This is fantastic. I always wanted to learn relativity. But almost all the books I borrowed were so deep targeted towards students who will think about that day in and out.
I think a lot of avenues of core sciences need books like these that are deeper than popular science but are accessible to someone who is not doing it for a living.
(Physicist here.) Looked at TOC, then skimmed some chapters. The special relativity part of the book is about as deep and denser than most undergraduate introductions, with less motivations and intuitive discussions. The general relativity part is dense by necessity (after all 95% of it is differential geometry, no way around it), and this book apart from being way more condensed isn’t in any way diluted.
It’s a compact book of essentials for mathematically minded people*. It’s not a shallower-than-textbooks accessible introduction one step up from pop science.
* If you only ever took some linear algebra and calculus for engineering students, this probably doesn’t describe you.
This is fair enough. Hopefully some will still find it useful. I will note that I took a special relativity course in university that did things the "slow and intuitive" way, and many years later realized I still had some fundamental misconceptions about it. Eventually learned it the right way mostly by osmosis. More examples and motivation doesn't necessarily lead to correct understanding.
Yes, I definitely think it will be useful for mathematically minded people who don’t find the “slow and intuitive” approach helpful. However, I suspect the fact that it is posted to HN where most of the audience would be people who only took some math for engineering students and that “popular physics book” is even mentioned on the homepage (it clearly says “unlike”, but for most textbooks this comparison isn’t even needed) might give rise to the misconception that it’s between pop science and other textbooks in terms of depth, as the root of this thread seems to demonstrate.
I recommend Sean Carroll’s “Biggest Ideas in the Universe” series on YouTube for modern physics. He’s a professor at CalTech. Each is 60-90 minutes with an accompanying 60-90 minute Q&A—don’t miss the latter, they’re just as interesting. You won’t get a PhD after watching them but he gets a lot mathier than e.g. a Kurzgesagt video while only assuming a high school education (basic calculus, trig, vectors).
Dr. Schuller’s lectures is one of the few recordings of this kind on YouTube that I can listen to without getting irritated by one thing or another. (Another one being the widely known physics lectures by L. Susskind.)
This is great and way better then the more long-winded textbook explanations. Math is around for a reason and it's good to see it at work by compressing a lot of material into still readable format.
@fferen Thank you for sharing your work. This is an excellent resource.
If you don’t mind, I have a question.
If I drive in a straight line on the Earth’s surface without stopping and I ignore mountains and oceans and other obstacles, then after 16 days, I will arrive at my starting point.
Why is this?
It’s because the Earth is a sphere.
This is a nice satisfying answer, whereas x^2+y^2+z^2=r^2, while perfectly accurate, is arguably less satisfying.
Given this context, my question is, for special relativity, why do time dilation and length contraction happen?
Ideally, I’m looking for an answer that has the same satisfying intuitive flavor as “Because the Earth is a sphere”, or at least is suggestive of that kind of answer.
Given this context, my question is, for special relativity, why do time dilation and length contraction happen?
Place two rulers right next to each other, and their length scales will agree. Now, place them at an angle, and have one ruler measure the other by orthogonal projection. Each of the observers represented by the rulers will conclude that the other one has 'contracted' by a factor given by the cosine of the angle.
Now, add a third ruler to complete the triangle. To go from one vertex to the opposite one, you can either follow along a single ruler, or via a bent path along two rulers. The symmetry has been broken, and the bent path will be objectively longer - that's the twin 'paradox'.
Things are more complicated than that because Minkowski space is non-Euclidean (for example, less time will pass for the travelling twin, ie the bent path will be the 'shorter' one), but if you want a simple analogy, I think that's a pretty decent one...
I'm not sure a satisfactory answer necessarily exists here. The earth is a sphere, but you are asking a question about a mechanical action whereas time dilation and friends are fundamental statements about the nature of measurement itself.
That’s exactly what often troubles students of special relativity - why should a physical quantity be defined by the way you are trying to measure it. (The mass is the mass, for example, no matter what device you construct to measure it. In special relativity, on the other hand, the very notion of time interval seems to be tied to observing how light travels between mirrors.)
Not the expert you’re looking for but here’s my intuitive reasoning:
Because light will always be moving at the speed of light for you. You move at 0C. As you speed up, time slows down to preserve that.
Thanks! I don't believe these two concepts are related. You get back to the same point because the earth is _globally_ a sphere. However, time dilation and length contraction are _local_ concepts: they happen even for very small motions.
I guess a global object in spacetime analogous to a sphere in space is the hyperboloid t^2 - x^2 = r^2. Moving on this hyperboloid corresponds to changing boost velocity. But unlike a sphere, it is not closed, so moving in one direction does not get you back to the same point.
Apologies, I didn't mean to suggest that the two concepts are related.
For SR, I'm looking for an answer to "why?" that only has same satisfying flavor as the sphere question. I want the same "aha!" feeling.
For example, if I stand up from the sofa and walk across the room and come back and sit down next to my friend, I want a deep intuitive sense that of course it must be the case that less time has passed for me than the amount of time that my friend has experienced. Why does this happen?
An answer like "t'=t/sqrt(1-v^2/c^2) describes what happens", while correct, is not satisfying.
Similarly, if I wave my hand in front of my face, I want it to seem obvious to me that less time must have passed for my hand than for the rest of my body.
Given your experience writing the book, you must have developed an intuitive sense for the behavior of the effects of relativity and why they happen, so I am wondering how you would translate that into words for a general audience.
Imagine the context where a random person with a minimal math background at a party was to ask you why less time passes in the sofa scenario, using an actual sofa to demonstrate it.
They stand up and walk away from you and return and sit back down next to you and they want you to explain to them why less time has passed for them. They want you to explain why the room around them got shorter in the direction that they were walking.
These are effects that, while undetectably small, really happened.
I see. I don't know of a great answer, but it comes down to the constant speed of light. The simplest clock is two parallel mirrors with light bouncing back and forth. If someone is moving, then the light seems to take longer to bounce between them, so their clock appears slower. Wiki has a a good explanation:
While this might not be the answer you are looking for, but I can present an argument about why time dilation must be true to explain electromagnetism. The experiment produces apparently paradoxical equations that simply happen to work out fine. Trying to explain this experiment forces us to accept special relativity.
Every question I have is predicted on ignorance of the complete picture, but I have dozens of questions. This is the perfect format for going through and re-formulating and re-writing it in ones own notes to hack through it.
I'm not sure who this is supposed to be useful to; I'm not sure how anybody can understand it unless they have completed an undergraduate degree in math AND already have a solid conceptual understanding of relativity.
> I'm not sure how anybody can understand it unless they have completed an undergraduate degree in math AND already have a solid conceptual understanding of relativity.
The prerequisites are clearly listed:
> Prerequisites: vector calculus and classical mechanics
Do you not find this accurate?
(Note that classical mechanics means the typical physics undergrad classical mechanics - where you know diff eq and things like Lagrangians and calculus of variations).
Any particular suggestions? Although it is compact, I have tried to carefully choose the wording and arrange the sections so each concept builds on the last. Also, I intentionally avoid unnecessary mathematical abstraction; for example, the abstract definition of manifolds using coordinate charts, and tangent vectors as differential operators. This really tripped me up when trying to learn GR from standard textbooks.
It works well as an outline from my point of view, but I think you have to assume the reader already understands the Lagrangian in classical physics (which is a stretch for even the comparatively well-educated Hacker News audience). Also, somewhere around 3.5 it gets too dense with material I don't already know. (Christoffel symbols? Riemann curvature tensor?)
And then in the General Relativity section, there's simply too much material covered for any of it to be explained in depth beyond the necessary equations. I would drop the fluids (at a minimum) to give more space for other topics.
The same can be said for, say, Lifshitz & Landau's textbooks. A little bit closer to earth, Axler's Linear Algebra Done Right is well-liked and popular but of relatively little use to people trying to do linear algebra right for the fist time.
For those of us with that background, it looks like it might be really useful though. I've only had a brief look so far, but as somebody who took all the pure courses in my maths degree, I've been looking for something a bit like this!
I disagree. Most of relativity comes from the fact that all objects have an average velocity of their fundamental constituents. The highest average velocity is C, when the object is a beam of electromagnetic energy. This is why it takes infinite energy to accelerate an object to the speed of light, you can add more photons to it, but the average will always be less than C, you just get it slightly closer.
The reason for time dilation and the other facets of relativity pretty much come down to the fact that objects either move or change, but cannot do both simultaneously. One can think of time passing inside a spacecraft or object as internal movement, as oppose to the external movement of the spacecraft throughout the cosmos. Each bit of energy provides h (Planck's constant) action, a measure of change of state. The more the object allocates toward external movement in space, the less it can allocate for internal movement / internal changes, which is what observer time really is. Even though I say external vs internal, I am not violating relativity. The reference frames are relative, we do not need an absolute reference frame.
Doubly special relativity deduces most of these elegant derivations by assuming there is a smallest quanta of energy possible, but nonetheless, these concepts can be derived just by understanding the role of mass as loops of energy, energy as an allowance for change of state, and time as the usage of energy for internal change of state, and movement as the usage of energy for external change of state.
Relativity is just the consequence of energy's connection to information and movement.
> average velocity of their fundamental constituents
An interesting (if unusual) viewpoint; I wonder if this could lead to (a more realistic) quantum gravity.
> objects either move or change, but cannot do both simultaneously
This does make time slowing down inside a moving object kind of obvious. (I wonder if this description based on action/energy quantization turns out to be equivalent, at a certain level, to the essentially geometric picture of the classical relativity.)
It is indeed a very geometrical picture since position is just as much of a state as spin is. Position just has a few huge dimensions as its degrees of freedom. We don't know _yet_ whether position is continuously valued or discrete.
Gravity being quantum would most likely indicate that position is also discrete.
Since all energy, being allocated for external movement or just internal time, produces the same amount of gravity, it appears that the act of any state change induces gravity. Einstein never really gave a physically understandable reason for why all change causes gravitation, but it's likely similar to a ship in water. If the ship is rocking really fast, the waves that emanate from it can pull objects towards it. The rocking is just state change and even for a rest mass, we know that it's a loop of energy, there really is no static storage of mass.. it's a process of energy containment and cyclic change.
For instance, your second paragraph: No, objects move or change at the same time, just at different rates. And an object can be moving in one frame of reference, and only "changing" (time passing) in another.
I made my point clear, the amount of change of state that an object can produce over time, is given by its energy. This is because action equals joule-seconds, so energy equals actions per second. Just because you can change your frame to see change differently (ie. stationary) doesn't invalidate the point. This principle applies just as well even as the frame is changed.
I think a lot of avenues of core sciences need books like these that are deeper than popular science but are accessible to someone who is not doing it for a living.