I think you can appreciate and enjoy science at quite a sophisticated level without being able to do science. I want to encourage people to treat science in the same kinds of way they would treat music or art or literature: as something to be enjoyed, not at a superficial level, but at quite a deep level, without necessarily being able to tell one Bunsen burner from another or integrate a function.
R Dawkins, Books do furnish a life: reading and writing science, 2021, pp 109-10
References
Tidally Locked and Loaded with Questions
Why are planets not tidally locked with the sun?
byu/Neotheo inaskscience
Richard Dawkins, Books do furnish a life, 2022 (paperback edition)
gravitational mysteries – part one, maybe
what happens when you fall for gravity…
I don’t understand gravity, and I doubt that memorising equations will be of much help.
Gravity, I’m told, is a killer. If I fall from a high cliff, or a multi-storey building, onto hard ground below, I’ll most certainly die, due to gravity (and carelessness, because I know what falling onto hard ground, even just from a standing position, can do to a person). So gravity should be treated with gravity.
But then, gravity has benefits. It keeps us on the ground, prevents us from flying away. In fact, gravity has essentially formed our bodily structure. We have muscular legs which with some small effort we can lift from the ground and plonk down in another place in a tiny ongoing battle with gravity, which we’ll eventually lose.
So I suppose it could be said that gravity is a given. An essential element in the development of all living things that creep over the earth and even fly in the sky just above it. We just have to deal with it.
And yet, I hear things about gravity that don’t make much sense to me. I hear that gravity pins humans to the Earth, but also pins our planet to the Sun, and pins the Moon to our planet. And yet it doesn’t. The Moon hasn’t fallen to the Earth in the way that my body would fall to Earth from a tall building. It circles the Earth. In fact it is spiralling slowly away from the Earth. Something else must be happening, surely?
So what do I do when I don’t know? I consult people who claim to know. And what do they say? Well, in terms of the Moon’s spiral, it’s about velocity. Here’s an explanation designed for children, or children at heart like me:
From Earth, it might look like the moon is stationary, meaning it is not moving, but in reality, each year the moon gets 3 cm [further] away from Earth. Without having the force of Gravity from earth [the] moon would have just floated away from us. The moon’s velocity and distance from Earth allow it to make a perfect balance between fall and escape.
In case the velocity of rotation of the moon was a little bit faster, it would have escaped the Earth’s Gravity. On the other hand, if it’s a little bit slower, it would have fallen on Earth. That’s why the moon doesn’t fall on Earth.
So that’s a good start, but why is the Moon revolving around the Earth at just such a speed that it keeps at (almost) the same distance? Isn’t that just too convenient? I also hear that the Moon is ‘tidally locked’ to the Earth, keeping the same ‘face’ to us all the time. That means it rotates on its axis over the same time-frame as a single orbit around Earth. Or nearly so, because the Moon’s orbit isn’t perfectly circular, which seems to be the case with every other orbit we know of. I suppose a precisely circular orbit would be a wonder, but then again…
Anyway, our Earth isn’t precisely globular either, and I’m betting it’s the same for the Moon, and every other planet and moon out there. I’m beginning to sense a pattern in this lack of a pattern. Or this approximation of a geometric pattern which doesn’t quite get there with the purity of mathematics.
Not that this is a bad thing. I’ve written previously about Milankovic cycles, variations in the eccentricity and tilt of Earth’s orbit around the Sun, which add spice to our planet’s climate. It’s like we use mathematics to understand the universe’s endless play with mathematics.
But getting back to that cliff fall. I’ve more than once heard the tale that Einstein’s ‘happiest thought’ was of such a scenario. Nothing to do with sadism or masochism, nothing to do with the landing. It occurred to him that, though the falling fellow might feel the force of the air swishing by him, he would not feel any ‘force’ of gravity. In a vacuum he wouldn’t feel any force at all. He might as well be stationary. Gravity, according to my good mate Wiki,
… is most accurately described by the general theory of relativity, proposed by Albert Einstein in 1915, which describes gravity not as a force, but as the curvature of spacetime, caused by the uneven distribution of mass, and causing masses to move along geodesic lines.
Which all sounds pretty radical, especially for 1915, when Fokkers had only just become a thing. So I get that mass is very unevenly distributed. At night we see clumps of stars here and there, with lots of apparently blank space in between. And though we can see for miles and miles and miles, this messy distribution of matter and space extends way beyond what we can see, perhaps even with our most inventive gadgetry. But ‘curvature of space-time’ still smacks of science fiction after all these decades.
Einstein had of course come up with this marriage of space and time 10 years earlier with his very special theory of relativity. So there are three dimensions of space and one of time. But are there? What exactly is dimensionality? Is it more than a human invention? In looking this up I’ve come up immediately with an essay ‘The invention of dimension’, on the naturephysics website. So that answers that question. Or does it? Here’s a quote from the start of the essay:
The modern concept of dimension started in 1863 with Maxwell, who synthesized earlier formulations by Fourier, Weber and Gauss. In doing so he added a nuance that we acknowledge today whenever we refer to the dimensions of, say, g (≈ 9.81 m s−2) as distance over time squared, rather than just the dimensional exponents (1, −2). By referring to the dimensions of a quantity, Maxwell seemed to imply that real things have natural dimensions. In the same spirit he designated units of mass, length and time as ‘fundamental units’.
Distance over time squared is a formula for constant acceleration, which again takes me back to gravity. When we fall from a cliff or a plane we constantly accelerate (leaving aside prevailing winds etc) until we hit the ground, but until that moment we’re not feeling any force upon us, according to Einstein. So acceleration isn’t a force? Apparently not. Is it the result of a force – the effect of a causal force? Well it can’t be an effect of gravity, because gravity isn’t a force.
So our acceleration in the above example is caused by a distortion of space-time which in turn is caused by the mass of planet Earth. But if we had fallen not from a plane but from a spacecraft much much further away, say the distance of the Moon from Earth, what would happen? Would we fall at all? We have satellites and a space station up there (I’m not exactly sure where), so would we just go into orbit like they do? Or are they carefully put into orbit by exquisitely precise mathematical calculations?
But, returning to Einstein’s not-so-happily falling fellow. The only thing he has to worry about is the landing. But the landing, and the force of the landing, is caused by the Earth’s mass. Presumably if we lived on a life-sustaining planet with the mass of Jupiter, which Dr Google tells me is over 300 times that of Earth, we’d be falling, or accelerating at a much faster rate (I’m tempted to say 300 times faster, but the mathematics is always more complicated). But then we couldn’t even live on Jupiter because our weight would be 300 times greater than that on Earth, just as the twelve men who walked on the Moon weighed, for a few days, only one sixth of what they weighed at home. So for life to have evolved on a planet like Jupiter (mass-wise) it would have to be made from very different stuff, molecularly. None of those heavy bones and dense tissues, like brains. An elephant’s brain weighs about 6 kilograms, and on Jupiter it would weigh 1800 kilos. So I suppose it’s important to think about planetary or lunar mass when we’re looking for extraterrestrial life, or alternatively, to think about different building blocks….
Anyway, it’s fascinating to note where thinking about gravity can take you, even when you know virtually eff all about the science. But I do want to learn more, and I’ll keep plugging away at it….
References
https://www.vedantu.com/physics/why-doesnt-the-moon-fall-into-the-earth#
https://en.wikipedia.org/wiki/Tidal_locking
aspects of climate change – Milankovic cycles
trying to understand special relativity – what fun!?
What?
Always being overly ambitious, and often reading stuff that’s beyond my ken, but perhaps not too far beyond, I read with intriguement this sentence from Sean Carroll’s book The Big Picture:
Einstein’s special relativity (as opposed to general relativity) is the theory that melds space and time together and posits the speed of light as an absolute limit on the universe .
Yeah, everybody who’s anybody knows that, so meet Mr Nobody. But I do try, sort of. Some months ago I bought Leonard Susskind’s Special Relativity and Classical Field Theory, read the first ten pages and then…
Anyway, I’ve still got the book. What I really need is my own personal tutor, who’ll drive the understanding into me, like a nail into diamond…
Susskind points out in his introduction that special relativity (SR) ‘is generally regarded as a branch of classical physics’, which is presumably his way of saying it’s not as much of a brainfuck as general relativity. It’s essentially derived from Maxwell’s work on electromagnetism and the constancy of the speed of light, and on the first page of the introduction I’m given this reassurance:
Some basic grounding in calculus and linear algebra should be good enough to get you through.
So I know that linear algebra has to do with lines and symbols, and calculus has something to do with teeth, so I suppose I’m halfway there.
Seriously, the concept of relativism, that all motion is relative, goes back to Galileo and Newton, and yet Maxwell, who introduced field theory to science, in the form of electromagnetic theory, predicted (via this theory, somehow) that light had a distinct, non-referential velocity – c = 299,792,458 metres per second, pretty much. Considering that we know of and can imagine many different frames of reference, this doesn’t seem to make sense. The concept of ether – the dark energy of the 19th century? – was hypothesised to somehow regularise the situation, as a possible medium through which light was ‘carried’. This sort of meant that no vacuum was really a vacuum, but rather a plenum of ether. The trouble with this concept was that it remained a concept, pretty much untestable. So in 1887 a famous experiment was carried out – the Michelson-Morley experiment, as an attempt to detect and capture the properties of ether. But, having sent the waves of light backwards, forwards, sideways, down, they could detect no difference in the speed (Wikipedia’s entry on the Michelson-Morley experiment, its background and subsequent experiments, and the effect upon Einstein, etc, etc, is comprehensive, complex and more than enough to make me wonder why I’m tackling this topic).
Anyway none of these experiments revealed anything like luminiferous ether, and all found no change in light’s speed, no matter how much they tried to torture it. At least I think that’s the case – experiments did get conflicting results, and Wikipedia describes it all in terms of mathematical equations that, sadly, are an alien language to me. All I’m really sure of is that the ether concept was pretty well debunked before Einstein came along. I believe Hendrik Lorentz, with his transformations, played a major role here.
So now I’ll switch to Sabine Hossenfelder’s video on special relativity, in the hope of achieving Enlightenment. Einstein’s idea of space-time, with time being a fourth dimension added to three-dimensional space, was first suggested by the brilliant mathematician Hermann Minkowski, but it was Einstein who built on his work, together with that of Lorentz and Henri Poincaré, to revolutionise our understanding of how our world, or universe, actually is. My feeling, which could be wrong, is that those other mathematicians were just doing mathematics, in the usual abstract way that mathematicians go about things, while Einstein recognised the real world applications – to put it over-simplistically, no doubt.
How all this relates to the famous E = mc² equation, I’m not sure, but it all brings back a childhood memory. I was in the back seat of the family car, driving along a freeway, with my two older siblings beside me. My guess is that I was around eight or nine. One of them said, something like, ‘well, if we go any faster we’ll start to lose weight, our mass will turn into energy, and if we get to the speed of light, we’ll disappear altogether, as Einstein says…’. Needless to say, I found this most discombobulating, but also intriguing. And I still do. That equation obviously relates mass and energy, but it relates these two variables to the square of a constant, c, the speed of light, a speed beyond which nothing can travel!?
So okay, I’ll return to Hossenfelder in a mo, and cite a little ‘nutshell’ piece from PBS, ‘Einstein’s Big Idea’:
Why… do you have to square the speed of light? It has to do with the nature of energy. When something is moving four times as fast as something else, it doesn’t have four times the energy but rather 16 times the energy—in other words, that figure is squared. So the speed of light squared is the conversion factor that decides just how much energy lies within a walnut or any other chunk of matter. And because the speed of light squared is a huge number—90,000,000,000 (km/sec)2—the amount of energy bound up into even the smallest mass is truly mind-boggling.
That almost makes me feel ashamed – I have far more energy bound up in me than in a walnut, so why do I feel so tired?
But Hossenfelder focuses on the space-time connection. Spatial co-ordinates are obviously very useful for locating everything, and for map-making. But those co-ordinates won’t tell us how to get to a destination in the least time. After all, there may be many pathways to take. But what Hossenfelder says next is a bit confusing:
If time becomes a co-ordinate, the same happens to time. If you give me your co-ordinates in space and in time, I will know where and when to find you. But the co-ordinates don’t tell me the length of the path that brings me to you, and they also don’t tell me how long it’ll take me to get there. But wait… if it’s 5am now, and I tell you we’re meeting at 5pm, then that’ll take 12 hours, right? No, wrong. Those 12 hours are your co-ordinate time. They are just convenient markers. They’re convenient for you because they agree with the time that actually passes for you. But how much time passes for me while I get to our meeting depends on how I get there. It’s just that the difference between the passage of your time and my time as I come to meet you is normally so small that we don’t notice. It would only become noticeable if I was moving close to the speed of light…
The numerous comments on Sabine’s video, to the effect that it’s the best explanation of special relativity they’ve ever heard, that at last they fully understand it, etc, fill me with whole teaspoonfuls of despair. What I think this passage means is that the wrongness of the 12 hours passing between one person and another vis-à-vis their meeting is a very very tiny wrongness due to the relatively small distance between them measured in relation to light-speed. I find it hard to call it wrong at all, but then I’m not light years away from anyone I know.
Co-ordinate time, which Sabine dismisses as a convenient marker, is surely the only time that matters for us non-physicist plebs, who are never going to consider the speed of light when making appointments. So…
It matters, I realise, if we planned to meet in a different galaxy, one in my neighbourhood but not in yours. So… let’s leave that one for now.
So now we have four dimensions, which clearly can’t be easily presented on a screen. On Sabine’s two-dimensional graph, representing space (x), time (y) and space-time (somewhere/everywhere), an immobile me can apparently be represented by a straight vertical line. Movement at constant velocity requires an angled line, with the speed of light conventionally set at 45 degrees. So far, so straightforward, almost, but then the bamboozling comes in, at least for me.
Space-time differs from space in one crucial way, which is how you calculate distances in it. If you have two dimensions of space, one called x and the other y, then calculating the length of a straight line between them is straightforward. You take the distances between the co-ordinate labels of x (call that delta x) and the same for y (call that delta y)…
That can be presented in an equation. I’m more or less allergic to even the simplest equations, and I’m doing Brilliant.org to cure myself. Not cured yet. Anyway,
Δ x = x2 − x1 and Δy = y2 − y1
The distance between the points (the Euclidean distance – which Sabine claims she learned in kindergarten – I hate her!) is the square root of the sum of the squares.
d = √[ (x – x)2 + (y – y)2] – (I couldn’t find an equation using the delta symbol, but I think I get it).
So, of course, space-time is quite different. There, we deal with co-ordinates in space and time, known as ‘events’. So, an extra layer of complexity. With two events, ‘each with a position in x and a time in t, and you want to calculate the space-time distance between them, then again you take the differences between the co-ordinate labels (Δ x and ∆ t). Then, as Sabine tells me, you take the square root of the square of ∆ t minus the square of Δ x divided by c (the speed of light). This is called the Lorentzian Distance, and the minus is apparently what it’s all about. It makes everything work out.
So, I’m not sure if I’m understanding this but I need to continue. So if you pick at random a reference point, say (0,0) and you map all points equidistant from that origin (in 2 dimensional space), you’ll map a circle. Different distances measured in these equidistant ways will map out circles with different radii. But, rather mind-blowingly, if we switch to space-time, ‘then all points at the same space-time distance from the origin are hyberbolae. You can’t move on those lines because that’d require you to move faster than light’. But you can move on the axes (according to Sabine), which would require a constant acceleration. I don’t understand this. Do I just have to accept it, like religious people have to accept their gods? Anyway, I’ll quote Sabine again:
The key to understanding space-time is now this. The time that passes for an observer moving on any curve in space-time is the length of that curve, calculated using this peculiar notion of Lorentzian distance that we just discussed. It’s called the ‘proper time’. It’s the proper way of calculating time according to Einstein. If you move between two events at a constant velocity, then the time which passes on the way is… the square root of ∆ t squared minus Δ x divided by c squared.
Okay, I’m waiting for a light-bulb moment, but I seem to be short of electricity. The ‘proper time’ is further defined as ‘the length of a curve in space-time’ and/or ‘the time that passes on that curve’. But why is this so? And why these hyperbolae? Something to do with space-time curvature perhaps?
Anyhow, let’s continue. If you’re not moving at a constant velocity, a more likely scenario, you break up the line into small pieces of straightness, sum them and ‘integrate over the curve’ (which I actually think I understand)….
One thing you shouldn’t do is confuse the co-ordinate time with the proper time. I must remember that…
But I’ve written enough, and I’ve had no Eureka moment. More later, perhaps.
References
Sean Carroll, The Big Picture, 2016
Leonard Susskind/Art Friedman, Special Relativity & Classical Field Theory: the Theoretical Minimum, 2017
https://en.wikipedia.org/wiki/Michelson–Morley_experiment