Can we make field lines for gravity?

Question

I wanted to know if we can draw field lines for the gravitational field

Why I think we should:

1. Field lines for electrostatics is defined as the path that the positive charge actually undergoes while in the influence of that electric field.

2. Field lines are also the lines tangent to which give the direction of electric field.

3. Interaction of the field lines create lateral and longitudinal pressure which help us understand why and how charges exert forces on each other.

I think that the same could be done for the gravitational fields, while in the electrostatics we have field lines both, pointing radially outwards and inwards, GFL(gravitational field lines) may just be pointing radially inwards

I want to know the reason for why we don't have field lines for gravitational fields.

Also if we can make them, can they curve like the EFLs?

Answer 1

Yes, although Micheal Faraday came up with the concept of field and field lines to describe electricomagnetic behaviour, you can represent any kind of force field with field lines, they are just lines that follow the direction of the force (can you picture earth's gravitational field lines?). Gravitational force field lines would also be quite similar to Electrical field lines, it's easy to see this in a very simple way by comparing Newton's gravitational force between two point-like objects ($G\frac{m_1 m_2}{r^2}$) and Coulomb's electrical force between two point-like charges ($k\frac{q_1 q_2}{r^2}$).

Answer 2

The equation $\phi(\vec{r})=\phi_0$ defines a family of (equipotential)-surfaces along which no work is required to move a mass. Therefore for all $\gamma$ lying entirely on this surface, we have $$ \int_{\gamma} d\vec{l}\cdot\vec{g}=0\Rightarrow \vec{g}\equiv 0\,\,\text{along}\,,\gamma\quad \text{or}\quad \vec{g}\perp {\hat{n}}\,, $$ where $\hat{n}$ is the local versor of the surface. We can now define field line as continuous curve, such that $$ d\vec{l} \propto \vec{g}\,, $$ where it is introduced the differential element of arc lenght. At this point you have a set of differential equations for the field lines (exploiting the previous one in your preferred coordinates system) $$ \frac{dx}{g_x}=\frac{dy}{g_y}=\frac{g_z}{d_x}=\lambda\,. $$ I have only written what you can find in electromagnetism books, for example Modern Electrodynamics, Zangwill, Sec. 3.3.4.