Squeeze mapping

Euclidean orthogonality is preserved by rotation in the left diagram; hyperbolic orthogonality with respect to hyperbola (B) is preserved by squeeze mapping in the right diagram

Spacetime geometry is conventionally developed as follows: Select (0,0) for a "here and now" in a spacetime. Light radiant left and right through this central event tracks two lines in the spacetime, lines that can be used to give coordinates to events away from (0,0). Trajectories of lesser velocity track closer to the original timeline (0,t). Any such velocity can be viewed as a zero velocity under a squeeze mapping called a Lorentz boost. This insight follows from a study of split-complex number multiplications and the diagonal basis which corresponds to the pair of light lines. Formally, a squeeze preserves the hyperbolic metric expressed in the form xy; in a different coordinate system. This application in the theory of relativity was noted in 1912 by Wilson and Lewis,[4] by Werner Greub,[5] and by Louis Kauffman.[6] Furthermore, the squeeze mapping form of Lorentz transformations was used by Gustav Herglotz (1909/10)[7] while discussing Born rigidity, and was popularized by Wolfgang Rindler in his textbook on relativity, who used it in his demonstration of their characteristic property.[8]

The term squeeze transformation was used in this context in an article connecting the Lorentz group with Jones calculus in optics.[9]

In fluid dynamics one of the fundamental motions of an incompressible flow involves bifurcation of a flow running up against an immovable wall. Representing the wall by the axis y = 0 and taking the parameter r = exp(t) where t is time, then the squeeze mapping with parameter r applied to an initial fluid state produces a flow with bifurcation left and right of the axis x = 0. The same model gives fluid convergence when time is run backward. Indeed, the area of any hyperbolic sector is invariant under squeezing.

For another approach to a flow with hyperbolic streamlines, see Potential flow § Power laws with n = 2.

In 1989 Ottino[10] described the "linear isochoric two-dimensional flow" as

${\displaystyle v_{1}=Gx_{2}\quad v_{2}=KGx_{1}}$

where K lies in the interval [−1, 1]. The streamlines follow the curves

${\displaystyle x_{2}^{2}-Kx_{1}^{2}=\mathrm {constant} }$

so negative K corresponds to an ellipse and positive K to a hyperbola, with the rectangular case of the squeeze mapping corresponding to K = 1.

Stocker and Hosoi[11] described their approach to corner flow as follows:

we suggest an alternative formulation to account for the corner-like geometry, based on the use of hyperbolic coordinates, which allows substantial analytical progress towards determination of the flow in a Plateau border and attached liquid threads. We consider a region of flow forming an angle of π/2 and delimited on the left and bottom by symmetry planes.

Stocker and Hosoi then recall Moffatt's[12] consideration of "flow in a corner between rigid boundaries, induced by an arbitrary disturbance at a large distance." According to Stocker and Hosoi,

For a free fluid in a square corner, Moffatt's (antisymmetric) stream function ... [indicates] that hyperbolic coordinates are indeed the natural choice to describe these flows.

The area-preserving property of squeeze mapping has an application in setting the foundation of the transcendental functions natural logarithm and its inverse the exponential function:

Definition: Sector(a,b) is the hyperbolic sector obtained with central rays to (a, 1/a) and (b, 1/b).

Lemma: If bc = ad, then there is a squeeze mapping that moves the sector(a,b) to sector(c,d).

Proof: Take parameter r = c/a so that (u,v) = (rx, y/r) takes (a, 1/a) to (c, 1/c) and (b, 1/b) to (d, 1/d).

Theorem (Gregoire de Saint-Vincent 1647) If bc = ad, then the quadrature of the hyperbola xy = 1 against the asymptote has equal areas between a and b compared to between c and d.

Proof: An argument adding and subtracting triangles of area 1⁄2, one triangle being {(0,0), (0,1), (1,1)}, shows the hyperbolic sector area is equal to the area along the asymptote. The theorem then follows from the lemma.

Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form logarithms of the asymptote index.

For instance, for a standard position angle which runs from (1, 1) to (x, 1/x), one may ask "When is the hyperbolic angle equal to one?" The answer is the transcendental number x = e.

A squeeze with r = e moves the unit angle to one between (e, 1/e) and (ee, 1/ee) which subtends a sector also of area one. The geometric progression

e, e², e³, ..., eⁿ, ...

corresponds to the asymptotic index achieved with each sum of areas

1,2,3, ..., n,...

which is a proto-typical arithmetic progression A + nd where A = 0 and d = 1 .

Following Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, Sophus Lie (1879) found a way to derive new pseudospherical surfaces from a known one. Such surfaces satisfy the Sine-Gordon equation:

${\displaystyle {\frac {d^{2}\Theta }{ds\ d\sigma }}=K\sin \Theta ,}$

where ${\displaystyle (s,\sigma )}$ are asymptotic coordinates of two principal tangent curves and ${\displaystyle \Theta }$ their respective angle. Lie showed that if ${\displaystyle \Theta =f(s,\sigma )}$ is a solution to the Sine-Gordon equation, then the following squeeze mapping (now known as Lie transform[13]) indicates other solutions of that equation:[14]

${\displaystyle \Theta =f\left(ms,\ {\frac {\sigma }{m}}\right).}$

Lie (1883) noticed its relation to two other transformations of pseudospherical surfaces:[15] The Bäcklund transform (introduced by Albert Victor Bäcklund in 1883) can be seen as the combination of a Lie transform with a Bianchi transform (introduced by Luigi Bianchi in 1879.) Such transformations of pseudospherical surfaces were discussed in detail in the lectures on differential geometry by Gaston Darboux (1894),[16] Luigi Bianchi (1894),[17] or Luther Pfahler Eisenhart (1909).[18]

It is known that the Lie transforms (or squeeze mappings) correspond to Lorentz boosts in terms of light-cone coordinates, as pointed out by Terng and Uhlenbeck (2000):[13]

Sophus Lie observed that the SGE [Sinus-Gordon equation] is invariant under Lorentz transformations. In asymptotic coordinates, which correspond to light cone coordinates, a Lorentz transformation is ${\displaystyle (x,t)\mapsto \left({\tfrac {1}{\lambda }}x,\lambda t\right)}$.

This can be represented as follows:

${\displaystyle {\begin{matrix}-c^{2}t^{2}+x^{2}=-c^{2}t^{\prime 2}+x^{\prime 2}\\\hline {\begin{aligned}ct'&=ct\gamma -x\beta \gamma &&=ct\cosh \eta -x\sinh \eta \\x'&=-ct\beta \gamma +x\gamma &&=-ct\sinh \eta +x\cosh \eta \end{aligned}}\\\hline u=ct+x,\ v=ct-x,\ k={\sqrt {\tfrac {1+\beta }{1-\beta }}}=e^{\eta }\\u'={\frac {u}{k}},\ v'=kv\\\hline u'v'=uv\end{matrix}}}$

where k corresponds to the Doppler factor in Bondi k-calculus, η is the rapidity.