For the umpteenth time, length contraction is a coordinate artifact.

It's the expression c2t2-x2 that remains invariant under Lorentz transformations:

c2t2-x2 =c2t'2-x'2

where, for example,

ct' = ct coshφ + x sinhφ

x' = ct sinhφ + x coshφ

y' = y

z' = z

and tanhφ = v/c; using standard identities the transformations can be recast in more ``traditional'' form.

This implies, in particular, that y2+z2 = y'2 + z'2 .

It does not imply that x2 +y2 is equal to x'2+y'2 or that x2 + z2 is equal to x'2+z'2 but these quantities aren't invariant under Lorentz transformations, that's why the fact that they aren't equal doesn't mean anything.

“Does the Pythagorean theorem conform to the length contraction implied by Einstein's special relativity?”

- that looks as rather vague question, if it isn’t pointed – in what right triangle the theorem is applied - i.e. what are the legs and hypotenuse?.

Returning to the special relativity - though yeah, really length of a moving body is contracted comparing with the case when the body is at rest [see below], however in the SR it is postulated that all/every inertial reference frames are absolutely completely equivalent and legitimate. From this postulate it follows, that if, say,:

- let there are two rods that have identical lengths, L, and are at rest in some inertial reference frame, and further one rod is accelerated up to a speed V along X-axis of the frame’s coordinate system, then:

- in the “host stationary” frame the length of the moving rod is contracted in Lorentz factor comparing with the rod that remains be at rest in the frame,

- however, simultaneously, the moving rod in a co-moving frame is at rest, and in this case, by the SR postulates above has in co-moving frame the length L, whereas the rod in the “host” frame is shorter than this moving rod, again, in the Lorentz factor.

I.e. in the SR every of the rods absolutely legitimately and really simultaneously is longer and shorter than the other one.

That is evident absurdity, however that isn’t an unique senseless consequence that follows from the SR postulates that there is no absolute Matter’s spacetime, and absolute equivalence of the frames above; the number of such consequences is arbitrary.

Really, of course, only one case can be: one of the rods is really shorter than the other.

From that from the postulates above at least one meaningless consequence follows [in this case see bold above] by completely rigorous “Proof by contradiction” it follows that Matter’s spacetime is absolute, and so there can exist the preferred “absolute” frames, which are at rest in the absolute 3D space of the [as that is rigorously scientifically shown in the Shevchenko-Tokarevsky’s Planck scale informational physical model

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics] utmost universal Matter’s spacetime that has [5]4D metrics (cτ,X,Y,Z, ct) [the real at least [4+4+1]4D spacetime metrics is (cτ,X,Y,Z, g,w,e,s,ct), however that is inessential in this case].

Correspondingly everything in Matter has real parameters – energies, momentums, angular momentums, etc., only in absolute frames; if a frame moves in the 3D space the measured values of the parameters aren’t real, however because of action of extremely mighty Galileo-Poincaré relativity principle, which is actualized, including, in Lorentz transformations, which, as that Poincaré has shown in 1905, form the “velocity group”, in most cases that is inessential; and, if a moving frame is “Lorentzian” one, i.e. if the distant clocks in the frame are synchronized in accordance with the transformations, using in the frame the measured “unreal” values at description and analysis of what happens in some material system gives the same results as simultaneous description and analysis of this system in an absolute frame,

- despite that, say, measured in a moving frame length of a rod that is at absolute rest is shorter than the length that it really has; i.e. measurements by instruments of a moving frame results are “artifacts”..

That, again, in most everyday physical practice cases isn’t essential, however not always; and at consideration of some physical problem it is more clear and so convenient if it is considered in an absolute frame.

As that, say, see explanation of how the real moving bodies contraction occurs in the SS&VT model above: that happens because of [having rest mass] particles are some 4D “gyroscopes”, axes of which are directed along their 4D motions directions – i.e. along 4D momentums Pdirection; since particles always move in the 4D space with metrics (cτ,X,Y,Z) only with 4D speeds of light, c, P=mc. If a particle is at absolute rest it moves with the speed of light only along cτ-axis having momentum P0=m0c, m0 is rest mass.

If such particle is impacted by a 3D space directed momentum, it moves also in the 3DXYZ space, say, along X-axis, so its momentum direction rotates in the (X, cτ) plane by Pythagoras theorem on the angle, if particles compose a rigid body, they rotate the body on this angle as a whole, and so the body’s real 3D projection by Pythagoras theorem is lesser then if the body is at rest in (1-V2/c2)1/2. Again, since all interactions on macroscale – and most in QM and Planck scales – happen in 3D space, this contraction is quite real.

More see the linked above paper, here note only also that for derivation of Lorentz transformations [and practically of all main equations in the SR] it is enough to know only Pythagoras theorem – however only if basing at that on the SS&VT model postulates, from which no any absurd consequences follow.

Cheers

The Pythagorean theorem applies to space in all inertial frames of special relativity. So, if one leg of a right triangle (x) is contracted in length and the other leg (y) remains invariant, then you can use the Pythagorean theorem to determine the contraction of the hypotenuse.

The only quantity that remains invariant among all frames of reference is space interval (dx)

The reason is that it is orthogonal to the motion, as shown in Figure 1.

the hypotenuse contracts but s^2=x^2+y^2 is still valid.

📷

Fixed Frame v=0 Moving Frame v > 0

Fig 1. Contraction of a right triangle according to Einstein's special relativity.

Fixed Frame v=0 Moving Frame v > 0

Fig 1. Contraction of a right triangle according to Einstein's special relativity.

The constant speed of light involves quantum (chord) space-time, and Einstein tried to understand a quantum space-time event using a clock, a ruler.

“…The Pythagorean theorem applies to space in all inertial frames of special relativity. So, if one leg of a right triangle (x) is contracted in length and the other leg (y) remains invariant, then you can use the Pythagorean theorem to determine the contraction of the hypotenuse.…..”

- The Pythagorean theorem applies to space anywhere, and not only in all inertial frames of special relativity; so if we say about this theorem in physics that should be some concrete cases, which should have some concrete physical sense. In that when a lag of some arbitrary right triangle is contracted its hypotenuse is contracted as well, there is practically no this sense.

However, again, if we consider how the main physical parameters of bodies, at least . momentums and energies, are measured in relatively moving frames, in this really fundamental physical case because of Matter is rather simple informational system that is based on a simple binary reversible logics, it turns out to be that to rigorously scientifically derive equations for these parameters, and of Lorentz transformations as well, it is sufficient to know the Pythagorean theorem;

- more see the SS post above and links in the post.

Cheers

The Pythagorean theorem can be stated as follows in a very general context (theory of Hilbert spaces): let H be a Hilbert space; let x,y be vectors belonging to H such that =0 (mutually orthogonal vectors); then = +. Let us now consider two four-vectors X,Y in Minkowski space-time. Let T= diag (1,-1,-1,-1). The scalar product of two four-vectors can be expressed as follows: s(X,Y)= where is the usual scalar product between vectors with real coefficients. Well, even in Minkowski space-time, it turns out that if =0, then = + i.e. s(X+Y, X+Y)=s(X,X)+s(Y,Y).

“…The Pythagorean theorem can be stated as follows in a very general context (theory of Hilbert spaces): let H be a Hilbert space; …. Let us now consider two four-vectors X,Y in Minkowski space-time.….”

- Pythagorean theorem is defined on Euclidian spaces with number of dimensions more 2, and all dimensions are principally [mathematically] real,

- and so the theorem principally isn’t defined on Minkowski 4D space, where either time or 3 space dimensions, are [mathematically] imaginary.

At that Matter’s spacetime fundamentally cannot have imaginary dimensions, and so utmost universal Matter’s spacetime has [5]4D metrics (cτ,X,Y,Z, ct) [the real at least [4+4+1]4D spacetime metrics is (cτ,X,Y,Z, g,w,e,s,ct), however that is inessential in this case],

- more, including that for derivation of main equations in fast bodies mechanics, i.e. Lorentz transformations, and of momentum [for E=mc2 it is necessary to solve a simple integral], it is enough to know only Pythagorean theorem

- provided, though, that the main postulates in the SS&VT Planck scale informational physical model are known and understood, of course.

More see the SS posts above and links in the posts.

Cheers