LOG#135. Modified gravity.


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:


The Coulomb force is strikingly similar to these equations


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


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:


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


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
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
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
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
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}
F_{NN}=F_N+F_{l-exp}=-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
\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
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:
Therefore, using the properties of the MONDian function at large distances, we get:
Now, we use the definition of centrifucal acceleration
to obtain
We substitute this value in the previous expression
and thus, we get
and a centrifugal acceleration

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

See  you in my next blog post!

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