# Relativity geometric analysis without spacetime

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)

• 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. – G. Smith Sep 13 '19 at 16:18
• '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. – dmckee --- ex-moderator kitten Sep 13 '19 at 22:00
• He said “I am self-studying Einstein's relativity.” – G. Smith Sep 13 '19 at 23:59
• 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 – PM 2Ring Sep 14 '19 at 8:52