Daniele Faccio Francesco Belgiorno Sergio Cacciatori - Analogue Gravity Phenomenology

Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA

Ted Jacobson

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The line element or metric ds 2 assigns a number to any infinitesimal displacement in spacetime. In a flat spacetime in a Minkowski coordinate system it takes the form where t is the time coordinate, x , y , z are the spatial Cartesian coordinates, and c is the speed of light. Hereafter I will mostly employ units with c =1 except when discussing analogue models (for which c may depend on position and time when using the Newtonian t coordinate) . When ds 2=0 the displacement is called lightlike , or null . The set of such displacements at each event p forms a double cone with vertex at p and spherical cross sections, called the light cone or null cone (see Fig. ). Events outside the light cone are spacelike related to p , while events inside the cone are either future timelike or past timelike related to p . For timelike displacements, ds 2 determines the square of the corresponding proper time interval .

The metric also defines the spacetime inner product g ( v , w ) between two 4-vectors v and w , that is, Here dt ( v )= v a a t = v t is the rate of change of the t coordinate along v , etc.

In a general curved spacetime the metric takes the form where { x } are coordinates that label the points in a patch of a spacetime (perhaps the whole spacetime), and there is an implicit summation over the values of the indices and . The metric components g are functions of the coordinates, denoted x in () at p and (ii) the first partial derivatives of the metric vanish at p . In two spacetime dimensions there are 9 independent second partials of the metric at a point. These can be modified by a change of coordinates x x , but the relevant freedom resides in the third order Taylor expansion coefficients ( 3 x / x x x ) p , of which only 8 are independent because of the symmetry of mixed partials. The discrepancy 98=1 measures the number of independent second partials of the metric that cannot be set to zero at p , which is the same as the number of independent components of the Riemann curvature tensor at p . So a single curvature scalar characterizes the curvature in a two dimensional spacetime. In four dimensions the count is 10080=20.

The Einstein equation has a unique (up to coordinate changes) spherical solution in vacuum for each mass, called the Schwarzschild spacetime .