1 Introduction
1.1 Going further
2 The Simplest Example of an Analogue Model
^{ $\text{1}$ } The need for a certain degree of caution regarding the allegedly straightforward physics of simple fluids might be inferred from the fact that the Clay Mathematics Institute is currently offering a US$1,000,000 Millennium Prize for significant progress on the question of existence and uniqueness of solutions to the Navier–Stokes equation. See http://www.claymath.org/millennium/ for details.
2.1 Background
^{ $\text{2}$ } In correct English, the word “dumb” means “mute”, as in “unable to speak”. The word “dumb” does not mean “stupid”, though even many native English speakers get this wrong.
2.2 Geometrical acoustics
$$\begin{array}{c}\frac{d\mathbf{x}}{dt}=c\mathbf{n}+\mathbf{v}.\end{array}$$ | (1) |
$$\begin{array}{c}-{c}^{2}d{t}^{2}+{\left(d\mathbf{x}-\mathbf{v}dt\right)}^{2}=0.\end{array}$$ | (2) |
$$\begin{array}{c}-[{c}^{2}-{v}^{2}]d{t}^{2}-2\mathbf{v}\cdot d\mathbf{x}dt+d\mathbf{x}\cdot d\mathbf{x}=0.\end{array}$$ | (3) |
$$\begin{array}{c}\text{g}={\Omega}^{2}\left[\begin{array}{cc}-({c}^{2}-{v}^{2})& -{\mathbf{v}}^{T}\\ -\mathbf{v}& \mathbf{I}\end{array}\right],\end{array}$$ | (4) |
^{ $\text{3}$ } For instance, whenever one has a system of PDEs that can be written in first-order quasi-linear symmetric hyperbolic form, then it is an exact non-perturbative result that the matrix of coefficients for the first-derivative terms can be used to construct a conformal class of metrics that encodes the causal structure of the system of PDEs. For barotropic hydrodynamics this is briefly discussed in [80] . This analysis is related to the behaviour of characteristics of the PDEs, and ultimately can be linked back to the Fresnel equation that appears in the eikonal limit.
2.3 Physical acoustics
$$\begin{array}{c}{\partial}_{t}^{2}\phi ={c}^{2}{\nabla}^{2}\phi .\end{array}$$ | (5) |
$$\begin{array}{c}\Delta \phi \equiv \frac{1}{\sqrt{-g}}{\partial}_{\mu}\left(\sqrt{-g}{g}^{\mu \nu}{\partial}_{\nu}\phi \right)=0.\end{array}$$ | (6) |
$$\begin{array}{c}{g}_{\mu \nu}(t,\mathbf{x})\equiv \frac{\rho}{c}\left[\begin{array}{ccc}-({c}^{2}-{v}^{2})& \mathtt{.}\mathtt{.}\mathtt{.}& -{\mathbf{v}}^{T}\\ \cdot \cdot \cdot \cdots \cdots \cdots & \cdot & \cdot \cdot \cdot \cdots \\ -\mathbf{v}& \mathtt{.}\mathtt{.}\mathtt{.}& \mathbf{I}\\ \end{array}\right].\end{array}$$ | (7) |
$$\begin{array}{c}{\partial}_{t}\rho +\text{}\mathsf{\nabla}\text{}\cdot (\rho \mathbf{v})=0,\end{array}$$ | (8) |
$$\begin{array}{c}\rho \frac{d\mathbf{v}}{dt}\equiv \rho \left[{\partial}_{t}\mathbf{v}+(\mathbf{v}\cdot \text{}\mathsf{\nabla}\text{})\mathbf{v}\right]=\mathbf{f}.\end{array}$$ | (9) |
$$\begin{array}{c}\mathbf{f}=-\text{}\mathsf{\nabla}\text{}p.\end{array}$$ | (10) |
$$\begin{array}{c}{\partial}_{t}\mathbf{v}=\mathbf{v}\times (\text{}\mathsf{\nabla}\text{}\times \mathbf{v})-\frac{1}{\rho}\text{}\mathsf{\nabla}\text{}p-\text{}\mathsf{\nabla}\text{}\left(\frac{1}{2}{v}^{2}\right).\end{array}$$ | (11) |
$$\begin{array}{c}h\left(p\right)={\int}_{0}^{p}\frac{d{p}^{\prime}}{\rho \left({p}^{\prime}\right)};sothat\text{}\mathsf{\nabla}\text{}h=\frac{1}{\rho}\text{}\mathsf{\nabla}\text{}p.\end{array}$$ | (12) |
$$\begin{array}{c}-{\partial}_{t}\phi +h+\frac{1}{2}(\text{}\mathsf{\nabla}\text{}\phi {)}^{2}=0.\end{array}$$ | (13) |
$$\begin{array}{ccc}\rho & =& {\rho}_{0}+\epsilon {\rho}_{1}+O\left({\epsilon}^{2}\right),\end{array}$$ | (14) |
$$\begin{array}{ccc}p& =& {p}_{0}+\epsilon {p}_{1}+O\left({\epsilon}^{2}\right),\end{array}$$ | (15) |
$$\begin{array}{ccc}\phi & =& {\phi}_{0}+\epsilon {\phi}_{1}+O\left({\epsilon}^{2}\right).\end{array}$$ | (16) |
$$\begin{array}{ccc}& & {\partial}_{t}{\rho}_{0}+\text{}\mathsf{\nabla}\text{}\cdot \left({\rho}_{0}{\mathbf{v}}_{0}\right)=0,\end{array}$$ | (17) |
$$\begin{array}{ccc}& & {\partial}_{t}{\rho}_{1}+\text{}\mathsf{\nabla}\text{}\cdot ({\rho}_{1}{\mathbf{v}}_{0}+{\rho}_{0}{\mathbf{v}}_{1})=0.\end{array}$$ | (18) |
$$\begin{array}{c}h\left(p\right)=h({p}_{0}+\epsilon {p}_{1}+O({\epsilon}^{2}\left)\right)={h}_{0}+\epsilon \frac{{p}_{1}}{{\rho}_{0}}+O\left({\epsilon}^{2}\right).\end{array}$$ | (19) |
$$\begin{array}{ccc}& & -{\partial}_{t}{\phi}_{0}+{h}_{0}+\frac{1}{2}(\text{}\mathsf{\nabla}\text{}{\phi}_{0}{)}^{2}=0.\end{array}$$ | (20) |
$$\begin{array}{ccc}& & -{\partial}_{t}{\phi}_{1}+\frac{{p}_{1}}{{\rho}_{0}}-{\mathbf{v}}_{0}\cdot \text{}\mathsf{\nabla}\text{}{\phi}_{1}=0.\end{array}$$ | (21) |
$$\begin{array}{c}{p}_{1}={\rho}_{0}\left({\partial}_{t}{\phi}_{1}+{\mathbf{v}}_{0}\cdot \text{}\mathsf{\nabla}\text{}{\phi}_{1}\right).\end{array}$$ | (22) |
$$\begin{array}{c}{\rho}_{1}=\frac{\partial \rho}{\partial p}{p}_{1}=\frac{\partial \rho}{\partial p}{\rho}_{0}({\partial}_{t}{\phi}_{1}+{\mathbf{v}}_{0}\cdot \text{}\mathsf{\nabla}\text{}{\phi}_{1}).\end{array}$$ | (23) |
$$\begin{array}{c}-{\partial}_{t}\left(\frac{\partial \rho}{\partial p}{\rho}_{0}({\partial}_{t}{\phi}_{1}+{\mathbf{v}}_{0}\cdot \text{}\mathsf{\nabla}\text{}{\phi}_{1})\right)+\text{}\mathsf{\nabla}\text{}\cdot \left({\rho}_{0}\text{}\mathsf{\nabla}\text{}{\phi}_{1}-\frac{\partial \rho}{\partial p}{\rho}_{0}{\mathbf{v}}_{0}({\partial}_{t}{\phi}_{1}+{\mathbf{v}}_{0}\cdot \text{}\mathsf{\nabla}\text{}{\phi}_{1})\right)=0.\end{array}$$ | (24) |
$$\begin{array}{c}{c}^{-2}\equiv \frac{\partial \rho}{\partial p}.\end{array}$$ | (25) |
$$\begin{array}{c}{f}^{\mu \nu}(t,\mathbf{x})\equiv \frac{{\rho}_{0}}{{c}^{2}}\left[\begin{array}{ccc}-1& ...& -{v}_{0}^{j}\\ \cdot \cdot \cdot \cdots & \cdot & \cdot \cdot \cdot \cdots \cdots \cdots \\ -{v}_{0}^{i}& ...& ({c}^{2}{\delta}^{ij}-{v}_{0}^{i}{v}_{0}^{j})\\ \end{array}\right].\end{array}$$ | (26) |
$$\begin{array}{c}{\partial}_{\mu}\left({f}^{\mu \nu}{\partial}_{\nu}{\phi}_{1}\right)=0.\end{array}$$ | (27) |
$$\begin{array}{c}\Delta \phi \equiv \frac{1}{\sqrt{-g}}{\partial}_{\mu}\left(\sqrt{-g}{g}^{\mu \nu}{\partial}_{\nu}\phi \right).\end{array}$$ | (28) |
$$\begin{array}{c}\sqrt{-g}{g}^{\mu \nu}={f}^{\mu \nu}.\end{array}$$ | (29) |
$$\begin{array}{c}det\left({f}^{\mu \nu}\right)=(\sqrt{-g}{)}^{4}{g}^{-1}=g.\end{array}$$ | (30) |
$$\begin{array}{c}det\left({f}^{\mu \nu}\right)={\left(\frac{{\rho}_{0}}{{c}^{2}}\right)}^{4}\cdot \left[(-1)\cdot ({c}^{2}-{v}_{0}^{2})-(-{v}_{0}{)}^{2}\right]\cdot \left[{c}^{2}\right]\cdot \left[{c}^{2}\right]=-\frac{{\rho}_{0}^{4}}{{c}^{2}}.\end{array}$$ | (31) |
$$\begin{array}{c}g=-\frac{{\rho}_{0}^{4}}{{c}^{2}};\sqrt{-g}=\frac{{\rho}_{0}^{2}}{c}.\end{array}$$ | (32) |
$$\begin{array}{c}{g}^{\mu \nu}(t,\mathbf{x})\equiv \frac{1}{{\rho}_{0}c}\left[\begin{array}{ccc}-1& ...& -{v}_{0}^{j}\\ \cdot \cdot \cdot \cdots & \cdot & \cdot \cdot \cdot \cdots \cdots \cdots \\ -{v}_{0}^{i}& ...& ({c}^{2}{\delta}^{ij}-{v}_{0}^{i}{v}_{0}^{j})\\ \end{array}\right].\end{array}$$ | (33) |
$$\begin{array}{c}{g}_{\mu \nu}\equiv \frac{{\rho}_{0}}{c}\left[\begin{array}{ccc}-({c}^{2}-{v}_{0}^{2})& ...& -{v}_{0}^{j}\\ \cdot \cdot \cdot \cdots \cdots \cdots & \cdot & \cdot \cdot \cdot \cdots \\ -{v}_{0}^{i}& ...& {\delta}_{ij}\\ \end{array}\right].\end{array}$$ | (34) |
$$\begin{array}{c}d{s}^{2}\equiv {g}_{\mu \nu}d{x}^{\mu}d{x}^{\nu}=\frac{{\rho}_{0}}{c}\left[-{c}^{2}d{t}^{2}+(d{x}^{i}-{v}_{0}^{i}dt){\delta}_{ij}(d{x}^{j}-{v}_{0}^{j}dt)\right].\end{array}$$ | (35) |
2.4 General features of the acoustic metric
$$\begin{array}{c}{f}^{\mu \nu}=\sqrt{-g}{g}^{\mu \nu}\end{array}$$ | (36) |
$$\begin{array}{c}{\eta}_{\mu \nu}\equiv (diag[-{c}_{light}^{2},1,1,1]{)}_{\mu \nu}.\end{array}$$ | (37) |
$$\begin{array}{c}{g}^{\mu \nu}\left({\nabla}_{\mu}t\right)\left({\nabla}_{\nu}t\right)=-\frac{1}{{\rho}_{0}c}<0.\end{array}$$ | (38) |
$$\begin{array}{c}{V}^{\mu}=\frac{(1;{v}_{0}^{i})}{\sqrt{{\rho}_{0}c}},\end{array}$$ | (39) |
$$\begin{array}{c}{g}_{\mu \nu}{V}^{\mu}{V}^{\nu}=g(V,V)=-1.\end{array}$$ | (40) |
$$\begin{array}{c}{\nabla}_{\mu}t=(1,0,0,0);{\nabla}^{\mu}t=-\frac{(1;{v}_{0}^{i})}{{\rho}_{0}c}=-\frac{{V}^{\mu}}{\sqrt{{\rho}_{0}c}}.\end{array}$$ | (41) |
$$\begin{array}{c}\tau =\int \sqrt{{\rho}_{0}c}dt,\end{array}$$ | (42) |
2.5 Dumb holes – ergo-regions, horizons, and surface gravity
$$\begin{array}{c}{K}^{\mu}\equiv (\partial /\partial t{)}^{\mu}=(1,0,0,0{)}^{\mu}.\end{array}$$ | (43) |
$$\begin{array}{c}{g}_{\mu \nu}(\partial /\partial t{)}^{\mu}(\partial /\partial t{)}^{\nu}={g}_{tt}=-[{c}^{2}-{v}^{2}].\end{array}$$ | (44) |
$$\begin{array}{c}d{s}^{2}=\frac{\rho}{c}\left[-{c}^{2}d{t}^{2}+(d\mathbf{x}-\mathbf{v}dt{)}^{2}\right].\end{array}$$ | (45) |
$$\begin{array}{c}{K}^{2}\equiv {g}_{\mu \nu}{K}^{\mu}{K}^{\nu}\equiv -\left|\right|\mathbf{K}|{|}^{2}=-\frac{\rho}{c}[{c}^{2}-{v}^{2}].\end{array}$$ | (46) |
$$\begin{array}{c}d{s}^{2}=\frac{\rho}{c}\left[-({c}^{2}-{v}^{2})d{t}^{2}-2\mathbf{v}\cdot d\mathbf{x}dt+(d\mathbf{x}{)}^{2}\right].\end{array}$$ | (47) |
$$\begin{array}{c}d\tau =dt+\frac{\mathbf{v}\cdot d\mathbf{x}}{{c}^{2}-{v}^{2}}.\end{array}$$ | (48) |
$$\begin{array}{c}d{s}^{2}=\frac{\rho}{c}\left[-({c}^{2}-{v}^{2})d{\tau}^{2}+\left\{{\delta}_{ij}+\frac{{v}^{i}{v}^{j}}{{c}^{2}-{v}^{2}}\right\}d{x}^{i}d{x}^{j}\right].\end{array}$$ | (49) |
$$\begin{array}{c}\text{}\mathsf{\nabla}\text{}\times \left\{\frac{\mathbf{v}}{({c}^{2}-{v}^{2})}\right\}=0,\end{array}$$ | (50) |
$$\begin{array}{c}\mathbf{v}\times \text{}\mathsf{\nabla}\text{}({c}^{2}-{v}^{2})=0.\end{array}$$ | (51) |
$$\begin{array}{c}{\mathbf{v}}_{FIDO}\equiv \frac{\mathbf{K}}{\left|\right|\mathbf{K}\left|\right|}=\frac{\mathbf{K}}{\sqrt{(\rho /c)[{c}^{2}-{v}^{2}]}}.\end{array}$$ | (52) |
$$\begin{array}{c}{\mathbf{A}}_{FIDO}\equiv ({\mathbf{v}}_{FIDO}\cdot \text{}\mathsf{\nabla}\text{}){\mathbf{v}}_{FIDO},\end{array}$$ | (53) |
$$\begin{array}{c}{\mathbf{A}}_{FIDO}=+\frac{1}{2}\frac{\text{}\mathsf{\nabla}\text{}\left|\right|\mathbf{K}|{|}^{2}}{\left|\right|\mathbf{K}|{|}^{2}}.\end{array}$$ | (54) |
$$\begin{array}{c}{\mathbf{A}}_{FIDO}=\frac{1}{2}\left[\frac{\text{}\mathsf{\nabla}\text{}({c}^{2}-{v}^{2})}{({c}^{2}-{v}^{2})}+\frac{\text{}\mathsf{\nabla}\text{}(\rho /c)}{(\rho /c)}\right].\end{array}$$ | (55) |
$$\begin{array}{c}\left|\right|{\mathbf{A}}_{FIDO}\left|\right|\left|\right|\mathbf{K}\left|\right|=\frac{1}{2}\mathbf{v}\cdot \text{}\mathsf{\nabla}\text{}({c}^{2}-{v}^{2})+O({c}^{2}-{v}^{2}),\end{array}$$ | (56) |
$$\begin{array}{c}{g}_{H}=\frac{1}{2}\frac{\partial ({c}^{2}-{v}^{2})}{\partial n}=c\frac{\partial (c-v)}{\partial n}.\end{array}$$ | (57) |
$$\begin{array}{c}\mathbf{v}={\mathbf{v}}_{\perp}+{\mathbf{v}}_{\parallel};where{\mathbf{v}}_{\perp}={v}_{\perp}\hat{\mathbf{n}}.\end{array}$$ | (58) |
$$\begin{array}{c}{L}^{\mu}=(1;{v}_{\parallel}^{i}).\end{array}$$ | (59) |
$$\begin{array}{c}\left|\right|\mathbf{L}|{|}^{2}=-\frac{\rho}{c}\left[-({c}^{2}-{v}^{2})-2{\mathbf{v}}_{\parallel}\cdot \mathbf{v}+{\mathbf{v}}_{\parallel}\cdot {\mathbf{v}}_{\parallel}\right]=\frac{\rho}{c}({c}^{2}-{v}_{\perp}^{2}).\end{array}$$ | (60) |
$$\begin{array}{c}{L}^{\alpha}{\nabla}_{\alpha}{L}^{\mu}={L}^{\alpha}({\nabla}_{\alpha}{L}_{\beta}-{\nabla}_{\beta}{L}_{\alpha}){g}^{\beta \mu}+\frac{1}{2}{\nabla}_{\beta}\left({L}^{2}\right){g}^{\beta \mu}.\end{array}$$ | (61) |
$$\begin{array}{c}{L}_{\mu}=\frac{\rho}{c}(-[{c}^{2}-{v}_{\perp}^{2}];{\mathbf{v}}_{\perp}).\end{array}$$ | (62) |
$$\begin{array}{c}{L}_{[\alpha ,\beta ]}=-\left[\begin{array}{ccc}0& ...& -{\nabla}_{i}\left[\frac{\rho}{c}({c}^{2}-{v}_{\perp}^{2})\right]\\ \cdot \cdot \cdot \cdots \cdots \cdots & \cdot & \cdot \cdot \cdot \cdots \\ +{\nabla}_{j}\left[\frac{\rho}{c}({c}^{2}-{v}_{\perp}^{2})\right]& ...& {\left(\frac{\rho}{c}{v}^{\perp}\right)}_{[i,j]}\\ \end{array}\right].\end{array}$$ | (63) |
$$\begin{array}{c}{L}^{\alpha}{L}_{[\beta ,\alpha ]}=\left({\mathbf{v}}_{\parallel}\cdot \text{}\mathsf{\nabla}\text{}\left[\frac{\rho}{c}({c}^{2}-{v}_{\perp}^{2})\right];{\nabla}_{j}\left[\frac{\rho}{c}({c}^{2}-{v}_{\perp}^{2})\right]+{v}_{\parallel}^{i}{\left(\frac{\rho}{c}{v}^{\perp}\right)}_{[j,i]}\right).\end{array}$$ | (64) |
$$\begin{array}{c}\left({L}^{\alpha}{L}_{[\beta ,\alpha ]}\right){|}_{horizon}=-\frac{\rho}{c}\left(0;{\nabla}_{j}({c}^{2}-{v}_{\perp}^{2})\right)=-\frac{\rho}{c}\frac{\partial ({c}^{2}-{v}_{\perp}^{2})}{\partial n}\left(0;{\hat{n}}_{j}\right).\end{array}$$ | (65) |
$$\begin{array}{c}{\nabla}_{\beta}\left({L}^{2}\right)=\left(0;{\nabla}_{j}\left[\frac{\rho}{c}({c}^{2}-{v}_{\perp}^{2})\right]\right).\end{array}$$ | (66) |
$$\begin{array}{c}{\nabla}_{\beta}\left({L}^{2}\right){|}_{horizon}=+\frac{\rho}{c}\left(0;{\nabla}_{j}({c}^{2}-{v}_{\perp}^{2})\right)=+\frac{\rho}{c}\frac{\partial ({c}^{2}-{v}_{\perp}^{2})}{\partial n}\left(0;{\hat{n}}_{j}\right).\end{array}$$ | (67) |
$$\begin{array}{c}({L}^{\alpha}{\nabla}_{\alpha}{L}_{\mu}{)}_{horizon}=+\frac{1}{2}\frac{\rho}{c}\frac{\partial ({c}^{2}-{v}_{\perp}^{2})}{\partial n}\left(0;{\hat{n}}_{j}\right),\end{array}$$ | (68) |
$$\begin{array}{c}({L}_{\mu}{)}_{horizon}=\frac{\rho}{c}\left(0;c{\hat{n}}_{j}\right).\end{array}$$ | (69) |
$$\begin{array}{c}({L}^{\alpha}{\nabla}_{\alpha}{L}_{\mu}{)}_{horizon}=+\frac{{g}_{H}}{c}({L}_{\mu}{)}_{horizon},\end{array}$$ | (70) |
$$\begin{array}{c}{g}_{H}=\frac{1}{2}\frac{\partial ({c}^{2}-{v}_{\perp}^{2})}{\partial n}=c\frac{\partial (c-{v}_{\perp})}{\partial n}.\end{array}$$ | (71) |
^{ $\text{5}$ } Henceforth, in the interests of notational simplicity, we shall drop the explicit subscript $0$ on background field quantities unless there is specific risk of confusion.
^{ $\text{6}$ } This discussion naturally leads us to what is perhaps the central question of analogue models – just how much of the standard “laws of black hole mechanics” [21, 423] carry over into these analogue models? Quite a lot but not everything – that's our main topic for the rest of the review.
^{ $\text{7}$ } Because of the background Minkowski metric there can be no possible confusion as to the definition of this normal derivative.
^{ $\text{8}$ } There are a few potential subtleties in the derivation of the existence Hawking radiation which we are for the time being glossing over, see Section 5.1 for details.
^{ $\text{9}$ } There is an issue of normalization here. On the one hand we want to be as close as possible to general relativistic conventions. On the other hand, we would like the surface gravity to really have the dimensions of an acceleration. The convention adopted here, with one explicit factor of $c$ , is the best compromise we have come up with. (Note that in an acoustic setting, where the speed of sound is not necessarily a constant, we cannot simply set $c\to 1$ by a choice of units.)
^{ $\text{10}$ } There are situations in which this surface gravity is a lot larger than one might naively expect [239] .
2.5.1 Example: vortex geometry
$$\begin{array}{c}{v}^{\hat{r}}\propto \frac{1}{r}.\end{array}$$ | (72) |
$$\begin{array}{c}{v}^{\hat{t}}\propto \frac{1}{r}.\end{array}$$ | (73) |
$$\begin{array}{c}\phi (r,\theta )=-Aln(r/a)-B\theta .\end{array}$$ | (74) |
$$\begin{array}{c}\mathbf{v}=-\text{}\mathsf{\nabla}\text{}\phi =\frac{(A\hat{r}+B\hat{\theta})}{r}.\end{array}$$ | (75) |
$$\begin{array}{c}d{s}^{2}=-{c}^{2}d{t}^{2}+{\left(dr-\frac{A}{r}dt\right)}^{2}+{\left(rd\theta -\frac{B}{r}dt\right)}^{2}+d{z}^{2}.\end{array}$$ | (76) |
$$\begin{array}{c}d{s}^{2}=-\left({c}^{2}-\frac{{A}^{2}+{B}^{2}}{{r}^{2}}\right)d{t}^{2}-2\frac{A}{r}drdt-2Bd\theta dt+d{r}^{2}+{r}^{2}d{\theta}^{2}+d{z}^{2}.\end{array}$$ | (77) |
$$\begin{array}{c}d{s}^{2}=-{c}^{2}(dt-\stackrel{~}{A}d\theta {)}^{2}+d{r}^{2}+(1-\stackrel{~}{B}){r}^{2}d{\theta}^{2}+d{z}^{2}.\end{array}$$ | (78) |
$$\begin{array}{c}{r}_{ergosphere}=\frac{\sqrt{{A}^{2}+{B}^{2}}}{c}.\end{array}$$ | (79) |
$$\begin{array}{c}{r}_{horizon}=\frac{\left|A\right|}{c}.\end{array}$$ | (80) |
2.5.2 Example: slab geometry
$$\begin{array}{c}d{s}^{2}\propto \frac{1}{v\left(z\right)c\left(z\right)}\left[-c(z{)}^{2}d{t}^{2}+{\left\{dz-v\left(z\right)dt\right\}}^{2}+d{x}^{2}+d{y}^{2}\right].\end{array}$$ | (81) |
$$\begin{array}{c}d{s}^{2}\propto \frac{1}{v\left(z\right)c\left(z\right)}\left[-\left\{c(z{)}^{2}-v(z{)}^{2}\right\}d{t}^{2}-2v\left(z\right)dzdt+d{x}^{2}+d{y}^{2}+d{z}^{2}\right].\end{array}$$ | (82) |
2.5.3 Example: Painlevé–Gullstrand geometry
$$\begin{array}{c}d{s}^{2}=-d{t}^{2}+{\left(dr\pm \sqrt{\frac{2GM}{r}}dt\right)}^{2}+{r}^{2}\left(d{\theta}^{2}+{sin}^{2}\theta d{\phi}^{2}\right).\end{array}$$ | (83) |
$$\begin{array}{c}d{s}^{2}=-\left(1-\frac{2GM}{r}\right)d{t}^{2}\pm \sqrt{\frac{2GM}{r}}drdt+d{r}^{2}+{r}^{2}\left(d{\theta}^{2}+{sin}^{2}\theta d{\phi}^{2}\right).\end{array}$$ | (84) |
$$\begin{array}{c}{t}_{PG}={t}_{S}\pm \left[4Marctanh\left(\sqrt{\frac{2GM}{r}}\right)-2\sqrt{2GMr}\right].\end{array}$$ | (85) |
$$\begin{array}{c}d{t}_{PG}=d{t}_{S}\pm \frac{\sqrt{2GM/r}}{1-2GM/r}dr.\end{array}$$ | (86) |
$$\begin{array}{c}d{s}^{2}\propto {r}^{-3/2}\left[-d{t}^{2}+{\left(dr\pm \sqrt{\frac{2GM}{r}}dt\right)}^{2}+{r}^{2}\left(d{\theta}^{2}+{sin}^{2}\theta d{\phi}^{2}\right)\right].\end{array}$$ | (87) |
^{ $\text{11}$ } The Painlevé–Gullstrand line element is sometimes called the Lemaître line element.
^{ $\text{12}$ } Similar constructions work for the Reissner–Nordstrom geometry [239] , as long as one does not get too close to the singularity. Likewise certain aspects of the Kerr geometry can be emulated in this way [401] .
2.6 Regaining geometric acoustics
$$\begin{array}{c}{h}_{\mu \nu}\equiv \left[\begin{array}{ccc}-({c}^{2}-{v}_{0}^{2})& ...& -{v}_{0}^{j}\\ \cdot \cdot \cdot \cdots \cdots \cdots & \cdot & \cdot \cdot \cdot \cdots \\ -{v}_{0}^{i}& ...& {\delta}^{ij}\\ \end{array}\right].\end{array}$$ | (88) |
$$\begin{array}{c}{h}^{\mu \nu}{\partial}_{\mu}\phi {\partial}_{\nu}\phi =0.\end{array}$$ | (89) |
$$\begin{array}{ccc}& & {h}_{\mu \nu}\frac{d{x}^{\mu}}{dt}\frac{d{x}^{\nu}}{dt}=0\end{array}$$ |
$$\begin{array}{ccc}& & \u27fa-({c}^{2}-{v}_{0}^{2})-2{v}_{0}^{i}\frac{d{x}^{i}}{dt}+\frac{d{x}^{i}}{dt}\frac{d{x}^{i}}{dt}=0\end{array}$$ |
$$\begin{array}{ccc}& & \u27fa\Vert \frac{d\mathbf{x}}{dt}-{\mathbf{v}}_{0}\Vert =c.\end{array}$$ | (90) |
$$\begin{array}{c}\frac{d{x}^{\mu}}{ds}=\left(\frac{dt}{ds};\frac{d{x}^{i}}{ds}\right).\end{array}$$ | (91) |
$$\begin{array}{c}{g}_{\mu \nu}\frac{d{x}^{\mu}}{ds}\frac{d{x}^{\nu}}{ds}=0.\end{array}$$ | (92) |
$$\begin{array}{c}-({c}^{2}-{v}_{0}^{2}){\left(\frac{dt}{ds}\right)}^{2}-2{v}_{0}^{i}\left(\frac{d{x}^{i}}{ds}\right)\left(\frac{dt}{ds}\right)+1=0.\end{array}$$ | (93) |
$$\begin{array}{c}\left(\frac{dt}{ds}\right)=\frac{-{v}_{0}^{i}\left(\frac{d{x}^{i}}{ds}\right)+\sqrt{{c}^{2}-{v}_{0}^{2}+{\left({v}_{0}^{i}\frac{d{x}^{i}}{ds}\right)}^{2}}}{{c}^{2}-{v}_{0}^{2}}.\end{array}$$ | (94) |
$$\begin{array}{c}T\left[\gamma \right]={\int}_{{\mathbf{x}}_{1}}^{{\mathbf{x}}_{2}}(dt/ds)ds={\int}_{\gamma}\frac{1}{{c}^{2}-{v}_{0}^{2}}\left\{\sqrt{({c}^{2}-{v}_{0}^{2})d{s}^{2}+({v}_{0}^{i}d{x}^{i}{)}^{2}}-{v}_{0}^{i}d{x}^{i}\right\}.\end{array}$$ | (95) |
2.7 Generalizing the physical model
2.7.1 External forces
2.7.2 The role of dimension
$$\begin{array}{c}\frac{\partial}{\partial {x}^{\mu}}\left({f}^{\mu \nu}\frac{\partial}{\partial {x}^{\nu}}\phi \right)=0,\end{array}$$ | (96) |
$$\begin{array}{c}{f}^{\mu \nu}=\left[\begin{array}{cc}-\rho /{c}^{2}& -\rho {v}^{j}/{c}^{2}\\ -\rho {v}^{i}/{c}^{2}& \rho \{{\delta}^{ij}-{v}^{i}{v}^{j}/{c}^{2}\}\end{array}\right],\end{array}$$ | (97) |
$$\begin{array}{c}{f}^{\mu \nu}=\sqrt{-g}{g}^{\mu \nu};g=\frac{1}{det\left({g}^{\mu \nu}\right)}\end{array}$$ | (98) |
$$\begin{array}{c}{g}^{\mu \nu}={\left(\frac{\rho}{c}\right)}^{-2/(d-1)}\left[\begin{array}{cc}-1/{c}^{2}& -{\mathbf{v}}^{T}/{c}^{2}\\ -\mathbf{v}/{c}^{2}& {\mathbf{I}}_{d\times d}-\mathbf{v}\otimes {\mathbf{v}}^{T}/{c}^{2}\end{array}\right],\end{array}$$ | (99) |
$$\begin{array}{c}{g}_{\mu \nu}={\left(\frac{\rho}{c}\right)}^{2/(d-1)}\left[\begin{array}{cc}-\left({c}^{2}-{v}^{2}\right)& -{\mathbf{v}}^{T}\\ -\mathbf{v}& {\mathbf{I}}_{d\times d}\end{array}\right].\end{array}$$ | (100) |
$$\begin{array}{c}{g}_{\mu \nu}=\left(\frac{\rho}{c}\right)\left[\begin{array}{cc}-\left({c}^{2}-{v}^{2}\right)& -{\mathbf{v}}^{T}\\ -\mathbf{v}& {\mathbf{I}}_{3\times 3}\end{array}\right].\end{array}$$ | (101) |
$$\begin{array}{c}{g}_{\mu \nu}={\left(\frac{\rho}{c}\right)}^{2}\left[\begin{array}{cc}-\left({c}^{2}-{v}^{2}\right)& -{\mathbf{v}}^{T}\\ -\mathbf{v}& {\mathbf{I}}_{2\times 2}\end{array}\right].\end{array}$$ | (102) |
2.7.3 Adding vorticity
^{ $\text{13}$ } Vorticity is automatically generated, for instance, whenever the background fluid is non-barotropic, and in particular when $\nabla \rho \times \nabla p\ne 0$ .
^{ $\text{14}$ } In references [141, 143, 139, 140, 145, 144, 138, 142] the author has attempted to argue that vorticity can be related to the concept of torsion in a general affine connexion. We disagree. Although deriving a wave equation in the presence of vorticity very definitely moves one beyond the realm of a simple Riemannian spacetime, adding torsion to the connexion is not sufficient to capture the relevant physics.
2.8 Simple Lagrangian meta-model
$$\begin{array}{c}\phi (t,\mathbf{x})={\phi}_{0}(t,\mathbf{x})+\epsilon {\phi}_{1}(t,\mathbf{x})+\frac{{\epsilon}^{2}}{2}{\phi}_{2}(t,\mathbf{x})+O\left({\epsilon}^{3}\right).\end{array}$$ | (103) |
$$\begin{array}{ccc}\mathcal{\mathcal{L}}({\partial}_{\mu}\phi ,\phi )& =& \mathcal{\mathcal{L}}({\partial}_{\mu}{\phi}_{0},{\phi}_{0})+\epsilon \left[\frac{\partial \mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)}{\partial}_{\mu}{\phi}_{1}+\frac{\partial \mathcal{\mathcal{L}}}{\partial \phi}{\phi}_{1}\right]\end{array}$$ |
$$\begin{array}{ccc}& +& \frac{{\epsilon}^{2}}{2}\left[\frac{\partial \mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)}{\partial}_{\mu}{\phi}_{2}+\frac{\partial \mathcal{\mathcal{L}}}{\partial \phi}{\phi}_{2}\right]\end{array}$$ |
$$\begin{array}{ccc}& +& \frac{{\epsilon}^{2}}{2}[\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}{\partial}_{\mu}{\phi}_{1}{\partial}_{\nu}{\phi}_{1}+2\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \phi}{\partial}_{\mu}{\phi}_{1}{\phi}_{1}\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \phi \partial \phi}{\phi}_{1}{\phi}_{1}]+O\left({\epsilon}^{3}\right).\end{array}$$ | (104) |
$$\begin{array}{c}S\left[\phi \right]=\int {d}^{d+1}x\mathcal{\mathcal{L}}({\partial}_{\mu}\phi ,\phi ),\end{array}$$ | (105) |
$$\begin{array}{c}{\partial}_{\mu}\left(\frac{\partial \mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)}\right)-\frac{\partial \mathcal{\mathcal{L}}}{\partial \phi}=0,\end{array}$$ | (106) |
$$\begin{array}{ccc}S\left[\phi \right]& =& S\left[{\phi}_{0}\right]+\frac{{\epsilon}^{2}}{2}\int {d}^{d+1}x[\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}{\partial}_{\mu}{\phi}_{1}{\partial}_{\nu}{\phi}_{1}\end{array}$$ |
$$\begin{array}{ccc}& & +\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \phi \partial \phi}-{\partial}_{\mu}\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \phi}\right\}\right){\phi}_{1}{\phi}_{1}]+O\left({\epsilon}^{3}\right).\end{array}$$ | (107) |
$$\begin{array}{c}{\partial}_{\mu}\left(\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}{\partial}_{\nu}{\phi}_{1}\right)-\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \phi \partial \phi}-{\partial}_{\mu}\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \phi}\right\}\right){\phi}_{1}=0.\end{array}$$ | (108) |
$$\begin{array}{c}\sqrt{-g}{g}^{\mu \nu}\equiv {f}^{\mu \nu}\equiv {\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}|}_{{\phi}_{0}}.\end{array}$$ | (109) |
$$\begin{array}{c}(-g{)}^{(d-1)/2}=-det\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}.\end{array}$$ | (110) |
$$\begin{array}{c}{g}^{\mu \nu}\left({\phi}_{0}\right)={{\left(-det\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}\right)}^{-1/(d-1)}|}_{{\phi}_{0}}{\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}|}_{{\phi}_{0}}.\end{array}$$ | (111) |
$$\begin{array}{c}{g}_{\mu \nu}\left({\phi}_{0}\right)={{\left(-det\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}\right)}^{1/(d-1)}|}_{{\phi}_{0}}{{\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}}^{-1}|}_{{\phi}_{0}}.\end{array}$$ | (112) |
$$\begin{array}{c}\left[\Delta \left(g\right({\phi}_{0}\left)\right)-V\left({\phi}_{0}\right)\right]{\phi}_{1}=0,\end{array}$$ | (113) |
$$\begin{array}{ccc}V\left({\phi}_{0}\right)& =& \frac{1}{\sqrt{-g}}\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \phi \partial \phi}-{\partial}_{\mu}\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \phi}\right\}\right)\end{array}$$ | (114) |
$$\begin{array}{ccc}& =& {\left(-det\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \left({\partial}_{\nu}\phi \right)}\right\}\right)}^{-1/(d-1)}\times \left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \phi \partial \phi}-{\partial}_{\mu}\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}\phi \right)\partial \phi}\right\}\right).\end{array}$$ | (115) |
2.9 Going further
3 History and Motivation
3.1 Modern period
3.1.1 The years 1981–1999
^{ $\text{15}$ } We emphasise: To get Hawking radiation you need an effective geometry, a horizon, and a suitable quantum field theory on that geometry.
^{ $\text{16}$ } Reference [172] is an attempt at connecting Hawking evaporation with the physics of collapsing bubbles. This was part of a more general programme aimed at connecting black hole thermodynamics with perfect fluid thermodynamics [173] .
3.1.2 The year 2000
3.1.3 The year 2001
3.1.4 The year 2002
3.1.5 The year 2003
3.1.6 The year 2004
3.1.7 The year 2005
3.2 Historical Period
^{ $\text{17}$ } Indeed historically, though not of direct relevance to general relativity, analogue models played a key role in the development of electromagnetism – Maxwell's derivation of his equations for the electromagnetic field was guided by a rather complicated analogue model in terms of spinning vortices of aether. Of course, once you have the equations in hand you can treat them in their own right and forget the model that guided you – which is exactly what happened in this particular case.
3.2.1 Optics
$$\begin{array}{c}[{g}_{effective}{]}_{\mu \nu}={\eta}_{\mu \nu}+\left[1-{n}^{-2}\right]{V}_{\mu}{V}_{\nu},\end{array}$$ | (116) |
$$\begin{array}{c}[{g}_{effective}{]}_{\mu \nu}={g}_{\mu \nu}+\left[1-\frac{1}{\epsilon \mu}\right]{V}_{\mu}{V}_{\nu},\end{array}$$ | (117) |
$$\begin{array}{c}[magneticpermitivity]\propto [electricpermeability].\end{array}$$ | (118) |
3.2.2 Acoustics
^{ $\text{18}$ } Indeed the results of Moncrief [268] are more general than those considered in the standard acoustic gravity papers that followed because they additionally permit a general relativistic curved background.
3.2.3 Electro-mechanical analogy
^{ $\text{19}$ } A recent attempt at connecting the electro-mechanical analogy back to relativity can be found in [434] .
3.3 Motivation
3.4 Going further
4 A Catalogue of Models
4.1 Classical models
4.1.1 Classical sound
4.1.2 Shallow water waves (gravity waves)
$$\begin{array}{ccc}{\partial}_{t}\phi +\frac{1}{2}(\text{}\mathsf{\nabla}\text{}\phi {)}^{2}=-\frac{p}{\rho}-gz-{V}_{\parallel}.& & \end{array}$$ | (119) |
$$\begin{array}{ccc}{\partial}_{t}\delta \phi +{\mathbf{v}}_{B}^{\parallel}\cdot \text{}\mathsf{\nabla}{\text{}}_{\parallel}\delta \phi =-\frac{p}{\rho}.& & \end{array}$$ | (120) |
$$\begin{array}{ccc}\delta \phi ={\sum}_{n=0}^{\infty}\frac{{z}^{n}}{n!}\delta {\phi}_{n}(x,y),& & \end{array}$$ | (121) |
$$\begin{array}{ccc}d{s}^{2}=\frac{1}{{c}^{2}}\left[-({c}^{2}-{v}_{B}^{\parallel 2})d{t}^{2}-2{\mathbf{v}}_{B}^{\parallel}\cdot \mathbf{d}\mathbf{x}dt+\mathbf{d}\mathbf{x}\cdot \mathbf{d}\mathbf{x}\right],& & \end{array}$$ | (122) |
$$\begin{array}{ccc}\delta {v}_{\perp}=-{h}_{B}{\nabla}_{\parallel}^{2}\delta {\phi}_{0}={\partial}_{t}\delta h+{\mathbf{v}}_{B}^{\parallel}\cdot \text{}\mathsf{\nabla}{\text{}}_{\parallel}\delta h=\frac{d}{dt}\delta h.& & \end{array}$$ | (123) |
^{ $\text{20}$ } Of course we now mean “gravity wave” in the traditional fluid mechanics sense of a water wave whose restoring force is given by ordinary Newtonian gravity. Waves in the fabric of spacetime are more properly called “gravitational waves”, though this usage seems to be in decline within the general relativity community. Be very careful in any situation where there is even a possibility of confusing the two concepts.
4.1.3 Classical refractive index
$$\begin{array}{ccc}& \text{}\mathsf{\nabla}\text{}\cdot \mathbf{B}=0,& \text{}\mathsf{\nabla}\text{}\times \mathbf{E}+{\partial}_{t}\mathbf{B}=0,\end{array}$$ | (124) |
$$\begin{array}{ccc}& \text{}\mathsf{\nabla}\text{}\cdot \mathbf{D}=0,& \text{}\mathsf{\nabla}\text{}\times \mathbf{H}-{\partial}_{t}\mathbf{D}=0,\end{array}$$ | (125) |
$$\begin{array}{c}{\partial}_{\alpha}\left({Z}^{\mu \alpha \nu \beta}{F}_{\nu \beta}\right)=0\end{array}$$ | (126) |
$$\begin{array}{c}{F}_{0i}=-{F}_{i0}=-{E}_{i},{F}_{ij}={\varepsilon}_{ijk}{B}^{k},\end{array}$$ | (127) |
$$\begin{array}{ccc}& & {Z}^{0i0j}=-{Z}^{0ij0}={Z}^{i0j0}=-{Z}^{i00j}=-\frac{1}{2}{\epsilon}^{ij};\end{array}$$ | (128) |
$$\begin{array}{ccc}& & {Z}^{ijkl}=\frac{1}{2}{\varepsilon}^{ijm}{\varepsilon}^{kln}{\mu}_{mn}^{-1};\end{array}$$ | (129) |
$$\begin{array}{c}\mathcal{\mathcal{L}}=\sqrt{-g}{g}^{\mu \alpha}{g}^{\nu \beta}{F}_{\mu \nu}{F}_{\alpha \beta}\end{array}$$ | (130) |
$$\begin{array}{c}{Z}^{\mu \nu \alpha \beta}=K\sqrt{-g}\left\{{g}^{\mu \alpha}{g}^{\nu \beta}-{g}^{\mu \beta}{g}^{\nu \alpha}\right\}\end{array}$$ | (131) |
$$\begin{array}{ccc}{g}_{\mu \nu}={\Omega}^{2}\left\{-1\oplus {g}_{ij}\right\}& & \end{array}$$ | (132) |
$$\begin{array}{ccc}& & {Z}^{0i0j}=-{Z}^{0ij0}={Z}^{i0j0}=-{Z}^{i00j}=-K\sqrt{-g}{g}^{ij};\end{array}$$ | (133) |
$$\begin{array}{ccc}& & {Z}^{ijkl}=K\sqrt{-g}\left\{{g}^{ik}{g}^{jl}-{g}^{il}{g}^{jk}\right\}\end{array}$$ | (134) |
$$\begin{array}{ccc}& & K\sqrt{-g}{g}^{ij}=\frac{1}{2}{\epsilon}^{ij};\end{array}$$ | (135) |
$$\begin{array}{ccc}& & K\sqrt{-g}\left\{{g}^{ik}{g}^{jl}-{g}^{il}{g}^{jk}\right\}=\frac{1}{2}{\varepsilon}^{ijm}{\varepsilon}^{kln}{\mu}_{mn}^{-1}.\end{array}$$ | (136) |
$$\begin{array}{c}K\sqrt{-g}{\varepsilon}_{ijm}{\varepsilon}_{kln}\left\{{g}^{ik}{g}^{jl}\right\}={\mu}_{mn}^{-1}.\end{array}$$ | (137) |
$$\begin{array}{c}{\varepsilon}_{ijm}{\varepsilon}_{kln}\left\{{X}^{ik}{X}^{jl}\right\}=2detX{X}_{mn}^{-1},\end{array}$$ | (138) |
$$\begin{array}{c}2K\frac{{g}_{ij}}{\sqrt{-g}}={\mu}_{ij}^{-1},\end{array}$$ | (139) |
$$\begin{array}{c}\frac{1}{2K}\sqrt{-g}{g}^{ij}={\mu}^{ij}.\end{array}$$ | (140) |
$$\begin{array}{c}2K\sqrt{-g}{g}^{ij}={\epsilon}^{ij},\end{array}$$ | (141) |
$$\begin{array}{ccc}{\epsilon}^{ij}& =& 4{K}^{2}{\mu}^{ij};\end{array}$$ | (142) |
$$\begin{array}{ccc}{g}^{ij}& =& \frac{4{K}^{2}}{det\text{}\mathit{\epsilon}\text{}}{\epsilon}^{ij};\end{array}$$ | (143) |
$$\begin{array}{ccc}{g}^{ij}& =& \frac{1}{4{K}^{2}det\text{}\mathit{\mu}\text{}}{\mu}^{ij}.\end{array}$$ | (144) |
$$\begin{array}{c}{g}^{ij}={\left[{\left\{\frac{\text{}\mathit{\mu}{\text{}}^{1/2}\text{}\mathit{\epsilon}\text{}\text{}\mathit{\mu}{\text{}}^{1/2}}{det\left(\text{}\mathit{\mu}\text{}\text{}\mathit{\epsilon}\text{}\right)}\right\}}^{1/2}\right]}^{ij}.\end{array}$$ | (145) |
^{ $\text{21}$ } The existence of this constraint has been independently re-derived several times in the literature. In contrast, other segments of the literature seem blithely unaware of this important restriction on just when permittivity and permeability are truly equivalent to an effective metric.
Eikonal approximation:
$$\begin{array}{ccc}{C}^{\mu \nu}={Z}^{\mu \alpha \nu \beta}{k}_{\alpha}{k}_{\beta}.& & \end{array}$$ | (146) |
$$\begin{array}{ccc}det\left({C}^{ij}\right)=\frac{1}{8}det\left(-{\omega}^{2}{\epsilon}^{ij}+{\varepsilon}^{ikm}{\varepsilon}^{jln}{\mu}_{mn}^{-1}{k}_{k}{k}_{l}\right).& & \end{array}$$ | (147) |
$$\begin{array}{ccc}det\left({C}^{ij}\right)=\frac{1}{8}det\left[-{\omega}^{2}{\epsilon}^{ij}+(det\text{}\mathit{\mu}\text{}{)}^{-1}({\mu}^{ij}{\mu}^{kl}{k}_{k}{k}_{l}-{\mu}^{im}{k}_{m}{\mu}^{jl}{k}_{l})\right].& & \end{array}$$ | (148) |
$$\begin{array}{c}{\stackrel{~}{k}}^{i}=[\text{}\mathit{\mu}{\text{}}^{1/2}{]}^{ij}{k}_{j}\end{array}$$ | (149) |
$$\begin{array}{c}[\text{}\stackrel{\mathbf{~}}{\mathit{\epsilon}}\text{}{]}^{ij}=det(\text{}\mathit{\mu}\text{}\left)\right[\text{}\mathit{\mu}{\text{}}^{-1/2}\text{}\mathit{\epsilon}\text{}\text{}\mathit{\mu}{\text{}}^{-1/2}{]}_{ij}\end{array}$$ | (150) |
$$\begin{array}{ccc}& & det\left({C}^{ij}\right)\propto det\left\{-{\omega}^{2}[\text{}\stackrel{\mathbf{~}}{\mathit{\epsilon}}\text{}{]}^{ij}+{\delta}^{ij}[{\delta}_{mn}{\stackrel{~}{k}}^{m}{\stackrel{~}{k}}^{n}]-{\stackrel{~}{k}}^{i}{\stackrel{~}{k}}^{j}\right\}.\end{array}$$ | (151) |
3 degenerate eigenvalues:
$$\begin{array}{c}\text{}\mathit{\epsilon}\text{}=\frac{tr\left(\text{}\mathit{\epsilon}\text{}\right)}{tr\left(\text{}\mathit{\mu}\text{}\right)}\text{}\mathit{\mu}\text{}and\stackrel{~}{\epsilon}=det\text{}\mathit{\mu}\text{}\frac{tr\left(\text{}\mathit{\epsilon}\text{}\right)}{tr\left(\text{}\mathit{\mu}\text{}\right)},\end{array}$$ | (152) |
$$\begin{array}{c}det\left({C}^{ij}\right)\propto {\omega}^{2}{\left\{{\omega}^{2}-\left[{\stackrel{~}{\epsilon}}^{-1}{\delta}_{mn}{\stackrel{~}{k}}^{m}{\stackrel{~}{k}}^{n}\right]\right\}}^{2}.\end{array}$$ | (153) |
$$\begin{array}{c}det\left({C}^{ij}\right)\propto {\omega}^{2}{\left\{{\omega}^{2}-\left[{g}^{ij}{k}_{i}{k}_{j}\right]\right\}}^{2},\end{array}$$ | (154) |
$$\begin{array}{c}{g}^{ij}=\frac{1}{\stackrel{~}{\epsilon}}[\text{}\mathit{\mu}\text{}{]}^{ij}=\frac{tr\left(\text{}\mathit{\mu}\text{}\right)[\text{}\mathit{\mu}\text{}{]}^{ij}}{tr\left(\text{}\mathit{\epsilon}\text{}\right)det\text{}\mathit{\mu}\text{}}=\frac{tr\left(\text{}\mathit{\epsilon}\text{}\right)[\text{}\mathit{\epsilon}\text{}{]}^{ij}}{tr\left(\text{}\mathit{\mu}\text{}\right)det\text{}\mathit{\epsilon}\text{}}.\end{array}$$ | (155) |
$$\begin{array}{c}{g}^{ij}=\frac{{\delta}^{ij}}{\epsilon \mu}.\end{array}$$ | (156) |
2 degenerate eigenvalues:
3 distinct eigenvalues:
Abstract linear electrodynamics:
Nonlinear electrodynamics:
$$\begin{array}{c}{F}_{\mu \nu}={F}_{\mu \nu}^{bg}+{f}_{\mu \nu}^{ph}.\end{array}$$ | (157) |
Summary:
4.1.4 Normal mode meta-models
$$\begin{array}{c}{\phi}^{A}(t,\stackrel{\u20d7}{x})={\phi}_{0}^{A}(t,\stackrel{\u20d7}{x})+\epsilon {\phi}_{1}^{A}(t,\stackrel{\u20d7}{x})+\frac{{\epsilon}^{2}}{2}{\phi}_{2}^{A}(t,\stackrel{\u20d7}{x})+O\left({\epsilon}^{3}\right).\end{array}$$ | (158) |
$$\begin{array}{ccc}\mathcal{\mathcal{L}}({\partial}_{\mu}{\phi}^{A},{\phi}^{A})& =& \mathcal{\mathcal{L}}({\partial}_{\mu}{\phi}_{0}^{A},{\phi}_{0}^{A})+\epsilon \left[\frac{\partial \mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)}{\partial}_{\mu}{\phi}_{1}^{A}+\frac{\partial \mathcal{\mathcal{L}}}{\partial {\phi}^{A}}{\phi}_{1}^{A}\right]\end{array}$$ |
$$\begin{array}{ccc}& +& \frac{{\epsilon}^{2}}{2}\left[\frac{\partial \mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)}{\partial}_{\mu}{\phi}_{2}^{A}+\frac{\partial \mathcal{\mathcal{L}}}{\partial {\phi}^{A}}{\phi}_{2}^{A}\right]\end{array}$$ |
$$\begin{array}{ccc}& +& \frac{{\epsilon}^{2}}{2}[\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial \left({\partial}_{\nu}{\phi}^{B}\right)}{\partial}_{\mu}{\phi}_{1}^{A}{\partial}_{\nu}{\phi}_{1}^{B}\end{array}$$ |
$$\begin{array}{ccc}& & +2\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial {\phi}^{B}}{\partial}_{\mu}{\phi}_{1}^{A}{\phi}_{1}^{B}+\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial {\phi}^{A}\partial {\phi}^{B}}{\phi}_{1}^{A}{\phi}_{1}^{B}]\end{array}$$ |
$$\begin{array}{ccc}& & +O\left({\epsilon}^{3}\right).\end{array}$$ | (159) |
$$\begin{array}{c}S\left[{\phi}^{A}\right]=\int {d}^{d+1}x\mathcal{\mathcal{L}}({\partial}_{\mu}{\phi}^{A},{\phi}^{A}).\end{array}$$ | (160) |
$$\begin{array}{ccc}S\left[{\phi}^{A}\right]& =& S\left[{\phi}_{0}^{A}\right]\end{array}$$ |
$$\begin{array}{ccc}& +& \frac{{\epsilon}^{2}}{2}\int {d}^{d+1}x[\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial \left({\partial}_{\nu}{\phi}^{B}\right)}\right\}{\partial}_{\mu}{\phi}_{1}^{A}{\partial}_{\nu}{\phi}_{1}^{B}\end{array}$$ |
$$\begin{array}{ccc}& & +2\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial {\phi}^{B}}\right\}{\partial}_{\mu}{\phi}_{1}^{A}{\phi}_{1}^{B}+\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial {\phi}^{A}\partial {\phi}^{B}}\right\}{\phi}_{1}^{A}{\phi}_{1}^{B}]\end{array}$$ |
$$\begin{array}{ccc}& +& O\left({\epsilon}^{3}\right).\end{array}$$ | (161) |
$$\begin{array}{ccc}& & {\partial}_{\mu}\left(\left\{\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial \left({\partial}_{\nu}{\phi}^{B}\right)}\right\}{\partial}_{\nu}{\phi}_{1}^{B}\right)+{\partial}_{\mu}\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial {\phi}^{B}}{\phi}_{1}^{B}\right)\end{array}$$ |
$$\begin{array}{ccc}& & -{\partial}_{\mu}{\phi}_{1}^{B}\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{B}\right)\partial {\phi}^{A}}-\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial {\phi}^{A}\partial {\phi}^{B}}\right){\phi}_{1}^{B}=0.\end{array}$$ | (162) |
$$\begin{array}{c}{f}_{AB}^{\mu \nu}\equiv \frac{1}{2}\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial \left({\partial}_{\nu}{\phi}^{B}\right)}+\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\nu}{\phi}^{A}\right)\partial \left({\partial}_{\mu}{\phi}^{B}\right)}\right).\end{array}$$ | (163) |
$$\begin{array}{ccc}{\Gamma}_{AB}^{\mu}& \equiv & +\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial {\phi}^{B}}-\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{B}\right)\partial {\phi}^{A}}\end{array}$$ |
$$\begin{array}{ccc}& & +\frac{1}{2}{\partial}_{\nu}\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\nu}{\phi}^{A}\right)\partial \left({\partial}_{\mu}{\phi}^{B}\right)}-\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial \left({\partial}_{\nu}{\phi}^{B}\right)}\right).\end{array}$$ | (164) |
$$\begin{array}{c}{K}_{AB}=-\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial {\phi}^{A}\partial {\phi}^{B}}+\frac{1}{2}{\partial}_{\mu}\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{A}\right)\partial {\phi}^{B}}\right)+\frac{1}{2}{\partial}_{\mu}\left(\frac{{\partial}^{2}\mathcal{\mathcal{L}}}{\partial \left({\partial}_{\mu}{\phi}^{B}\right)\partial {\phi}^{A}}\right).\end{array}$$ | (165) |
$$\begin{array}{c}{\partial}_{\mu}\left({f}_{AB}^{\mu \nu}{\partial}_{\nu}{\phi}_{1}^{B}\right)+\frac{1}{2}\left[{\Gamma}_{AB}^{\mu}{\partial}_{\mu}{\phi}_{1}^{B}+{\partial}_{\mu}\left({\Gamma}_{AB}^{\mu}{\phi}_{1}^{B}\right)\right]+{K}_{AB}{\phi}_{1}^{B}=0.\end{array}$$ | (166) |
$$\begin{array}{c}{\phi}^{A}\left(x\right)={\epsilon}^{A}\left(x\right)exp[-i\phi (x\left)\right],\end{array}$$ | (167) |
$$\begin{array}{c}\left\{{f}_{AB}^{\mu \nu}{\partial}_{\mu}\phi \left(x\right){\partial}_{\nu}\phi \left(x\right)+{\Gamma}_{AB}^{\mu}{\partial}_{\mu}\phi \left(x\right)+{K}_{AB}\right\}{\epsilon}^{B}\left(x\right)=0.\end{array}$$ | (168) |
$$\begin{array}{c}{f}_{AB}^{\mu \nu}{k}_{\mu}{k}_{\nu}+{\Gamma}_{AB}^{\mu}{k}_{\mu}+{K}_{AB},\end{array}$$ | (169) |
$$\begin{array}{c}F(x,k)\equiv det\left\{{f}_{AB}^{\mu \nu}{k}_{\mu}{k}_{\nu}+{\Gamma}_{AB}^{\mu}{k}_{\mu}+{K}_{AB}\right\}=0,\end{array}$$ | (170) |
$$\begin{array}{c}\mathcal{\mathcal{F}}\left(x\right)\equiv \left\{{k}_{\mu}\left|F\right(x,k)=0\right\}.\end{array}$$ | (171) |
$$\begin{array}{c}\mathcal{N}\left(x\right)\equiv \left\{{k}_{\mu}|det\left({f}_{AB}^{\mu \nu}{k}_{\mu}{k}_{\mu}\right)=0\right\}.\end{array}$$ | (172) |
$$\begin{array}{c}Q(x,k)\equiv det\left({f}_{AB}^{\mu \nu}\left(x\right){k}_{\mu}{k}_{\mu}\right).\end{array}$$ | (173) |
$$\begin{array}{c}Q(x,k)={Q}^{{\mu}_{1}{\nu}_{1}{\mu}_{2}{\nu}_{2}\cdots {\mu}_{N}{\nu}_{N}}\left(x\right){k}_{{\mu}_{1}}{k}_{{\nu}_{1}}{k}_{{\mu}_{2}}{k}_{{\nu}_{2}}\cdots {k}_{{\mu}_{N}}{k}_{{\nu}_{N}}.\end{array}$$ | (174) |
$$\begin{array}{c}{Q}^{{\mu}_{1}{\nu}_{1}{\mu}_{2}{\nu}_{2}\cdots {\mu}_{N}{\nu}_{N}}=\frac{1}{N!}{\epsilon}^{{A}_{1}{A}_{2}\cdots {A}_{N}}{\epsilon}^{{B}_{1}{B}_{2}\cdots {B}_{N}}{f}_{{A}_{1}{B}_{1}}^{{\mu}_{1}{\nu}_{1}}{f}_{{A}_{2}{B}_{2}}^{{\mu}_{2}{\nu}_{2}}\cdots {f}_{{A}_{N}{B}_{N}}^{{\mu}_{N}{\nu}_{N}}.\end{array}$$ | (175) |
$$\begin{array}{c}\mathcal{N}\left(x\right)\equiv \left\{{k}_{\mu}\left|Q\right(x,k)=0\right\}.\end{array}$$ | (176) |
$$\begin{array}{c}\mathcal{\mathcal{M}}\left(x\right)=\left\{{t}^{\mu}=\frac{\partial Q(x,k)}{\partial {k}_{\mu}}|{k}_{\mu}\in \mathcal{N}(x)\right\}.\end{array}$$ | (177) |
$$\begin{array}{c}{f}_{AB}^{\mu \nu}={h}_{AB}{f}^{\mu \nu}.\end{array}$$ | (178) |
$$\begin{array}{c}Q(x,k)=det\left({h}_{AB}\right)[{f}^{\mu \nu}{k}_{\mu}{k}_{\nu}{]}^{N}\end{array}$$ | (179) |
$$\begin{array}{c}{f}_{AB}^{\mu \nu}{f}_{BC}^{\alpha \beta}={f}_{AB}^{\alpha \beta}{f}_{BC}^{\mu \nu};thatis[{\mathit{f}}^{\mu \nu},{\mathit{f}}^{\alpha \beta}]=0.\end{array}$$ | (180) |
$$\begin{array}{c}{\overline{f}}_{AB}^{\mu \nu}=diag\{{\overline{f}}_{1}^{\mu \nu},{\overline{f}}_{2}^{\mu \nu},{\overline{f}}_{3}^{\mu \nu},...,{\overline{f}}_{N}^{\mu \nu}\}\end{array}$$ | (181) |
$$\begin{array}{c}Q(x,k){=}^{N}{\prod}_{A=1}\left[{\overline{f}}_{A}^{\mu \nu}{k}_{\mu}{k}_{\nu}\right].\end{array}$$ | (182) |
4.2 Quantum models
4.2.1 Bose–Einstein condensates
$$\begin{array}{ccc}i\u0127\frac{\partial}{\partial t}\widehat{\Psi}=\left(-\frac{{\u0127}^{2}}{2m}{\nabla}^{2}+{V}_{ext}(\mathbf{x})+\kappa \left(a\right){\widehat{\Psi}}^{\u2020}\widehat{\Psi}\right)\widehat{\Psi}.& & \end{array}$$ | (183) |
$$\begin{array}{c}\kappa \left(a\right)=\frac{4\pi a{\u0127}^{2}}{m}.\end{array}$$ | (184) |
$$\begin{array}{ccc}{\widehat{\phi}}^{\u2020}\widehat{\phi}\widehat{\phi}\simeq 2\langle {\widehat{\phi}}^{\u2020}\widehat{\phi}\rangle \widehat{\phi}+\langle \widehat{\phi}\widehat{\phi}\rangle {\widehat{\phi}}^{\u2020},& & \end{array}$$ | (185) |
$$\begin{array}{ccc}i\u0127\frac{\partial}{\partial t}\psi (t,\mathbf{x})& =& \left(-\frac{{\u0127}^{2}}{2m}{\nabla}^{2}+{V}_{ext}(\mathbf{x})+\kappa {n}_{c}\right)\psi (t,\mathbf{x})\end{array}$$ |
$$\begin{array}{ccc}& & +\kappa \left\{2\stackrel{~}{n}\psi (t,\mathbf{x})+\stackrel{~}{m}{\psi}^{*}(t,\mathbf{x})\right\};\end{array}$$ | (186) |
$$\begin{array}{ccc}& & \end{array}$$ |
$$\begin{array}{ccc}i\u0127\frac{\partial}{\partial t}\widehat{\phi}(t,\mathbf{x})& =& \left(-\frac{{\u0127}^{2}}{2m}{\nabla}^{2}+{V}_{ext}(\mathbf{x})+\kappa 2{n}_{T}\right)\widehat{\phi}(t,\mathbf{x})\end{array}$$ |
$$\begin{array}{ccc}& & +\kappa {m}_{T}{\widehat{\phi}}^{\u2020}(t,\mathbf{x}).\end{array}$$ | (187) |
$$\begin{array}{ccc}& & {n}_{c}\equiv {\left|\psi (t,\mathbf{x})\right|}^{2};{m}_{c}\equiv {\psi}^{2}(t,\mathbf{x});\end{array}$$ | (188) |
$$\begin{array}{ccc}& & \stackrel{~}{n}\equiv \langle {\widehat{\phi}}^{\u2020}\widehat{\phi}\rangle ;\stackrel{~}{m}\equiv \langle \widehat{\phi}\widehat{\phi}\rangle ;\end{array}$$ | (189) |
$$\begin{array}{ccc}& & {n}_{T}={n}_{c}+\stackrel{~}{n};{m}_{T}={m}_{c}+\stackrel{~}{m}.\end{array}$$ | (190) |
$$\begin{array}{c}\psi (t,\mathbf{x})=\sqrt{{n}_{c}(t,\mathbf{x})}exp[-i\theta (t,\mathbf{x})/\u0127],\end{array}$$ | (191) |
$$\begin{array}{ccc}& & \frac{\partial}{\partial t}{n}_{c}+\text{}\mathsf{\nabla}\text{}\cdot ({n}_{c}\mathbf{v})=0,\end{array}$$ | (192) |
$$\begin{array}{ccc}& & m\frac{\partial}{\partial t}\mathbf{v}+\text{}\mathsf{\nabla}\text{}\left(\frac{m{v}^{2}}{2}+{V}_{ext}(t,\mathbf{x})+\kappa {n}_{c}-\frac{{\u0127}^{2}}{2m}\frac{{\nabla}^{2}\sqrt{{n}_{c}}}{\sqrt{{n}_{c}}}\right)=0.\end{array}$$ | (193) |
$$\begin{array}{c}{V}_{quantum}=-{\u0127}^{2}{\nabla}^{2}\sqrt{{n}_{c}}/\left(2m\sqrt{{n}_{c}}\right),\end{array}$$ | (194) |
$$\begin{array}{c}{n}_{c}{\nabla}_{i}{V}_{quantum}\equiv {n}_{c}{\nabla}_{i}\left[-\frac{{\u0127}^{2}}{2m}\frac{{\nabla}^{2}\sqrt{{n}_{c}}}{\sqrt{{n}_{c}}}\right]={\nabla}_{j}\left[-\frac{{\u0127}^{2}}{4m}{n}_{c}{\nabla}_{i}{\nabla}_{j}ln{n}_{c}\right],\end{array}$$ | (195) |
$$\begin{array}{c}{\sigma}_{ij}^{quantum}=-\frac{{\u0127}^{2}}{4m}{n}_{c}{\nabla}_{i}{\nabla}_{j}ln{n}_{c}.\end{array}$$ | (196) |
$$\begin{array}{c}\rho \left[\frac{\partial}{\partial t}\mathbf{v}+(\mathbf{v}\cdot \text{}\mathsf{\nabla}\text{})\mathbf{v}\right]+\rho \text{}\mathsf{\nabla}\text{}\left[\frac{{V}_{ext}(t,\mathbf{x})}{m}\right]+\text{}\mathsf{\nabla}\text{}\left[\frac{\kappa {\rho}^{2}}{2{m}^{2}}\right]+\text{}\mathsf{\nabla}\text{}\cdot {\sigma}^{quantum}=0.\end{array}$$ | (197) |
$$\begin{array}{c}m\frac{\partial}{\partial t}\theta +\left(\frac{[\text{}\mathsf{\nabla}\text{}\theta {]}^{2}}{2m}+{V}_{ext}(t,\mathbf{x})+\kappa {n}_{c}-\frac{{\u0127}^{2}}{2m}\frac{{\nabla}^{2}\sqrt{{n}_{c}}}{\sqrt{{n}_{c}}}\right)=0.\end{array}$$ | (198) |
$$\begin{array}{ccc}\widehat{\phi}(t,\mathbf{x})=& & {e}^{-i\theta /\u0127}\left(\frac{1}{2\sqrt{{n}_{c}}}{\widehat{n}}_{1}-i\frac{\sqrt{{n}_{c}}}{\u0127}{\widehat{\theta}}_{1}\right),\end{array}$$ | (199) |
$$\begin{array}{ccc}& & {\partial}_{t}{\widehat{n}}_{1}+\frac{1}{m}\text{}\mathsf{\nabla}\text{}\cdot \left({n}_{1}\text{}\mathsf{\nabla}\text{}\theta +{n}_{c}\text{}\mathsf{\nabla}\text{}{\widehat{\theta}}_{1}\right)=0,\end{array}$$ | (200) |
$$\begin{array}{ccc}& & {\partial}_{t}{\widehat{\theta}}_{1}+\frac{1}{m}\text{}\mathsf{\nabla}\text{}\theta \cdot \text{}\mathsf{\nabla}\text{}{\widehat{\theta}}_{1}+\kappa \left(a\right){n}_{1}-\frac{{\u0127}^{2}}{2m}{D}_{2}{\widehat{n}}_{1}=0.\end{array}$$ | (201) |
$$\begin{array}{ccc}{D}_{2}{\widehat{n}}_{1}& \equiv & -\frac{1}{2}{n}_{c}^{-3/2}\left[{\nabla}^{2}\right({n}_{c}^{+1/2}\left)\right]{\widehat{n}}_{1}+\frac{1}{2}{n}_{c}^{-1/2}{\nabla}^{2}\left({n}_{c}^{-1/2}{\widehat{n}}_{1}\right).\end{array}$$ | (202) |
$$\begin{array}{c}{f}^{\mu \nu}(t,\mathbf{x})\equiv \left[\begin{array}{ccc}{f}^{00}& ...& {f}^{0j}\\ \cdot \cdot \cdot \cdots & \cdot & \cdot \cdot \cdot \cdots \cdots \cdots & {f}^{i0}& ...& {f}^{ij}\\ \end{array}\right].\end{array}$$ | (203) |
$$\begin{array}{c}{x}^{\mu}\equiv (t;{x}^{i})\end{array}$$ | (204) |
$$\begin{array}{c}{\partial}_{\mu}\left({f}^{\mu \nu}{\partial}_{\nu}{\widehat{\theta}}_{1}\right)=0.\end{array}$$ | (205) |
$$\begin{array}{ccc}{f}^{00}& =& -{\left[\kappa \left(a\right)-\frac{{\u0127}^{2}}{2m}{D}_{2}\right]}^{-1}\end{array}$$ | (206) |
$$\begin{array}{ccc}{f}^{0j}& =& -{\left[\kappa \left(a\right)-\frac{{\u0127}^{2}}{2m}{D}_{2}\right]}^{-1}\frac{{\nabla}^{j}{\theta}_{0}}{m}\end{array}$$ | (207) |
$$\begin{array}{ccc}{f}^{i0}& =& -\frac{{\nabla}^{i}{\theta}_{0}}{m}{\left[\kappa \left(a\right)-\frac{{\u0127}^{2}}{2m}{D}_{2}\right]}^{-1}\end{array}$$ | (208) |
$$\begin{array}{ccc}{f}^{ij}& =& \frac{{n}_{c}{\delta}^{ij}}{m}-\frac{{\nabla}^{i}{\theta}_{0}}{m}{\left[\kappa \left(a\right)-\frac{{\u0127}^{2}}{2m}{D}_{2}\right]}^{-1}\frac{{\nabla}^{j}{\theta}_{0}}{m}.\end{array}$$ | (209) |
$$\begin{array}{c}\sqrt{-g}{g}^{\mu \nu}={f}^{\mu \nu},\end{array}$$ | (210) |
$$\begin{array}{c}\Delta {\theta}_{1}\equiv \frac{1}{\sqrt{-g}}{\partial}_{\mu}\left(\sqrt{-g}{g}^{\mu \nu}{\partial}_{\nu}\right){\widehat{\theta}}_{1}=0,\end{array}$$ | (211) |
$$\begin{array}{c}{g}_{\mu \nu}(t,\mathbf{x})\equiv \frac{{n}_{c}}{m{c}_{s}(a,{n}_{c})}\left[\begin{array}{ccc}-\left\{{c}_{s}\right(a,{n}_{c}{)}^{2}-{v}^{2}\}& ...& -{v}_{j}\\ \cdot \cdot \cdot \cdots \cdots \cdots & \cdot & \cdot \cdot \cdot \cdots \\ -{v}_{i}& ...& {\delta}_{ij}\\ \end{array}\right].\end{array}$$ | (212) |
$$\begin{array}{c}{c}_{s}(a,{n}_{c}{)}^{2}=\frac{\kappa \left(a\right){n}_{c}}{m}.\end{array}$$ | (213) |
4.2.2 BEC models in the eikonal approximation
$$\begin{array}{ccc}{\widehat{\theta}}_{1}(t,\mathbf{x})& =& Re\left\{{\mathcal{A}}_{\theta}exp(-i\phi )\right\},\end{array}$$ | (214) |
$$\begin{array}{ccc}{\widehat{n}}_{1}(t,\mathbf{x})& =& Re\left\{{\mathcal{A}}_{\rho}exp(-i\phi )\right\}.\end{array}$$ | (215) |
$$\begin{array}{c}\omega =\frac{\partial \phi}{\partial t};{k}_{i}={\nabla}_{i}\phi .\end{array}$$ | (216) |
$$\begin{array}{ccc}{D}_{2}{\widehat{n}}_{1}& \equiv & -\frac{1}{2}{n}_{c}^{-3/2}[\Delta ({n}_{c}^{+1/2}\left)\right]{\widehat{n}}_{1}+\frac{1}{2}{n}_{c}^{-1/2}\Delta \left({n}_{c}^{-1/2}{\widehat{n}}_{1}\right)\end{array}$$ | (217) |
$$\begin{array}{ccc}& \approx & +\frac{1}{2}{n}_{c}^{-1}[\Delta {\widehat{n}}_{1}]\end{array}$$ | (218) |
$$\begin{array}{ccc}& =& -\frac{1}{2}{n}_{c}^{-1}{k}^{2}{\widehat{n}}_{1}.\end{array}$$ | (219) |
$$\begin{array}{c}{D}_{2}\to -\frac{1}{2}{n}_{c}^{-1}{k}^{2}.\end{array}$$ | (220) |
$$\begin{array}{ccc}{f}^{00}& \to & -{\left[\kappa \left(a\right)+\frac{{\u0127}^{2}{k}^{2}}{4m{n}_{c}}\right]}^{-1}\end{array}$$ | (221) |
$$\begin{array}{ccc}{f}^{0j}& \to & -{\left[\kappa \left(a\right)+\frac{{\u0127}^{2}{k}^{2}}{4m{n}_{c}}\right]}^{-1}\frac{{\nabla}^{j}{\theta}_{0}}{m}\end{array}$$ | (222) |
$$\begin{array}{ccc}{f}^{i0}& \to & -\frac{{\nabla}^{i}{\theta}_{0}}{m}{\left[\kappa \left(a\right)+\frac{{\u0127}^{2}{k}^{2}}{4m{n}_{c}}\right]}^{-1}\end{array}$$ | (223) |
$$\begin{array}{ccc}{f}^{ij}& \to & \frac{{n}_{c}{\delta}^{ij}}{m}-\frac{{\nabla}^{i}{\theta}_{0}}{m}{\left[\kappa \left(a\right)+\frac{{\u0127}^{2}{k}^{2}}{4m{n}_{c}}\right]}^{-1}\frac{{\nabla}^{j}{\theta}_{0}}{m}.\end{array}$$ | (224) |
$$\begin{array}{c}{f}^{00}{\omega}^{2}+({f}^{0i}+{f}^{i0})\omega {k}_{i}+{f}^{ij}{k}_{i}{k}_{j}=0.\end{array}$$ | (225) |
$$\begin{array}{c}-{\omega}^{2}+2{v}_{0}^{i}\omega {k}_{i}+\frac{{n}_{c}{k}^{2}}{m}\left[\kappa \left(a\right)+\frac{{\u0127}^{2}}{4m{n}_{c}}{k}^{2}\right]-({v}_{0}^{i}{k}_{i}{)}^{2}=0.\end{array}$$ | (226) |
$$\begin{array}{c}{\left(\omega -{v}_{0}^{i}{k}_{i}\right)}^{2}=\frac{{n}_{c}{k}^{2}}{m}\left[\kappa \left(a\right)+\frac{{\u0127}^{2}}{4m{n}_{c}}{k}^{2}\right].\end{array}$$ | (227) |
$$\begin{array}{c}\omega ={v}_{0}^{i}{k}_{i}\pm \sqrt{{c}_{s}^{2}{k}^{2}+{\left(\frac{\u0127}{2m}{k}^{2}\right)}^{2}}.\end{array}$$ | (228) |
$$\begin{array}{c}\delta \equiv \frac{\sqrt{1+\frac{{\lambda}_{c}^{2}}{4{\lambda}^{2}}}-1}{(1-{v}_{0}/{c}_{s})}\simeq \frac{1}{(1-{v}_{0}/{c}_{s})}\frac{{\lambda}_{c}^{2}}{8{\lambda}^{2}}.\end{array}$$ | (229) |
$$\begin{array}{c}{v}_{g}^{i}=\frac{\partial \omega}{\partial {k}_{i}}={v}_{0}^{i}\pm \frac{\left({c}^{2}+\frac{{\u0127}^{2}}{2{m}^{2}}{k}^{2}\right)}{\sqrt{{c}^{2}{k}^{2}+{\left(\frac{\u0127}{2m}{k}^{2}\right)}^{2}}}{k}^{i}.\end{array}$$ | (230) |
4.2.3 The Heliocentric universe
$$\begin{array}{c}d{s}^{2}=\frac{1}{(1-{\alpha}_{A}{\alpha}_{B}{U}^{2})}\left[-\left(1-{W}^{2}-{\alpha}_{A}{\alpha}_{B}{U}^{2}\right)d{t}^{2}-2\mathbf{W}\cdot \mathbf{d}\mathbf{x}dt+\mathbf{d}\mathbf{x}\cdot \mathbf{d}\mathbf{x}\right],\end{array}$$ | (231) |
$$\begin{array}{ccc}\mathbf{W}\equiv {\alpha}_{A}{\mathbf{v}}_{\mathbf{A}}+{\alpha}_{B}{\mathbf{v}}_{\mathbf{B}};\mathbf{U}\equiv {\mathbf{v}}_{\mathbf{A}}-{\mathbf{v}}_{\mathbf{B}};& & \end{array}$$ | (232) |
$$\begin{array}{ccc}{\alpha}_{A}\equiv \frac{{h}_{B}{\rho}_{A}}{{h}_{A}{\rho}_{B}+{h}_{B}{\rho}_{A}};{\alpha}_{B}\equiv \frac{{h}_{A}{\rho}_{B}}{{h}_{A}{\rho}_{B}+{h}_{B}{\rho}_{A}}.& & \end{array}$$ | (233) |
4.2.4 Slow light
$$\begin{array}{c}{k}^{2}-\frac{{\omega}^{2}}{{c}^{2}}\left[1+\chi \left(\omega \right)\right],\end{array}$$ | (234) |
$$\begin{array}{c}{v}_{g}=\frac{\partial \omega}{\partial k}=\frac{c}{\sqrt{1+\chi}+\frac{\omega}{2n}\frac{\partial \chi}{\partial \omega}};{v}_{ph}=\frac{\omega}{k}=\frac{c}{\sqrt{1+\chi}}.\end{array}$$ | (235) |
$$\begin{array}{c}\chi \left(\omega \right)=\frac{2\alpha}{{\omega}_{0}}\left(\omega -{\omega}_{0}\right)+O\left[{\left(\omega -{\omega}_{0}\right)}^{3}\right],\end{array}$$ | (236) |
$$\begin{array}{c}{v}_{g}=\frac{\partial \omega}{\partial k}\to \frac{c}{1+\alpha}\approx \frac{c}{\alpha};{v}_{ph}=\frac{\omega}{k}\to c.\end{array}$$ | (237) |
$$\begin{array}{c}\omega \to \gamma \left({\omega}_{0}-\mathbf{u}\cdot \mathbf{k}\right),\end{array}$$ | (238) |
$$\begin{array}{c}{g}^{\mu \nu}{k}_{\mu}{k}_{\nu}=0,\end{array}$$ | (239) |
$$\begin{array}{c}{k}_{\nu}=\left(\frac{{\omega}_{0}}{c},-\mathbf{k}\right),\end{array}$$ | (240) |
$$\begin{array}{c}{g}^{\mu \nu}=\left[\begin{array}{cc}1+\alpha {u}^{2}/{c}^{2}& \alpha {\mathbf{u}}^{T}/{c}^{2}\\ \alpha \mathbf{u}/{c}^{2}& -{\mathbf{I}}_{3\times 3}+4\alpha \mathbf{u}\otimes {\mathbf{u}}^{T}/{c}^{2}\end{array}\right].\end{array}$$ | (241) |
$$\begin{array}{c}{g}_{\mu \nu}=\left[\begin{array}{cc}A& B{\mathbf{u}}^{T}\\ B\mathbf{u}& -{\mathbf{I}}_{3\times 3}+C\mathbf{u}\otimes {\mathbf{u}}^{T}\end{array}\right],\end{array}$$ | (242) |
$$\begin{array}{ccc}A& =& \frac{1-4\alpha {u}^{2}/{c}^{2}}{1+({\alpha}^{2}-3\alpha ){u}^{2}/{c}^{2}-4{\alpha}^{2}{u}^{4}/{c}^{4}};\end{array}$$ | (243) |
$$\begin{array}{ccc}B& =& \frac{1}{1+({\alpha}^{2}-3\alpha ){u}^{2}/{c}^{2}-4{\alpha}^{2}{u}^{4}/{c}^{4}};\end{array}$$ | (244) |
$$\begin{array}{ccc}C& =& \frac{1-(4/\alpha +4{u}^{2}/{c}^{2})}{1+({\alpha}^{2}-3\alpha ){u}^{2}/{c}^{2}-4{\alpha}^{2}{u}^{4}/{c}^{4}}.\end{array}$$ | (245) |
$$\begin{array}{c}{g}_{\mu \nu}=\left[\begin{array}{cc}-[{c}_{eff}^{2}-{g}_{ab}{u}_{eff}^{a}{u}_{eff}^{b}]& [{u}_{eff}{]}_{i}\\ [{u}_{eff}{]}_{j}& [{g}_{eff}{]}_{ij}\end{array}\right],\end{array}$$ | (246) |
4.3 Going further
5 Lessons from Analogue Models
5.1 Hawking radiation
5.1.1 Basics