Can one understand Newton's law of gravitation using the holographic principle (or does such reasoning just amount to dimensional analysis)?

Following an argument similar to one given by Erik Verlinde, consider a mass $M$ inside a spherical volume of space of radius $R$.

The holographic principle says that the mass $M$ must be completely described on the surface of the sphere in terms of the energy in elementary entities or strings.

By the equipartition theorem, the total energy, $E=Mc^2$, on the surface is given by the number of strings, $N$, times the number of degrees of freedom per string, times $1/2\ k_BT$ per degree of freedom:

$$E = N \times d_f \times \frac{1}{2} k_B T.\ \ \ \ \ \ \ \ \ \ (1)$$

The number of strings on the surface is given by the Bekenstein-Hawking formula:

$$N = \frac{A}{4},$$

where $A$ is the sphere surface area in units of the Planck area $G\hbar/c^3$. In terms of the radius $R$ the number $N$ is given by:

$$N = \frac{\pi c^3 R^2}{G \hbar}.$$

By substituting into equation (1) we obtain an expression for temperature $T$:

$$M c^2 = \frac{\pi c^3 R^2}{G \hbar} \times d_f \times \frac{1}{2}k_BT$$

$$T = \frac{4}{d_f}\frac{\hbar}{2\pi ck_B}\frac{GM}{R^2}.$$

If there are $d_f=4$ degrees of freedom per string then the above formula gives the Unruh temperture for an object falling through the surface with acceleration $g$:

$$g = \frac{GM}{R^2}.$$

This of course is the acceleration given by Newton's law of gravity due to the presence of a mass $M$.

P.S. Could $d_f=4$ come from the 4 dimensions of spacetime?

This post imported from StackExchange Physics at 2015-03-23 09:23 (UTC), posted by SE-user John Eastmond