Then, relative to the laboratory, the velocity of a sound ray propagating, with respect to the fluid, along the direction defined by the unit vector , is
Solving this quadratic equation for as a function of provides a double cone associated with each point in space and time. This is associated with a conformal class of Lorentzian metrics [607, 622, 626, 624, 470]
The virtues of the geometric approach are its extreme simplicity and the fact that the basic structure is dimension-independent. Moreover, this logic rapidly (and relatively easily) generalises to more complicated physical situations.3
Living Rev. Relativity 14, (2011), 3
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