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LOG#135. Modified gravity.

Posted on 2013/09/18 by amarashikiSeptember 9, 2019

Let me write a review of some non-newtonian potentials and forces. There are some interesting potentials beyond the purely newtonian-like/coulombian-like potential energy and force. In 3 dimensions of space (3+1 spacetime), the gravitational potential energy and the universal law of gravity read:

$U_{N}=-G_N\dfrac{Mm}{r}$

$F_N=-G_N\dfrac{Mm}{r^2}e_r$

The Coulomb force is strikingly similar to these equations

$U_{em}=K_C\dfrac{Qq}{r}$

$F_C=K_C\dfrac{Qq}{r^2}e_r$

There, $G_N, K_C$ are the Newton-Cavendish constant and the Coulomb constant respectively. M and m are point masses, Q and q are point charges, $r$ is the distance between the point masses/point charges and $e_r$ is a unitary vector in the direction of the above two body force laws ( $F_N, F_C$ ). The potential energies are $U_N, U_{em}$ in the gravitational and electric cases.

Are there any other interesting potential energies and force laws beyond the “newtonian” force? Yes, of course! The first case is the harmonic oscillator (or Hooke law). The potencial energy for the elastic force reads:

$U_{el}=\dfrac{1}{2}k X^2=\dfrac{1}{2}k(X\cdot X)$

The force derived from this potential is called Hooke’s law:

$F_H=-kre_r=-k\vec r$

Indeed, the cosmological constant is a variation of the Hooke’s law for a negative $k$ , i.e., the positive cosmological constant is like an inverted harmonic oscillator or “negative” elastic medica. The equations for the cosmic repulsion potential energy and the associated Hooke’s law are (up to a multiplicative constant):

$U_{\Lambda}=-\dfrac{1}{2}\Lambda R^2=-\dfrac{1}{2}\Lambda (R\cdot R)$

$F_\Lambda=+\Lambda Re_R=+\Lambda \overrightarrow{R}$

Other interesting potential we can study is the screened potential or Yukawa potential:

$U_{\mathcal{Y}}=-\alpha_{Y} G_N\dfrac{Mme^{-r/a}}{r}$

The force provided by this potential energy is equal to

$F_{\mathcal{Y}}=-\alpha_Y\dfrac{GMm}{r^2}\left(1+\dfrac{r}{a}\right)e^{-r/a}u_r$

Other interesting potential is the logarithmic potential energy

$U_{p}=-\alpha U_0\log \left( \dfrac{r}{r_0}\right)$

and its force

$F_{p}=-\alpha \dfrac{U_0}{r}u_r$

The logarithmic potential appears in the quantum corrections due to the vacuum polarization triggered by Quantum Field Theories (e.g. Q.E.D.).

Other interesting non-newtonian potential is the gaussian potential

$U_G=-\alpha_g \dfrac{G_0}{2}e^{-r^2/\sigma^2}$

The associated force reads

$F_G=-\alpha_g \dfrac{G_0}{\sigma^2}re^{-r^2/\sigma^2}u_r=-\alpha_g \dfrac{G_0}{\sigma^2}e^{-r^2/\sigma^2}\overrightarrow{r}$

The string potential energy reads:

$U_s=-\sigma r$

and its force will be

$F_s=\sigma u_r$

Finally, I would like to add two generalized potencials from the previous case. The newtonian potential in d spacelike dimensions (or Van der Waals molecular potential if $d=n>1$ ) reads:

$U_N=-G_N\dfrac{Mm}{r^d}$

The hyperharmonic (hyperelastic) oscillator, with $n>1$ , $n\in \mathbb{Z^+}$ , is given by

$U_{hel}=Ar^{2n}=A(r\cdot r)^n, \;\; n>1$

The hyperstring (odd dimensional like) potential energy, with $n>1$ , $n\in\mathbb{Z}^+$ , is given by

$U_{hyps}=-\sigma_{hyps} r^{2n+1},\;\; n>1$

Finally, we can also have the exponential potential

$U_{exp}=-U_0e^{-r/r_0}$

If we focus on the gravitational potential (although the electromagnetic potential can be done by analogy), the presence of non-newtonian potentials induce a (local) spatial variation of the gravitational force. Let’s see the cases given above:

Yukawa potential

$F_{NN}=F_N+F_Y=-G_N\dfrac{Mm}{r^2}u_r-\alpha_YG_N\dfrac{Mme^{-r/a}}{r^2}\left(1+\dfrac{r}{a}\right)u_r=-\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left[1+\alpha_Ye^{-r/a}\left(1+\dfrac{r}{a}\right)\right]$

Logarithmic potential

$F_{NN}=F_N+F_p=-G_N\dfrac{Mm}{r^2}u_r+\alpha_p G_N\dfrac{Mm}{r}u_r=-\overline{G_N }(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left(1-\alpha_p r\right)$

The exponential potential

$F_{NN}=F_N+F_Y=-G_N\dfrac{Mm}{r^2}u_r-\alpha_{exp} G_NMm\dfrac{1}{r_0}e^{-r/r_0}u_r=\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left[1+\left(\dfrac{\alpha_{exp}r^2e^{-r/r_0}}{r_0}\right)\right]$

The gaussian potential

$F_{NN}=F_N+F_G=-G_N\dfrac{Mm}{r^2}u_r-\alpha_g \dfrac{G_NMm}{\sigma^2}re^{-r^2/\sigma^2}u_r=-\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left(1+\dfrac{\alpha_{g}r^3e^{-r^2/\sigma^2}}{\sigma^2}\right)$

The string potential

$F_{NN}=F_N+F_s=-G_N\dfrac{Mm}{r^2}u_r+\sigma_s G_NMm u_r=-\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left(1-\sigma_s r^2\right)$

The elastic/Hooke like potential

$F_{NN}=F_N+F_{el}=-G_N\dfrac{Mm}{r^2}u_r-kG_NMmr u_r=-\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left(1+kr^3\right)$

The antielastic/cosmological constant like/antigravitational potential

$F_{NN}=F_N+F_{\Lambda}=-G_N\dfrac{Mm}{r^2}u_r+\Lambda G_NMmr u_r=-\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left(1-\Lambda r^3\right)$

The log-exponential potential

The log-exponential potential is given by

$U_{lexp}=\alpha_{lexp}\log (r)e^{-r/r_0}$

Then,

$F_{NN}=F_N+F_{lexp}=-G_N\dfrac{Mm}{r^2}u_r-\left(\alpha_{lexp}G_NMm\dfrac{1}{r}e^{-r/r_0}-\alpha_{lexp}G_NMm\log (r)\dfrac{1}{r_0}e^{-r/r_0} \right)u_r$

$F_{NN}=-\overline{G_N}(r)\dfrac{Mm}{r^2}u_r$

$\overline{G_N}(r)=G_N\left(1+\alpha_{lexp}re^{-r/r_0}-\alpha_{lexp}\log (r)r^2\dfrac{1}{r_0}e^{-r/r_0}\right)$

Thus, we observe that we can obtain local modifications of the newtonian gravity adding an extra piece, and we get a non-newtonian force. The general result is that we obtain a gravitational constant depending of the distance (i.e., a locally varying $G_N=G_N(r)$ ) in non-newtonian gravity. Indeed, this class of theories in which we modify gravity are called MOdified Gravity (MOG). MOND (MOdified Newtonian Dynamics) can be also used as a synonim but, it is applied better to the modification of the Newton law of inertia rather than gravity. Namely, instead of
$F=ma$
we get something like
$F=m\mu \left(\dfrac{a}{a_0}\right)$
where $\mu (x)$ is the so called Mondian function. It is generally defined as:

$\mu (x)=\begin{cases}a, \mbox{if}\;\;\ a>>a_0, \;\;\; r<<r_0\\ \dfrac{a}{a_0},\mbox{if}\;\;\; a\approx a_0, \;\;\; r>>r_0\end{cases}$

Let us demonstrate that MOND produces the right rotation curve for long distances. Suppose we balance the gravitational force with the MOND inertial law:

$m\dfrac{G_NM}{r^2}=m\mu\left(\dfrac{a}{a_0}\right)a$

Therefore, using the properties of the MONDian function at large distances, we get:

$\dfrac{G_NM}{r^2}=\dfrac{a}{a_0}=\dfrac{a^2}{a_0}$

$\dfrac{G_NM}{r^2}=\dfrac{a^2}{a_0}$

Now, we use the definition of centrifucal acceleration

$a=\dfrac{v^2}{r}$

to obtain

$a^2=\dfrac{v^4}{r^2}$

We substitute this value in the previous expression

$\dfrac{G_NM}{r^2}=\dfrac{a^2}{a_0}=\dfrac{v^4}{r^2a_0}$

and thus, we get

$\boxed{v_{MOND}=\sqrt[4]{G_NMa_0}}$

and a centrifugal acceleration

$\boxed{a_{MOND}=\dfrac{v^2}{r}=\dfrac{\sqrt{G_NMa_0}}{r}}$

Dark Matter versus MOND? Dark Matter AND MOND? What do you think?

See you in my next blog post!

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LOG#135. Modified gravity.

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