A modified gravitational potential is given by \(V=-\dfrac{GM}{r}+\dfrac{A}{r^{2}}\). If the constant A is expressed in terms of gravitational constant (G), mass (M) and velocity of light (c), then from dimensional analysis, A is,
\(\dfrac{G^{2}M^{2}}{c^{2}}\)
Every term in a physical equation must carry the same dimensions. Gravitational potential is potential energy per unit mass, so the term containing A must have the dimensions of energy per unit mass.
Step 1 — Dimensions of potential: \([V]=[L^{2}T^{-2}]\).
Step 2 — Since A/r2 has the same dimensions, \([A]=[V][r^{2}]=[L^{4}T^{-2}]\).
Step 3 — From V = −GM/r, the product GM has dimensions \([GM]=[L^{3}T^{-2}]\), and c has dimensions [LT−1].
Step 4 — Testing the square of GM divided by c2: \(\left[\dfrac{G^{2}M^{2}}{c^{2}}\right]=\dfrac{[L^{6}T^{-4}]}{[L^{2}T^{-2}]}=[L^{4}T^{-2}]\), which matches A exactly.
The ratio GM/c2 gives \(\left[\dfrac{GM}{c^{2}}\right]=[L]\), a length only. The inverse square of the speed of light gives \(\left[\dfrac{1}{c^{2}}\right]=[L^{-2}T^{2}]\). A dimensionless constant cannot be added to a potential divided by r2, because A must carry L4T−2.
\(\dfrac{G^{2}M^{2}}{c^{2}}\) option is correct.
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