Relativistic velocities and dimensions
Roland Sprenger, Germany 28.04.2024
Abstract
In the reference system of an observer photons and any objects at light speed have only two dimensions. The Minkowski area of a very thin rod at relativistic velocity rotates within the x-t-area and at the same time it turns into a fifth dimension.
Paragraphs:
1. Two dimensions of objects at light speed
2. The implicit Pythagorean theorem
3. Rotation within the x-t-area
4. Rotation into the fifth dimension
5. Conservation of the Minkowski-area and of mass
6. Summary and outlook
1. Two dimensions of objects at light speed
Relativistic length contraction shrinks the length l´ of a rod in its rest system S´ which is moved at velocity v in the direction of l´ in the reference system S of an observer . The length l in S is
l = l´ √ (1 - v²/c²)
When an object is moved at vacuum light speed c, v = c , its length in moving direction in framework S is zero:
l = l´ √ (1 - 1) = 0 .
This means that photons and other quanta moved at light speed are infinitely short in their moving direction. So they are real two-dimensional objects in our three-dimensional world.
According to the relativity principle the stars and galaxies of the universe are two-dimensional too in the rest system of a photon. It makes no difference how large the length l´ is, even if it was millions of light years long, the factor zero contracts it to zero: l = l´ ∙ 0 = 0 .
Because of the time dilatation Δt = (Δt´) / √ (1 - v²/c²)
time does not pass within the galaxies flying by at light speed, for
Δt´= Δt √ (1 - v²/c²) = Δt √ (1 - 1) = 0
applies for any time interval Δt, irrelevant how large it might be. Within the galaxies in framework S´ time does not exist, everything stands still, being observed from a photon at the relative speed c. Time passes only within the photons rest system S.
Again, this phenomenon is symmetric according to the relativity principle: Regarded from our laboratory system time does not exist for the two-dimensional photons. Emitted in a galaxy a billion light years away they are absorbed in our telescope at once from their point of view (extreme of the twin paradox). Length contraction has shrinked the distance in their rest system to zero. For objects at light speed the three-dimensional space is a plane.
What consequence does the relativity of mass have then? From
What consequence does the relativity of mass have then? From
m = (m0) / √ (1 - v²/c²) or m0 = m √ (1 - v²/c²)
follows that the rest mass of objects moved at vacuum light speed always has to be zero. Again, this applies not only to photons in a laboratory system but reversed also for galaxies in the rest system of a photon. That is, it is part of the nature of mass – and because of E = m c² of energy too – to disappear, when it is observed from a framework moved with vacuum light speed.
The summery is, that photons and galaxies are related symmetrically to each other concerning rest mass, time and the number of dimensions. The respective object moved at speed c is two-dimensional, timeless and without rest mass.
The timelessness means that the objects can be found on the same time-coordinate in the four-dimensional spacetime, independent from their direction of movement in space. And they are located at all places of their course at the same time, respectively all places coincide.
The contradiction that a course in system S with variable coordinates of place and time in system S´ becomes a point with constant coordinates can only be resolved by projection or turning in a higher dimensional spacetime. In fig. 1 is shown how in a five-dimensional space that contradiction can disappear.
Fig. 1: In this complemented Minkowski diagram the black line represents the movement of a photon with variable coordinates x and ict in framework S (blue). The green axes are located in the drawing level, whereas the red w-axis stands orthogonal on it. In the green coordinate system for the photon applies t´= 0 ; only the location coordinate x´ is variable. In the w-ict´ plane the orthogonal projection of the black line results in the point (0|0). This plane is the rest system of the photon.
2. The implicit Pythagorean theorem
A very thin rod of the length l be moved at the constant relativistic speed v in its longitudinal direction.
From l = l´ √ (1 - v2/c2 )
follows l2 = l´2 (1 - v2/c2 ) = l´2 - (l´ v/c)2
and from that l2 + (l´ v/c)2 = l´2 , (1)
the Pythagorean theorem. This can be interpreted as a rotation of the rod by an angle φ , see fig. 2.
Fig. 2: The lengths l and l´ of a rod moved at a relativistic velocity v in its longitudinal direction in the reference systems S and S´
For the rotation angle applies sin φ = v/c (2)
and cos φ = l / (l´) .
For quanta with v = c follows sin φ = 1 , i.e. φ = 90° .
From Δt´ = Δt √ (1 - v²/c²) follows (Δt´)2 = (Δt)2 ∙ (1 - v2/c2 ) = (Δt)2 - (Δt ∙ v/c)2
and from that (Δt´)2 + (Δt ∙ v/c)2 = (Δt) 2 , (3)
the Pythagorean theorem again, see fig. 3.
Fig. 3: The times Δt and Δt´ in which the rod moves forward by its length l in the frameworks S and S´
Again applies sin φ = v/c and in addition cos φ = (Δt´) / Δt .
The rotation angles thus are equal. But for the times the hypotenuse is in reference system S, whereas for the lengths it is in reference system S´.
Correspondingly, the formula for the relativistic mass m0 = m √ (1 - v²/c²)
delivers m02 = m2 (1 - v2/c2 ) = m2 - (m v/c)² ,
so m02 + (m v/c)2 = m2 , (4)
again, according to the Pythagorean theorem, with the hypotenuse in framework S, see fig. 4.
Fig. 4: The masses m0 and m of a body moved at relativistic speed in its rest system S´ and in the system S of an observer
With sinφ=v/c it is the same turning angle φ again.
Moreover applies cos φ = m0 / m (5)
and thus m0 / m = (Δt´) / Δt (6)
and cos φ = √ (1 - v2/c2) (7)
3. Rotation in the x-t plane
These results are summarized as follows:
A very thin rod at rest with length l, which is observed for a time Δt , is represented in an x-t diagram by a right-angled area (Minkowski area, fig. 5). The one-dimensional “rod” thus becomes a two-dimensional object of the space-time.
Fig. 5: The grey area contains all pairs of coordinates of a resting rod of length l in the time interval Δt , so it represents it.
If the rectangle is turned around the coordinate origin by the angle φ while maintaining the point (0| Δt) , see fig. 6, both of the triangles for length (fig. 2) and time (fig. 3) in S and S´ appear together. The sections Δt´ and l´ of the two triangles define a second rectangle (depicted green). The pairs of coordinates of its area represent the rod in its rest system S´ if the rod is moved in S at the constant speed v = c ∙ sin φ in its longitudinal direction.
Fig. 6: The Minkowski areas of a rod of length l, watched for a time Δt and moved with v = c ∙ sin φ in the reference system S of an observer and in its rest system S´ together with the triangles from fig. 2 and fig. 3
Watching the same rod at equal times Δt at different constant velocities means that the angle φ changes and the length l´ in its rest system stays constant. For v = 0 applies φ = 0 and l´ = l . So constant l´ means l´ = l for all velocities respectively angles φ, see fig. 7. The constant observation time Δt causes reduction of the Minkowski area at growing φ .
Fig. 7: Reduction of the Minkowski area which represents a rod of length l´ at growing constant velocities. The time of observation Δt and l´ are constant. On the x-axis length contraction is visible.
For all areas applies A´ = l´ ∙ Δt´ with constant l´ = l , so A´ = l ∙ Δt´ .
A´ = l ∙ Δt √ (1 - v2/c2 )
From (7) follows A´ = A ∙ cos φ (8)
For photons and quanta with v = c i.e. φ = 90° follows A´ = 0 .
The size of the Minkowski area depends on the angle φ this way:
For all areas applies A´ = l´ ∙ Δt´ with constant l´ = l , so A´ = l ∙ Δt´ .
A´ = l ∙ Δt √ (1 - v2/c2 )
From (7) follows A´ = A ∙ cos φ (8)
For photons and quanta with v = c i.e. φ = 90° follows A´ = 0 .
4. Rotation into a fifth dimension
Equation 8 and figure 7 now are interpreted in that way, that the areas A´ are orthogonal projections of the area A, which has rotated around the respective x´-axis into a fifth dimension (w-axis), on the x-t plane (fig. 8 and 9). In case of v = c , e.g. photons, then the area A stands orthogonal on the x-t plane, with its side l´ lying on the t-axis.
Fig. 8: A three-dimensional space with the directions x, t and w (5. dimension); in side view the area A and its projection A´ in w-direction on the x-t plane; equation 8 is valid.
The area A´ is flipped up by the same angle φ into the fifth dimension by which the area A is turned to the area A´. Flipping and turning happen together each by the angle φ, see fig. 9.
Fig. 9: The area A´ as a projection of the area A´´ from the fifth dimension to the x-t plane parallel to the w-axis and the dotted blue lines; the area A´´ as result of a double turn or flip-turn by φ at the velocity v = c sin φ
Even at the speed of light, a length l does not disappear despite length contraction to zero in the laboratory system; it moves into the five-dimensional space-time with a flip-turn. Then the x'-axis runs in the direction of the t-axis and the t''-axis in the direction of the w-axis. Whether the character of a dimension is spatial or temporal is therefore relative and depends on the speed. For all speeds between 0 and c, it is both spatial and temporal.
5. Preservation of the Minkowski area and mass
As the area A´ is the orthogonal projection of the area A´´ (fig.9 and 10) analogous to fig. 8 applies
A´ : A´´ = cos φ
and with (6) A´´ = A´ : cos φ = A ∙ cos φ : cos φ
i.e. A´´ = A (9)
During the entire rotation-folding process, the area of A remains the same or constant, which is consistent with a more complete representation of the world. A' becomes smaller in relation to A by cos φ as the angle increases, while at the same time A'' becomes larger in relation to A' by cos φ, which balances each other out. During the flip-turn, the two right corner points of A in Fig. 9 move on spirals on the cylinder with the radius l and the w-axis as the central axis to the front upper corner points of the red-blue cuboid.
The dimension of A is ms. The conservation of A requires that the dimension is also preserved. Then the t'-axis must also follow the folding into the 5th dimension. It is shown in Fig. 9 as the t'' axis.
The fact that a law of conservation applies to the surface A is only natural insofar as it represents a body (in a time Δt), even if it is only "one-dimensional" (a "very thin" rod). In the case of a three-dimensional body, the lengths in the y and z directions are not changed by the movement. As a result of the flip-turn of the x-length l, it is located in an x''-y''-z'' space whose y'' and z'' axes are orthogonal to the x''-t'' plane (with x'' = x'), i.e. in five-dimensional space-time, which is neither graphically representable nor conceivable.
According to eq. 6, mass behaves in the same way as time and from eq. 5 follows
m0 = m ∙ cos φ
If the rest mass m0 in S' is called m' ( m0 := m´), then the mass in S'' be m''. Since the time Δt' is stretched by the factor cos φ to Δt´´= Δt´ / cos φ when changing from S' to S'' according to fig. 10 and the mass behaves like the time,
applies m´´ = m´ / cos φ
and thus m´´ = (m ∙ cos φ) / cos φ ,
so m´´ = m (10)
Fig. 10: Projection of the surfaces A' and A'' onto the t'-w-plane in the direction of the x'-axis; both have the "height" l.
The mass remains unchanged during the flip-turn in five-dimensional space-time. The relativistic mass only exists in four-dimensional space-time.
According to E = m c² , the energy of the body of mass m remains preserved at the flip-turn too.
6. Summary and outlook
The conclusions and interpretations presented point to two-dimensional and five-dimensional aspects of the world. Although noticeable effects only occur at relativistic speeds, in principle the flip-turn into the 5th dimension occurs at any speed. Since everything in the world is in motion, e.g. the thermal motion of molecules or electrons in atoms (at approx. 1/100 c), the world is five-dimensional, even if only in a "thin layer". This may explain orbitals, probability waves, the red shift of the galaxy spectra and other phenomena. In [1], the 5th dimension is used to resolve the paradox that the resulting speed of two speeds of light is again the speed of light, and the speed addition in five-dimensional space-time is described. By all this the hypothese of Theodor Kaluza of the existence of a fifth dimension [2] is supported.
References
[1] R. Sprenger: Relativistic addition of velocities in R5. In: DOI 10.5281/zenodo.10965934, 2024, www.zenodo.org
[2] T. Kaluza: Zum Unitätsproblem der Physik. In: Sitzungsberichte Preußische Akademie der Wissenschaften, 1921, S. 966–972, www.archive.org.