Bound on the photon circular orbit, for generic static and spherically symmetric spacetimes normally relativity, with arbitrary Fmoc-leucine-d3 PPAR spacetime dimensions. The result can then be quickly specialized towards the case of four spacetime dimensions. As a starting point, we will assume the following metric ansatz for describing a static and spherically symmetric d-dimensional spacetime generally relativity, which reads, ds2 = -e(r) dt2 e(r) dr2 r2 d2-2 . d (1)Galaxies 2021, 9,3 ofSubstitution of this metric ansatz inside the Einstein’s field equations, with anisotropic Gedunin web perfect fluid because the matter source, yields the following field equations for the unknown functions, (r) and (r), in d spacetime dimensions, r e- (d – three) 1 – e- = (8 )r2 , r e- – (d – 3) 1 – e- = (8 p -)r2 , (2) (three)where `prime’ denotes derivative with respect for the radial coordinate r. It has to be noted that we’ve incorporated the cosmological constant within the above analysis. The differential equation for (r), presented in Equation (2), might be promptly integrated, since the left hand side in the equation is expressible as a total derivative term, except for some overall issue, top to, e- = 1 – 2m(r) – r2 ; d -3 ( d – 1) rrm(r) = MH rHdr (r)r d-2 .(four)Here, MH denotes the mass in the black hole, with its horizon radius getting rH . This scenario is very much equivalent towards the case of black hole accretion, where (r) and p(r) are, respectively, the energy density and stress of matter fields accreting onto the black hole spacetime. Becoming spherically symmetric, we are able to merely focus on the equatorial plane plus the photon circular orbit around the equatorial plane arises as a solution to the algebraic equation, r = 2. Analytical expression for is usually derived from Equation (three), whose substitution into the equation r = 2, yields the following algebraic equation, 8 pr2 – r2 (d – three) 1 – e- = 2e- , (5)that is independent of (r) and dependent only on (r) and matter variables. At this stage, it will likely be useful to define the following quantity,Ngr (r) -8 pr2 r2 – (d – 3) (d – 1)e- ,(six)such that on the photon circular orbit rph , we’ve Ngr (rph) = 0, which follows from Equation (five). Employing the resolution for e- , when it comes to the mass m(r) and also the cosmological continuous , from Equation (4), the function Ngr (r), defined in Equation (6), yields, 2m(r) – r2 d -3 ( d – 1) r m (r) – 8 pr2 , r d -Ngr (r) = -8 pr2 r2 – (d – three) (d – 1) 1 -= two – two( d – 1)(7)that is independent of the cosmological constant . It is additional assumed that each the power density (r) plus the pressure p(r) decays sufficiently rapidly, in order that, pr2 0 and m(r) constant as r . Therefore, from Equation (7) it promptly follows that,Ngr (r) = 2 .(8)Note that this asymptotic limit of Ngr (r) is independent with the presence of higher dimension, as well as with the cosmological constant and can play an essential function inside the subsequent analysis. It can be attainable to derive a handful of exciting relations and inequalities for the matter variables and also for the metric functions, on and close to the horizon. The first of such relations can be derived by adding the two Einstein’s equations, written down in Equations (two) and (3), which yields, e- r= eight ( p ) .(9)Galaxies 2021, 9,four ofThis relation should hold for all probable choices on the radial coordinate r, which includes the horizon. The horizon, by definition, satisfies the condition e-(rH) = 0, thus if is assumed to become finite at the location in the horizon, it follows that, (rH) p (rH) = 0 . (10)In ad.
