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I'm an engineer and I'm used to classical mechanics "spatial-vectors" approach, which allows a geometric analysis of rigid bodies motion. In this context, as you know, it is very useful to imagine (and draw) each vector as an arrow.

I am self-studying Einstein's relativity and the abstract "four-vectors". Though I understand the necessity of introducing the time dimension, It is clear that these entities are not intuitive at all: not only they have four dimensions, but their metric is also pseudo-euclidean (in best case). This makes it impossible to draw them or even imagine their shape.

Do you think it would be possible to adopt a more intuitive approach to relativity, which keeps the usual 3d-spatial-vectors but (for example) foresees a "graphical contraction" of vector-arrows depending on the frame? Have you ever heard of a similar approach anywhere? (Clearly the concept of spacetime curvature would decay, but maybe it could be replaced by a less elegant / more practical solution)

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  • $\begingroup$ Four-vectors have $(ct,x,y,z)$ components just like three-vectors have $(x,y,z)$ components. The most common way to draw them is to suppress one or two spatial dimensions so you get a 2D or 3D diagram. The fact that their length may be zero or negative doesn’t matter when drawing. (Stop thinking that a vector is “a length plus a direction”.) Go draw $(E, E,0,0)$, $(E, E/2,0,0)$, $(E, 0,0,0)$, $(E, -E/2,0,0)$, and $(E, -E,0,0)$ in the $tx$ plane. Make $t$ go upwards like everybody does. Those are possible energy-momentum vectors of various particles. $\endgroup$ – G. Smith Sep 13 '19 at 16:18
  • $\begingroup$ 'Make t go upwards like everybody does." as long as you're doing relativity. If you're drawing Feynmann diagrams then you should make $t$ go to the right like everyone does. $\endgroup$ – dmckee --- ex-moderator kitten Sep 13 '19 at 22:00
  • $\begingroup$ He said “I am self-studying Einstein's relativity.” $\endgroup$ – G. Smith Sep 13 '19 at 23:59
  • $\begingroup$ I think your proposal hides (or at least obscures) the intimate connection between space and time in relativity. You may find robphy's diagrams, using rotated graph paper and light-clock diamonds, more intuitive than the standard spacetime diagrams. For example, see physics.stackexchange.com/a/383363/123208 $\endgroup$ – PM 2Ring Sep 14 '19 at 8:52
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You can certainly draw four-vectors. That's done all of the time, using the fancy-sounding name "spacetime diagram". It is true that lengths under the spacetime metric behave differently, but that doesn't prevent the drawing of the diagram and, with some practice, many people do generate intuition for these types of figures, even in complicated spacetimes. That's related to your second paragraph.

For your last paragraph, I guess the only answer can be "maybe"? If you have a different formulation of relativity that passes all of the solar system tests or is somehow provably the same as the accepted theory while casting a more intuitive light on the results, please write it up! I don't believe that there's any broad-based, serious effort to do that, however, if you're asking if it's a hot research topic. It seems unlikely to succeed given the complexity of the equations and also unnecessary at this point since the equations in increasingly complicated situations can be adequately solved numerically to achieve practical results that are needed for experiment. (It's only in the last decade or so that the equations could be solved numerically, so before that attempts to "simplify" were more critical to the advancement of the field.)

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A space-time diagram is the graphical tool used to accomplish exactly this. For purposes of illustration, only one dimension of space is customarily represented (on the x-axis) and time is on the +y axis. If a particle is moving at constant positive velocity in x, it traces a line that runs from the origin upwards to the right at some angle and the slope of that line is its velocity. This is called its world line.

If we have an observer who is watching the particle move, and if (s)he is moving relative to the particle, then that observer is also following a sloped world line.

Solving problems in special relativity on a space-time diagram then involves graphically mapping events in time and space from one sloped line to the other according to a set of rules, and the math of special relativity is contained in those mapping rules.

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