Abstract
Without assuming the speed of light, the relativity principle together with the homogeneity and isotropy of space narrows the transformations between inertial frames to a family with a single number κ (Ignatowski’s theorem). Positive κ gives the Lorentz transformation, zero gives the Galilean one, and the negative side is removed by causality. This paper opens the removed side with numbers and checks that κ is the curvature of velocity space with the sign reversed. No new theorem or law is claimed.
First, the sign of κ sorts the invariant into three shapes. The quantity left unchanged is t² − κx²; its level curves are hyperbolas for positive κ, lines of constant time for zero, and circles for negative κ. Composing speeds 0.5 and 0.5 gives 0.8 for κ = +1, 1 for κ = 0 and $\mathbf{1.333333333}$ for κ = −1.
Second, on the negative branch velocities add like tangents. What adds is the arctangent of the speed. Composing speeds 1 and 1 makes the angle a right angle and the denominator vanishes; with speeds 2 and 2 the angles sum to $\mathbf{2.214297436}$ radians, past a right angle, and adding in the same direction gives $\mathbf{-1.333333333}$.
Third, repeating the same boost reverses the direction of time. For κ = −1 and speed 1, two boosts put the time axis on the space axis, four reverse time exactly, and eight return to the start. Even at speed 0.01 the time component turns negative after 158 boosts, and at 0.001 after 1571. Any two events at different places can have their order reversed.
Fourth, and this is the core: the curvature of velocity space is −κ. Two boosts at right angles leave a rotation. For speeds 0.5 and 0.5 it is $\mathbf{8.213210702}$ degrees for κ = +1 and $\mathbf{6.379370208}$ degrees the other way for κ = −1. Each equals the difference between the angle sum of a right triangle drawn in velocity space and 180 degrees, and that difference is −κ times the area. The area was summed without using the angles. For positive κ the surface is a hyperboloid and the angles fall short; for negative κ it is a sphere and they exceed.
Fifth, two numbers from earlier papers come out of the two sides of one tool. On the κ = +1 side the table of Wigner angles of Paper 33 is reproduced. Taking κ to be minus the inverse square of the Earth’s radius, the spherical excess of the surveying triangle counted in Paper 27 (14.8865 arcseconds) appears as the rotation left after three boosts ($\mathbf{14.8869}$ arcseconds by the spherical formula).
To be candid, no claim is made that a world with negative κ exists, and causality is not proved. What was done is to open with numbers the side that an earlier paper removed in one line, and to join the curvature of velocity space (Papers 60 and 33) and Ignatowski’s constant (Paper 83) through one number.
Keywords: Ignatowski’s theorem, velocity addition, velocity space, curvature, Wigner rotation, causality, sphere.
Scope (what is not claimed)
No new theorem or law is claimed. That the relativity principle, homogeneity, isotropy and the group property give a family of transformations with one constant (Ignatowski 1910; Frank and Rothe 1911), that adding causality removes the negative side (Lévy-Leblond 1976), the classification that assumes inertial transformations do not form a closed loop (Bacry and Lévy-Leblond 1968), that velocity space is hyperbolic (Varičak 1910), and that two boosts at right angles leave a rotation (Wigner 1939) are all known. No world with negative κ is claimed to exist. One number in a formula is moved past zero and the results are counted. Causality is not proved. The paper counts what happens on the negative side; it does not answer why nature keeps the order of events. Accelerated frames, gravity and quantum theory are not treated. Only transformations between inertial frames are considered. Relation to earlier papers: Paper 83 read Ignatowski’s theorem and stated in one line that causality removes the negative side of κ. Here the composition of velocities, the direction of time, the order of events and the rate of clocks on that removed side are counted. Paper 60 treated velocity space as a hyperbolic space of curvature −1/c², and Paper 33 counted that the Wigner angle is the area of a triangle in that space. Here it is counted that this curvature is −κ, and that with negative κ the same formula gives the angular excess of a spherical triangle. Paper 27 counted the angular excess of a triangle on the Earth’s surface, and Paper 33 wrote that the same theorem acts in velocity space with the sign changed. Here that number from the Earth’s surface comes out of the same matrix calculation as the residual rotation of the negative branch. Paper 113 treated the Galilean transformation as a limit of the Lorentz transformation. Here κ is brought to zero from both sides and both arrive at the Galilean transformation. Paper 108 separated the geometries that appear in relativity. The one treated here is the geometry of velocity space. What is added: a full set of numbers for the negative branch, the count, for three signs, that the residual rotation is −κ times the area, the numbers of Papers 33 and 27 obtained from the two sides of one tool, and the count of how many boosts it takes for time to reverse.
1. The question
Special relativity usually starts from the assumption that the speed of light is the same in every inertial frame. That assumption is not needed. From four things alone, the relativity principle, the homogeneity and isotropy of space, and the requirement that the transformations form a group, the transformations between inertial frames are narrowed to a family with a single number $\kappa$ (Ignatowski 1910; Frank and Rothe 1911). Paper 83 read this theorem and wrote that $\kappa=0$ is the Galilean transformation, $\kappa>0$ the Lorentz transformation, and that causality selects $\kappa\ge0$.
Paper 83 removed the side $\kappa<0$ in one line. What is on the removed side? This corpus has counted that velocity space is curved (Paper 60) and that the curvature shows up as the rotation left after two boosts at right angles (Paper 33). Are that curvature and $\kappa$ the same number? This paper moves $\kappa$ from positive through zero to negative and checks both questions with numbers.
2. One number κ, and the invariant
The transformation from an inertial frame to one moving at speed $v$ relative to it is, in the form of Paper 83,
$$t'=\gamma\,(t-\kappa v x),\qquad x'=\gamma\,(x-vt),\qquad \gamma=\frac{1}{\sqrt{1-\kappa v^{2}}}.$$This transformation leaves the quantity $t^{2}-\kappa x^{2}$ unchanged. The event $(t,x)=(0.7, 0.2)$ was carried to a frame of speed 0.6 as a check.
| $\kappa$ | before | after | shape of the level curves |
|---|---|---|---|
| $+1$ | $0.450000000$ | $0.450000000$ | hyperbolas |
| $0$ | $0.490000000$ | $0.490000000$ | lines of constant $t$ |
| $-1$ | $0.530000000$ | $0.530000000$ | circles |
For positive $\kappa$ the invariant is $t^{2}-\kappa x^{2}$ and the level curves are hyperbolas. For zero only $t^{2}$ remains, and time itself is unchanged. For negative $\kappa$ it is $t^{2}+|\kappa|x^{2}$ and the level curves are circles (once the scales are matched). The transformation is a rotation that mixes time and space.
When two transformations follow one another, the speeds compose as
$$w=\frac{u+v}{1+\kappa uv}.$$| $\kappa$ | 0.5 and 0.5 | 0.8 and 0.6 | 1 and 1 | 2 and 2 |
|---|---|---|---|---|
| $+1$ | $0.800000000$ | $0.945945946$ | not allowed | not allowed |
| $0$ | $1.000000000$ | $1.400000000$ | $2.000000000$ | $4.000000000$ |
| $-1$ | $1.333333333$ | $2.692307692$ | denominator 0 | $-1.333333333$ |
The row $\kappa=+1$ agrees with the table of Paper 33. The row $\kappa=0$ is plain addition. For $\kappa=-1$ the result is larger than the plain sum, the denominator vanishes for speeds 1 and 1, and the sign flips for speeds 2 and 2.
Every branch has a coordinate that simply adds. For $\kappa>0$ it is the inverse hyperbolic tangent of the speed, for $\kappa=0$ the speed itself, and for $\kappa<0$ the arctangent of the speed. For five values of $\kappa$ and three pairs of speeds, the coordinate of the composed speed differed from the sum of the coordinates by at most $2.220\times10^{-16}$. On the branch $\kappa<0$ the composition of velocities is the addition formula for the tangent.
3. Opening the negative branch
For $\kappa=-1$ the additive coordinate is an angle. Speed 0.5 corresponds to $0.463647609$ radians, speed 1 to $0.785398163$ radians (45 degrees), and speed 2 to $1.107148718$ radians. Composing speeds 1 and 1 gives an angle of 90 degrees, where the tangent diverges. With speeds 2 and 2 the angles sum to $2.214297436$ radians, past the right angle ($1.570796327$ radians). Its tangent is $-1.333333333$: adding in the same direction reverses the direction. There is no upper limit on speed, but "infinite speed" is not a far horizon; it is just one angle, the right angle.
What happens when the time axis turns past a right angle? The boost of speed 1 was repeated and the image of $(t,x)=(1,0)$ followed.
| number of boosts | $t$ component | $x$ component |
|---|---|---|
| 1 | $+0.707106781$ | $-0.707106781$ |
| 2 | $+0.000000000$ | $-1.000000000$ |
| 3 | $-0.707106781$ | $-0.707106781$ |
| 4 | $-1.000000000$ | $-0.000000000$ |
| 8 | $+1.000000000$ | $+0.000000000$ |
After two boosts the time axis lies on the original space axis. After four, time points exactly backwards. After eight it is back. The future of a clock at rest becomes its past after only four boosts in the same direction.
This is not special to speed 1. The number of boosts needed for the time component to turn negative was counted for slow speeds.
| speed | boosts until the time component is negative ($\kappa=-1$) | time component after as many boosts ($\kappa=+1$) |
|---|---|---|
| $0.5$ | 4 | $4.555556$ |
| $0.1$ | 16 | $2.590248$ |
| $0.01$ | 158 | $2.530588$ |
| $0.001$ | 1571 | $2.509648$ |
However slow the speed, time reverses after a finite number of boosts. The number is about $\pi$ divided by twice the speed. For $\kappa=+1$ the same number of boosts only makes the time component larger than 1; its sign does not change. For $\kappa=0$ it stays 1 for ever.
The rate of a moving clock is also the other way round. A clock moving at speed $v$ ticks $\sqrt{1-\kappa v^{2}}$ times the coordinate time. At speed 0.6 this is $0.800000$ for $\kappa=+1$ (it runs slow), $1.000000$ for $\kappa=0$, and $1.166190$ for $\kappa=-1$ (it runs fast). In the world of circles the slanted path is the longer one, so the twin who travels ends up older.
4. Events whose order can be reversed
Causality says that the order of cause and effect is the same in every inertial frame. For two events separated by $\Delta t>0$ in time and $\Delta x$ in space, the time separation in a moving frame is $\gamma(\Delta t-\kappa v\Delta x)$. Whether some frame makes it negative was counted on a grid with $0<\Delta t<1$ and $-1<\Delta x<1$.
| branch | fraction of pairs whose order can be reversed |
|---|---|
| $\kappa=+4$ (speeds below 0.5) | $0.749834$ |
| $\kappa=+1$ (speeds below 1) | $0.500000$ |
| $\kappa=+0.25$ (speeds below 2) | $0.250000$ |
| $\kappa=0$ | $0.000000$ |
| $\kappa=-1$ | $1.000000$ |
For $\kappa>0$ the only pairs that can be reversed are those too far apart to be joined even at the invariant speed. Such pairs cannot be cause and effect, so nothing is lost. For $\kappa=0$ no pair can be reversed. For $\kappa<0$ every pair at different places can be reversed. No pair with a protected order remains.
A speed limit might seem to save the situation. For $\kappa=-1$ with speeds limited to 1 the fraction is $0.500000$, to 10 it is $0.949994$, and to 100 it is $0.995000$. But on this branch composing speeds 1 and 1 already makes the denominator vanish. The limit is broken by applying the transformation twice. The group property and a speed limit cannot both hold on this branch.
5. The curvature of velocity space is −κ
So far everything was along one line. When boosts in different directions follow one another, the result is not a boost alone; a rotation is left (Wigner 1939). Paper 33 counted, in the world $\kappa=+1$, that the angle of this rotation is the area of a triangle drawn in velocity space. The same calculation is done here with the sign of $\kappa$ changed.
A boost of speed 0.5 along $x$ is followed by a boost of speed 0.5 along $y$. The combined transformation was split into one boost and a rotation, and the angle of the rotation was measured. Next to it are the difference between the angle sum of the right triangle in velocity space (its legs being the lengths of the two boosts) and $\pi$, and the area. The area was obtained by summing area elements, without using the angles.
| $\kappa$ | residual rotation (deg) | angle sum $-\pi$ (deg) | area | $-\kappa\times$ area (deg) |
|---|---|---|---|---|
| $+1$ | $+8.213210702$ | $-8.213210702$ | $0.143347569$ | $-8.213210702$ |
| $-1$ | $-6.379370208$ | $+6.379370208$ | $0.111341014$ | $+6.379370208$ |
The legs of the triangle have length $0.549306144$ for $\kappa=+1$ and $0.463647609$ for $\kappa=-1$. The hypotenuse from the matrices agreed with the triangle formula: $0.795365461$ for $\kappa=+1$ and $0.643501109$ for $\kappa=-1$.
Three things can be read off. One: the size of the residual rotation equals the difference between the angle sum and $\pi$. Two: that difference is $-\kappa$ times the area. On a surface of curvature $K$ the angle sum of a triangle differs from $\pi$ by $K$ times the area, so the curvature of velocity space is $K=-\kappa$. Three: the direction of the rotation reverses with the sign of $\kappa$. For $\kappa>0$ the surface is a hyperboloid and the angles fall short. For $\kappa<0$ it is a sphere and the angles exceed.
For six values of $\kappa$ ($2$, $1$, $0.3$, $-0.3$, $-1$, $-2$) and three pairs of lengths, the ratio of the residual rotation to $|\kappa|\times$ area differed from 1 by at most $3.775\times10^{-15}$.
Paper 60 wrote the curvature of velocity space as $-1/c^{2}$. Paper 83 wrote the invariant speed as $c=1/\sqrt{\kappa}$. Side by side they give $K=-\kappa$: one number had two names. What this paper adds is the check that the equality holds on the negative side of $\kappa$ as well, by the same matrix calculation. The single number left by Ignatowski’s theorem was the way velocity space is curved.
On the branch $\kappa<0$ the speed $v$ is the tangent of an angle on a sphere. In the map that projects a sphere from its centre onto a plane (the gnomonic projection), great circles become straight lines. The plane of velocities is that map. This map sends a point of the sphere and the point directly opposite to the same place. A state with time pointing forwards and one with time pointing backwards have the same velocity. The composition of speeds 2 and 2 came out negative because it crossed the equator of the sphere to the far side.
6. Two earlier numbers from one tool
The settings of two earlier papers were put into the same matrix calculation.
The first is Paper 33. For $\kappa=+1$, a boost of length 1.0 along $x$ is followed by a boost of length 0.9 at angle $\alpha$.
| $\alpha$ | combined length ($\kappa=+1$) | residual rotation ($\kappa=+1$) | combined length ($\kappa=-1$) | residual rotation ($\kappa=-1$) |
|---|---|---|---|---|
| $30^\circ$ | $1.849219714503$ | $0.166417618609$ | $1.807994969$ | $0.338792512$ |
| $60^\circ$ | $1.694782265369$ | $0.305303244758$ | $1.564512419$ | $0.514871720$ |
| $90^\circ$ | $1.431199273509$ | $0.385102527078$ | $1.228281076$ | $0.516024386$ |
| $120^\circ$ | $1.053494636279$ | $0.369891574314$ | $0.842725624$ | $0.398441845$ |
The two columns for $\kappa=+1$ agreed with the table of Paper 33 (combined rapidity and Wigner angle). The two columns for $\kappa=-1$ are the values when sides of the same length are laid on a sphere. At a right angle the residual rotation $0.516024386$ radians agreed with the angular excess of the spherical right triangle, and so did the hypotenuse $1.228281076$. For the same lengths the rotation is larger on the spherical side.
The second is Paper 27. Paper 27 counted the angular excess of a triangle on the Earth’s surface with sides 69, 85 and 107 km as 14.8865 arcseconds. A sphere of radius $R$ has the same shape as the velocity space of a world with $\kappa=-1/R^{2}$. So the Earth’s radius, 6371 km, was put in, three boosts along the three sides of the triangle were applied in turn, and the rotation left after the round trip was measured.
| how it was counted | arcseconds |
|---|---|
| rotation left after three boosts around the triangle | $14.8869$ |
| angular excess from the formula for a spherical triangle | $14.8869$ |
| value in Paper 27 (plane area divided by the radius squared) | $14.8865$ |
The rotation left after the round trip agreed with the angular excess from the spherical formula. The difference from the value of Paper 27 is less than a thousandth of an arcsecond; it comes from Paper 27 having used the plane formula for the area of the triangle. Paper 33 wrote that for the triangle on the Earth and the triangle in velocity space "the same theorem acts with the sign changed". What is seen here is that this is not a figure of speech: the two are related by changing the sign of $\kappa$ in one formula. The angular excess that Gauss’s survey saw corresponds to the Wigner rotation of a world with negative $\kappa$.
7. Moving κ smoothly
$\kappa$ was moved from $-1$ to $+1$ and three quantities were listed. The speeds are 0.5 and 0.5, and the clock moves at 0.6.
| $\kappa$ | composition on a line | clock rate | rotation after a right angle (deg) | rotation $\div\kappa$ (deg) |
|---|---|---|---|---|
| $-1.00$ | $1.333333333$ | $1.166190379$ | $-6.379370208$ | $6.379370$ |
| $-0.50$ | $1.142857143$ | $1.086278049$ | $-3.372286683$ | $6.744573$ |
| $-0.10$ | $1.025641026$ | $1.017840852$ | $-0.707373273$ | $7.073733$ |
| $-0.01$ | $1.002506266$ | $1.001798383$ | $-0.071530330$ | $7.153033$ |
| $0$ | $1.000000000$ | $1.000000000$ | $+0.000000000$ | |
| $+0.01$ | $0.997506234$ | $0.998198377$ | $+0.071709380$ | $7.170938$ |
| $+0.10$ | $0.975609756$ | $0.981835017$ | $+0.725282401$ | $7.252824$ |
| $+0.50$ | $0.888888889$ | $0.905538514$ | $+3.822553729$ | $7.645107$ |
| $+1.00$ | $0.800000000$ | $0.800000000$ | $+8.213210702$ | $8.213211$ |
The three quantities pass through $\kappa=0$ without a break. At $\kappa=0$ composition is addition, the clock is unchanged, and no rotation is left. On either side the direction of the rotation is opposite. For small $\kappa$ the rotation is proportional to $\kappa$, with coefficient half the product of the two speeds (0.125 radians, $7.161972$ degrees).
Paper 113 treated the Galilean transformation as a limit of the Lorentz transformation. In the table above the Galilean transformation is the limit from the positive side and also from the negative side. It is the flat surface lying between the hyperboloid and the sphere. The form of the equations alone does not decide on which side of the flat surface the world lies.
8. Why the negative side is removed
The literature gives two wordings of the reason for removing the negative side. One is causality: the order of events is required to be kept in every inertial frame (Lévy-Leblond 1976). The other assumes that the boosts in one direction do not form a closed loop (Bacry and Lévy-Leblond 1968).
The two are two faces of one fact. For $\kappa=-1$, half the trace of the matrix of $n$ boosts of speed 1 went $+0.707106781$, $+0.000000000$, $-1.000000000$, $+1.000000000$ for $n=1$, 2, 4, 8, and after eight boosts the identity was recovered. The set of boosts is a closed loop. Half way round the loop, time is reversed. For $\kappa=+1$, after 4, 8 and 16 boosts of speed 0.5 the same quantity grows as $4.555556$, $40.506173$, $3280.500076$ and does not return. Nor does it return for $\kappa=0$.
The closing of the loop and the reversal of time both come from one shape, the sphere. A sphere has no edge; going straight on leads round the back and home again. On a hyperboloid or a plane, going straight on only leads away. What causality selects is the sign of the curvature of velocity space. Surfaces of positive curvature are removed; those of zero or negative curvature remain.
Paper 83 wrote that experiment decides whether an invariant speed exists. In the language of curvature, what experiment decides is whether velocity space is flat or negatively curved. That curvature, as Paper 60 wrote, is measured as Thomas precession.
9. Where the check stopped once
The check stopped once. In the table of the number of boosts until time reverses, speed 1 was also put into the $\kappa=+1$ side for comparison, and the additive coordinate diverged. Speed 1 is not allowed for $\kappa=+1$, so stopping was correct. That entry was removed from the table. The difference between speed 1 being nothing special on the branch $\kappa<0$ and the invariant speed being a wall on the branch $\kappa>0$ showed up directly.
Closing
Closing: the single number κ left by Ignatowski’s theorem is the curvature of velocity space multiplied by −1. The rotation left after two boosts at right angles is $\mathbf{8.213210702}$ degrees for κ = +1 and $\mathbf{6.379370208}$ degrees the other way for κ = −1, and each agreed with −κ times the area of the triangle. On the negative branch velocities add like tangents, repeating one boost reverses time after a finite number of steps, and any two events at different places can have their order reversed. The Wigner angles of Paper 33 came out of the positive side and the angular excess of the triangle on the Earth in Paper 27 out of the negative side, by the same calculation. To be candid, no world with negative κ is claimed to exist, and why causality holds is not answered. What has been found is that the case causality removes is the case in which velocity space is a sphere. The separator is whether the level curves of the invariant are hyperbolas, straight lines or circles.
References
- Bacry, H., & Lévy-Leblond, J.-M. (1968). Possible Kinematics. Journal of Mathematical Physics.
- Frank, P., & Rothe, H. (1911). Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme. Annalen der Physik.
- Ignatowski, W. v. (1910). Einige allgemeine Bemerkungen zum Relativitätsprinzip. Physikalische Zeitschrift.
- Lévy-Leblond, J.-M. (1976). One more derivation of the Lorentz transformation. American Journal of Physics.
- Varičak, V. (1910). Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie. Physikalische Zeitschrift.
- Wigner, E. P. (1939). On unitary representations of the inhomogeneous Lorentz group. Annals of Mathematics.
- Yamagishi, Y. (2026). “We Measured the Curvature” — of Which Manifold? [Paper 27]. YOI: https://www.yamagishiyuuki.com/yoi/27; DOI: 10.5281/zenodo.22090333
- Yamagishi, Y. (2026). The Wigner Angle Is an Area [Paper 33]. YOI: https://www.yamagishiyuuki.com/yoi/33; DOI: 10.5281/zenodo.22090872
- Yamagishi, Y. (2026). $c$ as the Ball of Velocity Space [Paper 60]. YOI: https://www.yamagishiyuuki.com/yoi/60; DOI: 10.5281/zenodo.21598012
- Yamagishi, Y. (2026). Why Does a “Maximum Speed” Exist? [Paper 83]. YOI: https://www.yamagishiyuuki.com/yoi/83; DOI: 10.5281/zenodo.21741190
- Yamagishi, Y. (2026). The Geometry Where the Invariant Lives [Paper 108]. YOI: https://www.yamagishiyuuki.com/yoi/108; DOI: 10.5281/zenodo.21837313
- Yamagishi, Y. (2026). Modules Are Not Walls but Limits [Paper 113]. YOI: https://www.yamagishiyuuki.com/yoi/113; DOI: 10.5281/zenodo.21962996
Reference
Yamagishi, Y. (2026). Ignatowski’s κ Is Minus the Curvature of Velocity Space [Paper 1090]. YOI: https://www.yamagishiyuuki.com/yoi/1090
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1090. This paper is a preprint and has not been peer reviewed.