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I am reading Erik Verlinde's paper "Emergent Gravity and the Dark Universe". Equations 2.12 and 2.13 give the area $A(r)$ and volume $V(r)$ respectively. Do $d$ dimensions comprise of both space and time dimensions? Also, $\Omega_{d-2}$ is the volume of a $d-2$ dimensional unit sphere. Here, are we counting the dimensions of the bulk of the sphere or its boundary. For example, is the regular notion of a sphere a 2-dim or 3-dim object?

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Here $d$ is the number of dimensions of spacetime, and $d-2$ is the number of angular (i.e., non-radial) spatial dimensions. $\Omega_{d-2}$ is the integral over those angular dimensions. When $d=4$, it is the area of the two-dimensional surface of the usual unit sphere, namely $4\pi$.

Here a “$d-2$ dimensional unit sphere” means the set $x_1^2+…+x_{d-1}^2=1$, not the set $x_1^2+…+x_{d-1}^2\le 1$. The “regular notion of a sphere” is a 2-dimensional object, a “2-sphere”, which can be considered the boundary of a 3-dimensional ball. A “3-sphere” is not a 3-dimensional ball but rather the boundary of a 4-dimensional ball.

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