Put all your velocity into the time component and you can't move in space. Conversely, if you put all of your velocity into the spatial components, you will freeze in time like a photon.
It's like a CS guy responding to "hashtables are O(1) lookup" with a wall-of-pedantry about different implementations, complete with complexity analysis by evaluation of recursion equations and whatnot.
For the same patch of spacetime with "a photon" in it, different observers can calculate different photon numbers and different photon energies.[2] That is to say that these properties are not always conserved under a change of systems of coordinates (trivially, when we have two observers with different observables, we can fix a coordinate system's origin on either of them, but that doesn't make either "right"). Indeed, the properties of the photon that survives such changes are: they locally move at c, they have no intrinsic mass, but they do have momentum (and thus contribute to the stress-energy-momentum tensor).
The intrinsic mass is the same as the rest mass (a quantity that remains the same in all frames of reference related by Lorentz transformations). The intrinsic masslessness of photons is required for the gauge invariance of the Feynman amplitudes of QED or the Standard Model. More detailed explanation would involve a trip through an explanation of the Ward identity[1] which gets even harder when curved spacetime is in play.
I'm sure you've already discovered that the topic of photons' frames of reference comes up a lot in much harder-science forums than HN, and hopefully you've found a decent treatment of that on e.g. physics.stackexchange.com or physicsforums.com. If you find a decent link, maybe someone (and probably I) would appreciate it if you attach it to this thread because it is likely to come up again someday. :-)
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[1] https://www.wikiwand.com/en/Ward%E2%80%93Takahashi_identity
[2] redshifting is the clearest case of photon energy change, and can arise from uniform relativistic motion, relative acceleration, metric expansion, or real gravitation. Extremely relatively accelerated observers will disagree on particle counts generally, with the Unruh effect serving as a partial formalization.
We don't use inertial frames of reference in General Relativity because in the presence of real gravity, there are (strictly speaking) none anywhere. [1]
There are however static spacetimes that admit inertial frames of reference. Flat spacetime aka Minkowski spacetime is an example. That is the spacetime of Special Relativity, and in Special Relativity inertial frames of reference are extremely useful. However, the defining feature of flat spacetime is that there is no gravity anywhere in it.
We can talk about coordinate conditions [2] where those generalize a set of activities that pick out a specific choice of coordinates (and consequently an origin for the coordinates), a choice of units, and some other choices that one can freely make.
The photon's "own reference frame" could be specified as for example keeping it at the origin of a set of flat Cartesian coordinates (x=y=z=0=const) and letting it move against the t coordinate. This is unlikely to be useful, and can be made useless with various choices of units. However, one can say conclusively that the photon's spatial coordinate velocity is zero.
However, coordinate velocity isn't physical: it goes away by changing the system of coordinates. For example, in this system, your coordinate velocity is always exactly c. And so is the moon's. And so is the Andromeda galaxy's. But we can see how unphysical that is simply by fixing coordinates with you always at the origin, or the sun always at the origin, and noticing that the only thing moving with a coordinate velocity of c in those systems of coordinates is the photon.
Indeed, in General Relativity comparing velocities is extremely tricky for objects not occupying the same point in spacetime because it is very easy to be misled by what you're being told by the coordinates and the choices "hidden" within them. The usual advice is to avoid such comparisons (cf. Baez [3]).
Nowadays pretty much every relativist will tell you that Special Relativity emerges from General Relativity (the more fundamental theory) as a special case in the limit where gravity is weak, even though historically Special Relativity came first and informed the development of General Relativity. (They'll also probably advise you to calculate using Special Relativity forms of physical formulae where you can do so!)
However, where gravity is non-negligible or where one is tempted to use a broader set of coordinates than e.g. spherical or Cartesian coordinates on a local patch of sufficiently flat spacetime, Special Relativity is simply inappropriate. Intuitions from Special Relativity about how an object moving at exactly light speed (or even extreeeeeeemely close to it over sufficiently long intervals, like with an extragalactic ultra high energy cosmic ray[4]) are likely to be misleading; instead, one should use the toolset of General Relativity.
Unfortunately that toolset requires complicated mathematics. [5]
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[1] We can define locally inertial frames of reference (LIFs), and the Lorentzian structure of spacetime (four dimensions, three of one sign and one of the opposite sign) guarantees that we can do this in many cases, especially in infinitesimally small regions around a point, or in a small region along a geodesic. (I can explain this further if you are very interested, but its fairly technical and grinding it down to something suitable for HN may take some iteration. I don't even have a link to a decent explanation for what e.g. Fermi coordinates are, why they're useful, and how to use them :( so maybe I'd be breaking new ground ;) ). Some LIFs can be more extensive when gravity is sufficiently weak in that region: an Earth-based "laboratory frame" in Special Relativity is really just a LIF without admitting it; particle colliders typically don't really have to consider the influence of the gravity of bodies like the Earth, the Moon, the Sun, and so on, even if one has a view of the ocean (and its tides) out of one of the lab's windows.
[2] https://www.wikiwand.com/en/Coordinate_conditions goes pretty deep on this
[3] http://math.ucr.edu/home/baez/einstein/node2.html third paragraph
[4] https://www.wikiwand.com/en/Ultra-high-energy_cosmic_ray
[5] where one is dealing with generalized matter (light and massive particles, etc) in general curved spacetime, one might want to take the initial values approach in http://fanfreluche.math.univ-tours.fr/notes/geroch/geroch_no... (or if you like) ch. 27. onwards of https://books.google.co.uk/books?id=NOJ9AgAAQBAJ&printsec=fr... but also in more recent work -- Special Relativity is essentially a framework which has done all this heavy lifting for the user.
I like Carroll's [ https://www.preposterousuniverse.com/spacetimeandgeometry/ ] and indeed, you get to deal with intervals and worldlines in chapter 1.
It assumes you know or are ready to learn some differential calculus and how to read a formula with an integral but it (maybe a bit steeply) teaches tensors (and some aspects of vectors and scalars) across the first couple of chapters. Carroll provides some (quasi-)samples under the "Lecture Notes" tab, but the book itself has benefited from editing. He also supplies links to alternatives that can be had for free-as-in-beer.
https://en.wikipedia.org/wiki/The_Road_to_Reality
It is quite long and dense but explains the math from first princples like you want.
The one where everything always move at c, and only the direction of movement changes is one among many possible (and indistinguishable) descriptions, and not a very useful one.
It certainly does though -- the existence of many valid models and simple mappings between them implies a 'deeper model' at play, and putting one particular model above all others as the 'correct' is actually discouraging the reader from getting towards the deeper truth.
If you say 'the sun is stationary and the planets revolve around it' is the only valid description of the solar system, you would be wrong, and you're also making it harder for a person to understand relativity down the line.
No one is putting a particular model above all others, one model is simply being used to explain the relationship between C and time, it's not incorrect just because other models are also valid as long as they all explain the same relationship between C and time.
Not necessarily. You can have completely different formulations of the same physics. Lagrangian, Hamiltonian, and Newtonian dynamics are different models of classical motion. Does that imply there's a "deeper model" of classical motion? I wouldn't say so.
I'm sorry this instantly turned you off.
The only equation in my comment is the Minkowski metric which is the metric for Special Relativity, and should be familiar to everyone who has done any SR at all.
Moreover, it's just a a generalization of the Euclidean metric as follows:
The Euclidean interval for
spatial
dimension equation for Cartesian coordinates in flat space
0 dS^2 = 0
1 dS^2 = dx^2
2 dS^2 = dx^2 + dy^2
3 dS^2 = dx^2 + dy^2 + dz^2
(mostly-plusses, flat spacetime)
spacetime
dimension equation for Cartesian coordinates
2 dS^2 = dx^2 - cdt^2
3 dS^2 = dx^2 + dy^2 - cdt^2
4 dS^2 = dx^2 + dy^2 + dz^2 - cdt^2
The interesting difference is this: in Euclidean space, straight paths are shorter than curved paths, but in flat spacetime, it is curved paths that are shorter.
If you prefer wikipedia, https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval which you can edit, or perhaps you can add to the Spacetime page on the simple English wikipedia.
I will offer a perspective on this exchange...
I think people who have survived many-many math based courses often have an immediate and aggressive response to diving into the minutae of a quantitative topic before they have grokked the intuition behind it. This is a defence mechanism built up from hours and hours of wasted time in lectures where the topic has moved on before the student/s have really developed the basics and are ready for the detailed stuff.
Hours and hours of wasted life.
When a person with this defence mechanism sees a noobie about to fall into the same horrible cycle, this will trigger some aggression. For example: downvoting your post- they are trying to protect the noobs.
So if you are interested in reaching as many people as possible, please don't give up. I think your teaching effectiveness could be improved by finding ways to engage people at their (lower) level of understanding and trying to help then incrementally improve their mental models.
I hope you don't mind if I pick up your comment as an invitation to go even more meta than you. :-)
> if you are interested in reaching as many people as possible
I'm not sure I am, even if it's "as many people as possible on HN who open this discussion". I'm guessing that most people who will read down a thread on a topic like this have some interest in it, and probably have a little math, or some search-fu, or perhaps even a little physics, but little exposure to General Relativity (one can earn a Ph.D. in physics without ever having to walk through a comma-goes-to-semicolon exercise let alone deal with exceptions to that procedure, but I'm not writing for e.g. solid state physics Ph.D.s here and hopefully they already know how to look beyond an HN thread or Nature News link if they want to know more about SN-BH or similar stellar collisions).
However, I don't want to alienate people on either side of that -- neither the experts nor the enthusiastic-but-allergic-to-mathematical-physics readers.
> please don't give up
Thank you again. If you have any concrete suggestions (now or in some future thread) about how to help engage the latter group, I'll gladly read them.
However,
> before they have grokked the intuition behind it
the problem is that intuitions like "the shortest path between two points is a straight line" are based on Euclidean geometry, which is probably much more often taught rather than discovered by a student sua sponte, although once taught experimental validation is easy. But in Euclidean (well, Minkowski) spacetime, curved paths are shorter. I think that pretty much nobody would have any chance of discovering that feature of spacetime on her or his own, or intuiting it from planar geometry. However, it's easy enough to teach by explaining what a line element is, and what the line element of Minkowski spacetime is. Once that is absorbed and is familiar enough that reading and drawing spacetime diagrams isn't a chore, then one might expect intuitions like "one can resolve the twin paradox by observing that the travelling twin takes a more-curved path through spacetime than the non-travelling twin". But even there, people sometimes stumble on understanding that that statement is demonstrably true under any choice of coordinates, not just ones which hold the non-travelling twin at the spacelike origin (from which the traveller departs and to which the traveller returns) throughout. And even then, where does one's intuition take one when one or both twins experiences significant real gravity?
One option of course is to shrug off opportunities to try to write into words what one would normally describe using a formula. I'm sure that's not what you're suggesting (but others in this topic seem to).
Another is to give a reply that is neither correct nor detailed but which is at least more correct. Maybe that helps a little, but I doubt it advances anyone's understanding rather than be memorized as a slogan or factoid.
Yet another might be a pointer to a standard textbook. Since they tend to be chunky and expensive (and I can't even guess about availability at a local public library rather than a major reference library open to the general public), I'm not sure that's so helpful either, unless the pointer is to a pirate scan. :-)
Penultimately, this is unpaid pseudonomymous fun. I think ELIx (FSVO x) is a good challenge for the explainer too, especially for extremely abstract topics, otherwise why bother? From this perspective, what's a decent choice for "x" on HN? (We surely can agree that it will be different than "x" on e.g. physics.stackexchange.com; in fact I think that is close to your central point.)
Finally, in comparison to the previous paragraph other models exist, e.g. http://backreaction.blogspot.com/p/talk-to-physicist_27.html - I am reasonably sure Sabine Hossenfelder would be happy to negotiate on publishing a transcript or summary of a conversation on her blog or elsewhere (perhaps even as a comment on HN :) ) and I am even more sure the quality of her or her associates' answers will be better than mine.
If you don't care about that, then I think that we should agree that nobody profits by you or I replying to one another's HN comments in future.
I think your comment is too long and too technical, but it's correct and explain why the original comment is wrong. So I upvoted it.
The problem is that is too long and too technical, so the main ideas are lost and many people prefer some wrong handwaving.
Some recommendations:
* Try to keep the answer shorter. Link to the Wikipedia article if possible. Perhaps ignore part of the original comment to make the reply shorter.
* Try to avoid too much technical jargon. This is a technical forum. You can assume everyone has Calculus 101 and Algebra 101, and a few more elementary courses, but nobody has advanced knowledge of all the topics.
It's difficult to explain in simple words why I don't like the original comment. The answer is strangely in one of your other comments.
> "Sure, you can always choose useless systems of coordinates."
It's too short! I'd try to add some details: The problem with "everything moves through spacetime at c" is that you must "sum" the squares of distance "x" in the laboratory reference frame and the time in the reference frame of the moving spaceship "s" (and fix the units, sorry). So you get a nice property that is valid if you use properties that are measure in different reference frames. But 99.99% of the time, life it's much better if you don't mix measurements in different reference frames.
[My version still needs a lot of fixes, fix the units, perhaps replace "reference frame" with "astronaut", add at least a formula, ... I guest that with 1 or 2 more hours I can fix it, but this is a deeply nested thread that nobody will read and I have to go to eat ... https://xkcd.com/386/ ]
I read it.
> too technical
Too technical for the comment or too technical for HN period?
> Link to the Wikipedia article if possible
I do link to things that go into further detail in many of my comments.
However, I think the relevant (English-language) wikipedia articles for this thread are bound to be much longer and more technical. The simple English wikipedia versions, to the extent they even exist, are generally not worth linking to.
Do you think it would have been right to simply drop https://www.wikiwand.com/en/Line_element#/General_spacetime as the bulk of the content of my original disliked-by-you comment?
> I'd try to add some details
The details were already in the replied-to message that people seem to either like or dislike. :-)
> I guest that with 1 or 2 more hours I can fix it
FWIW, I only took a few minutes. Is it likely that anyone would spend an hour or two on any HN comment (especially a down-thread one) as opposed to a blog posting, article or comment elsewhere?
Indeed some of my original motivation is exactly https://xkcd.com/386/ however the point wasn't just to be right, but also to try to advance someone's (not necessarily the original writer's) understanding of a line element and that line elements are meaningful (and in particular typically nonzero) for light.
No. Everything has its own worldline through spacetime, and between two events at point p and q on a worldline through a given spacetime we can measure the interval dS between p and q. When we normalize the interval against a set of coordinates and a chosen metric signature (here +++-) we can have three types of interval: dS^2 = 0 is lightlike, dS^2 > 0 is spacetlike and dS^2 < 0 is timelike.
A concrete example using the Minkowski metric for a set of Cartesian coordinates dS^2 = dx^2 + dy^2 + dz^2 - cdt^2. If we have a test object that always remains at the (x=0,y=0,z=0) origin of the coordinates then as the "t" coordinate increases with the passage of time, -cdt^2 is the only nonzero component of dS^2. From t=0 to t=10000 (where t is in, say, seconds) is perfectly timelike interval. However, any way we vary x, y, and z, (measuring the coordinate distances in, say, light-seconds) if the changes are small compared to the constant factor c, we will have a timelike interval. Light itself, conversely, follows a lightlike interval. If we restrict a beam of light to move only on the x axis, then we have (in (light-)seconds and seconds) x=c, t=1; x=2c, t=2; x=3c, t=3; and so forth; the -c factor cancels out the change in x at each step, so dS^2 = 0.
But bear in mind here that the Minkowski metric is just one of many known exact solutions to the Einstein Field Equations, and there are many many many known approximate solutions. Moreover, we are free to use arbitrary coordinates. The Minkowski metric looks different in spherical polar coordinates, for example. We are also free to use arbitrary units. We can even use the metric signature (-,-,-,+) if we like. However, when we take all of these into account, we're left with the same distinction based on the interval: they're either lightlike, timelike, or spacelike.
A lightlike worldline is one in which intervals on the worldline are always light-like; a timelike worldine is one in which intervals on the worldline are always spacelike.
We have strong evidence and stronger theoretical reasoning to expect that massless objects will always have lightlike worldlines (and that light itself is massless) while massive objects will always have timelike worldlines.
So:
> Electromagnetic waves have no mass, they don't travel in time, so the entire portion of their travel takes place in space
No, they have lightlike worldlines. An interval between any two points on the wave's worldline will be lightlike. This generally means that changes in the spacelike coordinates will exactly match the change in the timelike coordinate multiplied by the constant factor c. However, under most reasonable choices of coordinates, the "t" coordinate will certainly vary from point to point along its worldline.
However, one has free choice to decide which axis is timelike or spacelike, and different choices may seem like the natural ones to different observers.
In order to cope with these sets of choices we write down the laws of physics in a generally covariant manner. This has been one of the greatest successes of relativity; any proposed theory that cannot be written down in generally covariant form is almost certainly unphysical in some way.
Lastly, the value of "c" is determined empirically, and will vary depending on one's choice of units. Relativists will often use a system of units in which c is set to unity (c=1), for example, in order to simplify the form of equations.
> (GPS satellites do have to account for this)
The theory side of GPS relies upon covariance matrices.