The fast non-commutative sharp drop

TL;DR

This paper presents an exact solution to a membrane matrix model describing a fast-moving, axially symmetric drop with a sharp tip, advancing understanding of dynamic membrane configurations.

Contribution

It introduces a novel exact solution for a membrane matrix model representing a rapidly moving sharp-tipped drop.

Findings

Provides an explicit membrane matrix solution for a fast-moving drop.

Models the world volume of a sharp-tipped axially symmetric drop.

Enhances theoretical understanding of dynamic membrane configurations.

Abstract

An exact GH membrane matrix model solution is given that corresponds to the world volume swept out by a fast moving axially symmetric drop with a sharp tip.

Equations

$$\begin{array}[]{l}\ddot{X}_{i}=-\sum_{j=1}^{d}\big{[}[X_{i},X_{j}],X_{j}\big{]},\\[7.11317pt] \sum_{i}[X_{i},\dot{X}_{i}]=0,\quad X_{i}^{\dagger}=X_{i}\end{array}$$

$$(t^2+x^2+y^2-z^2)(t+z{)}^2=C<0.$$

$$(t^2-x^2-y^2-z^2)=C(t+z{)}^6>0$$

$$(t^2+x^2+y^2-z^2)-6C\sqrt{x^2+y^2}(t+z{)}^2+3C^2(t+z{)}^4=0$$

$$3C^2(\delta t{)}^3-(\delta t)-2t=0;$$

$$v(x):=\frac{r}{(t+z)^{2}}\big{(}1-\sqrt{\frac{2}{3}+\frac{z^{2}-t^{2}}{3r^{2}}}\big{)}=C<0,$$

$$\begin{array}{rl}r(t,z)& =(t+z{)}^2(3C+\sqrt{6C^2+f(z;t)})\\ f(z;t)& =\frac{z-t}{(z+t{)}^3},f^{\prime }(z)=\frac{4t-2z}{(t+z{)}^4}\end{array}$$

$$\begin{array}[]{l}\tilde{r}(\tilde{t},\kappa:=\frac{t+z}{-C})=3\kappa^{2}(\sqrt{\frac{2}{3}+g(\kappa,\tilde{t})}-1)\\[7.11317pt] g(\kappa)=\frac{1}{9}(\frac{1}{\kappa^{2}}-\frac{2\tilde{t}}{\kappa^{3}}),\quad g^{\prime}(\kappa)=\frac{1}{9}(\frac{6\tilde{t}-2\kappa}{\kappa^{4}})\end{array}$$

$$\begin{array}{rl}\frac{\partial r}{\partial z}& =2(t+z)(3c+\sqrt{})+\frac{2t-z}{(t+z{)}^2\sqrt{}}\\ & \approx -2\sqrt{\frac{z-t}{z+t}}+\frac{2t-z}{\sqrt{z^2-t^2}}\\ & =\frac{z}{\sqrt{z^2-t^2}}\searrow -\infty \end{array}$$

$$\begin{split}\frac{\partial r}{\partial z}&\approx-2\delta t(3c+\sqrt{\hphantom{xx}})+\frac{t+\delta t}{(\delta t)^{2}\sqrt{\hphantom{xx}}}\\ &=\frac{1}{(\delta t)^{2}\sqrt{\hphantom{x}}}\big{(}t+\delta t-2(\delta t)^{3}(\underbrace{3C\sqrt{\hphantom{xx}}+\sqrt{\hphantom{xx}}^{2})}_{\rightarrow 0}\big{)}\\ &\rightarrow\frac{t+\delta t}{(\delta t)^{2}(-3C)}=\text{finite}>0,\end{split}$$

$$\ddot{R}=R(R{R}^{\prime }{)}^{\prime };$$

$$\begin{split}\ddot{x}_{i}&=\sum_{j=1}^{2}\left\{\{x_{i},x_{j}\},x_{j}\right\}=\Delta x_{i}\\ \sum_{j}\{x_{j},\dot{x}_{j}\}&:=\sum_{j}\big{(}\frac{\partial x_{j}}{\partial\mu}\frac{\partial\dot{x}_{j}}{\partial\psi}-\frac{\partial x_{j}}{\partial\psi}\frac{\partial\dot{x}_{j}}{\partial\mu}\big{)}=0\end{split}$$

$$\begin{array}{rl}{\zeta }^{\prime }& =\dot{R}{R}^{\prime }\\ \dot{\zeta }& =\frac{1}{2}({\dot{R}}^2+R^2{R}^{\prime 2})\end{array}$$

$$\ddot{\zeta }=\sum _j\left\{\{\zeta ,x_i\},x_i\right\}(=(R^2{\zeta }^{\prime }{)}^{\prime }).$$

$$\begin{array}{rl}R& =\sqrt{2}\frac{\mu }{\tau }+\epsilon {\tau }^2\\ \zeta & ={\epsilon }^2{\tau }^3-\frac{{\mu }^2}{{\tau }^3}+2\sqrt{2}\epsilon \mu ,\end{array}$$

$$\begin{split}x_{1}&=(\sqrt{2}\frac{\mu}{\tau}+\varepsilon\tau^{2})\cos\psi=\frac{\sqrt{2}}{\tau}\big{(}\bar{x}_{1}:=(\mu+\frac{\varepsilon\tau^{3}}{\sqrt{2}})\cos\psi\big{)}\\ x_{2}&=(\sqrt{2}\frac{\mu}{\tau}+\varepsilon\tau^{2})\sin\psi=\frac{\sqrt{2}}{\tau}\big{(}\bar{x}_{2}:=(\mu+\frac{\varepsilon\tau^{3}}{\sqrt{2}})\sin\psi\big{)}\end{split}$$

$${\bar{x}}_0:=\{{\bar{x}}_1,{\bar{x}}_2\}=\mu +\frac{\epsilon {\tau }^3}{\sqrt{2}}=:\mu +e$$

$$\{{\bar{x}}_{\alpha },{\bar{x}}_{\beta }\}={\epsilon }_{\alpha \beta }{}^{\gamma }{\bar{x}}_{\gamma },-q:={\bar{x}}_0^2-{\bar{x}}_1^2-{\bar{x}}_2^2=0,$$

$$\{h_1,y\}=h_2,\{y,h_2\}=h_1,\{h_1,h_2\}=0,$$

$${\bar{x}}_1=(y+e)h_1,{\bar{x}}_2=(y+e)h_2,{\bar{x}}_0=y+e$$

$${\bar{x}}_0^2-{\bar{x}}_1^2-{\bar{x}}_2^2=0$$

$$\begin{array}[]{l}[H_{1},Y]=iH_{2},\;[Y,H_{2}]=iH_{1},\;[H_{1},H_{2}]=0\\[4.26773pt] H^{2}_{1}+H^{2}_{2}=\mathbb{1}\end{array}$$

$$\begin{array}{rl}{X}_1& :=\frac{1}{2}(H_{1}Y+Y{H}_1)+e{H}_1=:Y_1+e{H}_1\\ X_2& :=\frac{1}{2}(H_{2}Y+Y{H}_2)+e{H}_2=:Y_2+e{H}_2\\ X_0& :=Y+e1=:Y_0+e{H}_3.\end{array}$$

$$[X_{\alpha },X_{\beta }]=i{\epsilon }_{\alpha \beta }{}^{\gamma }{X}_{\gamma },[Y_{\alpha },Y_{\beta }]=i{\epsilon }_{\alpha \beta }{}^{\gamma }{Y}_{\gamma },$$

$$\begin{array}{rl}[Y_{\alpha },H_{\beta }]+[H_{\alpha },Y_{\beta }]& =i{\epsilon }_{\alpha \beta }{}^{\gamma }{H}_{\gamma }\\ {\eta }^{\alpha \beta }(H_{\alpha }{Y}_{\beta }+Y_{\alpha }{H}_{\beta })& =0\\ H_{1}Y{H}_1+H_{2}Y{H}_2& =Y\\ H_1{Y}^2{H}_1+H_2{Y}^2{H}_2& =Y^2+1;\end{array}$$

$$\begin{array}{rl}-Q& :=Y_0^2-Y_1^2-Y_2^2=-\frac{1}{4}1\\ -Q^{\prime }& :=X_0^2-X_1^2-X_2^2\\ & =(Y_0+e{)}^2-(Y_1+e{H}_1{)}^2-(Y_2+e{H}_2{)}^2=-\frac{1}{4}1.\end{array}$$

$$\begin{array}{rl}\hat{\zeta }& \stackrel{?}{=}{\epsilon }^2{\tau }^{3}1-\frac{Y^2}{{\tau }^3}+2\sqrt{2}\epsilon Y-\frac{1}{12{\tau }^3}\\ \hat{\Delta }\hat{\zeta }& =\frac{2}{{\tau }^2}\Delta (2\sqrt{2}\epsilon Y-\frac{Y^2}{{\tau }^3})\\ & =-12\frac{Y^2}{{\tau }^5}+6{\epsilon }^2\tau -\frac{1}{{\tau }^5}\\ \ddot{\hat{\zeta }}& =6{\epsilon }^2\tau -12\frac{Y^2}{{\tau }^5}-\frac{1}{{\tau }^5}\\ \hat{\square }\hat{\zeta }& =({\partial }_t^2-\widehat{\Delta )}\hat{\zeta }=0;\end{array}$$

$$\begin{split}\Delta Y&:=-\big{[}[Y,X_{j}],X_{j}\big{]}=-\big{[}[X_{i},X_{j}],X_{j}\big{]}\\[4.26773pt] &=-i\varepsilon_{0j}\,^{k}[X_{k},X_{j}]=\varepsilon_{0j}\,^{k}\varepsilon_{kj}\,^{0}X_{0}\\[4.26773pt] &=2(Y+e\mathbb{1})\\[4.26773pt] \Delta Y^{2}&:=-\big{[}Y[X_{0},X_{j}],X_{j}\big{]}-\big{[}[X_{0},X_{j}]Y,X_{j}\big{]}\\[4.26773pt] &=Y\Delta Y+(\Delta Y)Y-2[X_{0},X_{1}]^{2}-2[X_{0},X_{2}]^{2}\\[4.26773pt] &=4Y(Y+e)+\underbrace{2X_{2}^{2}+2X_{1}^{2}}_{2(X_{0}=Y+e)^{2}+2/4}\\ &=6Y^{2}+8Ye+2e^{2}+\frac{1}{2}.\\[4.26773pt] \end{split}$$

$$Y|n>=(n+\kappa )|n>,W|n>=,W^{†}|n>=|n+1>$$

$$\begin{array}{rl}Z& :=X_1+i{X}_2=\frac{1}{2}(WY+YW)+eW\\ Z|n>& =Z_n|n-1>,Z^{†}|n>=Z_{n+1}|n+1>\\ Z_n& =n-\frac{1}{2}+e+\kappa .\end{array}$$

Full text

Exact solutions of non-linear differential equations, as useful as they can be, are generally difficult to get, and for the case of the membrane matrix model,

[TABLE]

(the $X_{i}\,(i=1,\ldots d)$ being time-dependent hermitean matrices) not many are known. In [2] it was shown that (time-independent) representations of $so(1,d=2)$ can be used to give non-commutative analogues of the ‘minimal’ (zero-mean curvature) time-like infinitely extended 3-manifolds defined by [3]

[TABLE]

A non-commutative analogue of the (for fixed $t$) compact variant

[TABLE]

was described in [4].

In this note I would like to give exact solutions to (1) that correspond to the one-parameter family of zero-mean-curvature $\mathfrak{M}_{3}\subset\mathbb{R}^{1,3}$ defined by

[TABLE]

which for $C<0$ describes a fast moving ($z=-t,\,r=0$ with velocity $-1$), for each fixed (say, positive) $t>\frac{-1}{9C}>0$ ‘compact’ axially symmetric drop with a sharp tip at $z=-t-\delta t<-t$,

[TABLE]

note that $\mathfrak{M}_{3}^{C<0}$ may also be written as the set of points where

[TABLE]

resp.

[TABLE]

resp. (letting $r=\frac{\tilde{r}}{-C}$, $t=\frac{\tilde{t}}{-C}$, $z=\frac{\tilde{z}}{-C}$; this scale-‘symmetry’ could of course be used already above to effectively put $C=-1$; in particular: $v(x)$ should be a solution of the level-set minimality condition without restricting to $\mathfrak{M}_{3}$)

[TABLE]

(for positive $\kappa$, $g$ has a maximum, for $\kappa=3\tilde{t}$, i.e. $z=2t$, with value $\frac{1}{3}\cdot\frac{1}{81\tilde{t}^{2}}$; which is the reason that for $\tilde{t}>\frac{1}{9}$ only negative $\kappa$ can make (8) positive; $g^{\prime}$ being positive there restricts $\kappa$ to the interval $[-\kappa_{M},0)$, with $3\kappa^{3}_{M}-\kappa_{M}-2\tilde{t}=0$, $g(-\kappa_{M})=\frac{1}{3}$, which has a unique positive solution $\kappa_{M}>0$ for any given $\tilde{t}>0$). For $z\nearrow-t$

[TABLE]

(as it should, for a closed surface to be round, at the top) while for $z\searrow-t-\delta t$

[TABLE]

meaning that at the bottom the drop is sharp, rather than round (though getting less and less sharp as $t\rightarrow\infty$); to show that $\frac{\partial r}{\partial z}$ is a monotonically decreasing function in the interval $[-t-\delta t,-t)$ (so, in particular, will have only one zero, at height $\hat{z}$ - where the radius $r$ becomes maximal) is cumbersome (though, at least for large enough $t$, presumably true).

As realized already more than 40 years ago [1], axially symmetric time-like minimal 3 manifolds can be described by solutions $R(\tau,\mu)$ of the simple-looking PDE

[TABLE]

$x_{1}=R\cos\psi$ and $x_{2}=R\sin\psi$ will then satisfy

[TABLE]

with respect to a $(\mu,\psi)$ Poisson bracket, and $\zeta=t-z$, (re)constructed via

[TABLE]

will also be in the kernel of the non-linear wave-operator $\square=\partial^{2}_{t}-\Delta$,

[TABLE]

The solution of (13) corresponding to the ‘fast moving sharp drop’ (4) is (with $\varepsilon=4C\leq 0$, cp.[5])

[TABLE]

i.e

[TABLE]

will solve (12). Defining

[TABLE]

one has (with $\varepsilon_{012}=+1$, and raising indices with respect to $\eta^{\alpha\beta}=$ diag$(1,-1,-1)$ metric)

[TABLE]

which together with the equally crucial property $\ddot{x}_{i}=\frac{2}{\tau^{2}}x_{i}$ almost trivially implies (12). Compared to previously discussed solutions (e.g. (2) in [2]), where $so(1,2)$ appears as well, the new, highly interesting, feature of (16)/(18) is the $\tau$-dependence of the $so(1,2)$ generators; though trivial from the Poisson point of view, just corresponding to a ‘constant’ shift of $\mu$ by $e$, it actually implies the existence of the following underlying structure, following from $h_{1}=\cos\psi$, $h_{2}=\sin\psi$, $y=\mu$ satisfying the $e(2)$ relations

[TABLE]

and $h_{1}^{2}+h_{2}^{2}=1$.

[TABLE]

with

[TABLE]

then necessarily satisfy (18), for any (central) $e$.

This clearly suggests to start with 3 elements $H_{1}$, $H_{2}$, and $Y$ satisfying

[TABLE]

(i.e. a ‘fuzzy cylinder’ [6]) and define, as non-commutative analogues of (20)

[TABLE]

It is straight forward to deduce (alone from (22)) that both the $X_{\alpha}$ and the $Y_{\alpha}$ are representations of $so(1,2)$, i.e.

[TABLE]

noting/verifying in particular

[TABLE]

while $\hat{X}_{1}=\frac{\sqrt{2}}{\tau}X_{1}(\tau)$ and $\hat{X}_{2}=\frac{\sqrt{2}}{\tau}X_{2}(\tau)$ then automatically satisfy (1) (the ‘quantum’ - calculation trivially following the classical one) there is one significant quantum reflection of the classical singularity111in [7] non-commutative analogues $\hat{\sum}(t)$ of a 1-parameter family of compact 2 dimensional surfaces $\sum(t)$ going through a singularity, $\sum(t_{0})$, were found ($\sum_{t>t_{0}}$ being spheres, topologically, and $\sum_{t<t_{0}}$ being tori) with $\hat{\sum}(t_{0})$ being ‘perfectly fine’, except exhibiting a sudden change in representation/dimension of the finite dimensional matrices representing $\hat{\sum}(t)$. As in [2], for a family of time-like regular faces in $\mathbb{R}^{1,3}$, a perfect match was found between classical and quantum Casimirs, it is intriguing to attribute the change form [math] to $\frac{1}{4}$ (cp.(26)) to a reflection of singularities of (4) (presumably the sharp edge, possibly the light-like line), namely the change of Casimir-value(s) from zero (cp.(21)) to

[TABLE]

What about (related to Lorentz-invariance, and classically necessary to reconstruct the wordvolume)222the last term, put in with hindsight, to cancel the otherwise occurring discrepancy between $\ddot{\hat{\zeta}}$ and $\hat{\Delta}\hat{\zeta}$ (which inherits an ‘anomalous term from the right-hand side of (26)) is a pure quantum effect; because of the numerical value (the factor $\frac{1}{12}$), and having in mind the general ‘reconstruction-algebra’ (generalizing the Virasoro algebra to arbitrarily extended objects) that was discovered in [8] it is tempting to speculate that there is a central-extension interpretation to it; in any case the existence of $\hat{\zeta}$ satisfying (27) is highly significant, as it is linked to the issue of Lorentz invariance of the GH-BFSS matrix model (which most people believe to/be proven to/not be realizable for (1); I disagree/am mildly optimistic)

[TABLE]

which, using (24) and (22), indeed follows from

[TABLE]

It is instructive to also check by a direct calculation that the quantized embedding coordinates $\hat{X}_{1}$, $\hat{X}_{2}$ and $\hat{\zeta}$ are annihilated by the ($\hat{X}_{i}$-dependent) quantum wave operator $\square$, by taking concrete representations of (22), resp. (letting $W:=H_{1}+iH_{2}$, $W^{\dagger}=H_{1}-iH_{2}$)

[TABLE]

$n\in\mathbb{Z}$, $\kappa\in[0,1)$, note that $WW^{\dagger}=1=W^{\dagger}W$

[TABLE]

[TABLE]

acting an $|n>$ gives $2Z^{3}_{n}-Z^{2}_{n-1}Z_{n}-Z_{n}Z^{2}_{n+1}=-2Z_{n}$ i.e. (for $Z_{n}\neq 0$) the condition

[TABLE]

which is satisfied for (30) (any $e$ and $\kappa$); in fact $Z_{n}=\pm n+$any constant will work, resp. (general solution of the simple recursion relation (32))

[TABLE]

Verifying (28) on the other hand is more involved :

[TABLE]

gives ( assuming $\hat{\zeta}|n>=\zeta_{n}|n>$ )

[TABLE]

[TABLE]

here taking $\kappa=0$ for simplicity, gives

[TABLE]

which matches $\frac{\tau^{2}}{2}\ddot{\zeta}_{n}=\frac{\tau^{2}}{2}(6\varepsilon^{2}\tau-\frac{12n^{2}}{\tau^{5}}+0-\frac{1}{\tau^{5}})$; note that without the quantum effect (the last term in (27)) (37) (the last term arising from (36), no matter what) would not be equal to $\frac{\tau^{2}}{2}\ddot{\zeta}_{n}$.

Acknowledgement: I would like to thank J.Arnlind, T.Damour, J.Eggers, M.Kontsevich, V.Roubtsov and V.Sokolov for discussions.