When Diving Into The Details Of Gravitational Theories

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Before diving into the details of gravitational theories with anisotropic scaling, we discuss some important ingredients of the theory and why they are essential. 2.1.1 Higher order terms The non-renormalizability of General Relativity means that it is an effective theory and the Einstein-Hilbert action contains only the terms relevant at low energies. Then one naturally is tempted to add higher order curvature terms to the action thereby making the theory applicable at high energies. This possibility was first explored in 1962 by R. Utiyama and Bryce S. DeWitt [33]. They noticed that the action of quan- tum gravity should contain functionals of higher derivatives of metric tensor besides the Einstein-Hilbert action. But is such a theory renormalizable? This question was answered in affirmitve in 1977 by Kellogg Stelle [34]. He showed that the theory is renomalizable with quadraditic curvature invariants. However due to the presence of higher time derivatives, such a theory has the negative norm state called ghosts which allow the probability to be negative and hence breaks the unitarity. In fact, as back as in 1850, Mikhail Ostrogradsky showed that presence of time derivatives higher than two will lead to the problem of ghosts [35]. Combining all these ideas, Horava added only the terms containing higher spacial derivatives while keeping the time derivatives to second order. Also the power-counting renormalizability restricts the number of spatial derivatives to six. This

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