# Gravitational field

In classical mechanics , the gravitational field (also called gravity field ) is the force field that is caused by the gravitation of masses . The field strength of the gravitational field are for each location that part caused by gravitational acceleration due to. It can be calculated from the spatial distribution of the masses using Newton's law of gravitation . ${\ displaystyle {\ vec {g}}}$

The Einstein field equations of general relativity describes gravity no more than force field, but as a curvature of spacetime . In rotating frames of reference, such as the one connected to the earth, the gravitational field consists of the gravitational field and the centrifugal acceleration . A vivid model of the gravitational field is the potential funnel, in which balls or coins roll on a three-dimensional funnel surface and simulate the movement in the plane perpendicular to the funnel axis.

## Potential and field

Gravitational potential (red curve) and acceleration (blue) against the distance from the center of the earth. In contrast to the gravitational potential, the gravitational potential is usually set to zero at infinity.

Belonging to the gravitational field potential is gravitational potential . If the mass density is known, its value at the location can be determined by solving the Poisson equation${\ displaystyle \ Phi ({\ vec {r}})}$${\ displaystyle {\ vec {r}}}$ ${\ displaystyle \ rho ({\ vec {r}})}$

${\ displaystyle \ Delta \ Phi ({\ vec {r}}) = 4 \ pi G \ rho ({\ vec {r}})}$,

where is the constant of gravity and the Laplace operator . So the potential to an approximately point-shaped or radially symmetric body of the mass of , for example, ${\ displaystyle G}$${\ displaystyle \ Delta}$${\ displaystyle M}$

${\ displaystyle \ Phi (r) = - {\ frac {GM} {r}} (+ \ Phi _ {\ infty})}$.

Here the potential is in infinity. It is a freely selectable integration constant and is usually arbitrarily set to zero. (For a detailed derivation see Potential (Physics) ). ${\ displaystyle \ Phi _ {\ infty}}$

If you multiply the potential by the mass of a body , you get its potential energy ${\ displaystyle m}$

${\ displaystyle V ({\ vec {r}}) = m \, \ Phi ({\ vec {r}})}$.

The gravitational field can be written as a gradient field of the gravitational potential : ${\ displaystyle {\ vec {g}}}$${\ displaystyle \ Phi}$

${\ displaystyle {\ vec {g}} ({\ vec {r}}) = - \ nabla \ Phi ({\ vec {r}})}$

The force generated by the field on a body of mass is then ${\ displaystyle {\ vec {F}} _ {\ mathrm {G}}}$${\ displaystyle m}$

${\ displaystyle {\ vec {F}} _ {\ mathrm {G}} ({\ vec {r}}) = m \, {\ vec {g}} ({\ vec {r}})}$.

## Field strength

The field strength of the gravitational field is called gravitational field strength or gravitational acceleration . It is independent of the sample mass (i.e. the mass of the body under consideration, which is in the gravitational field). If there are no other forces acting, the exact acceleration of a test mass is in the field. ${\ displaystyle {\ vec {g}}}$${\ displaystyle {\ vec {g}}}$

A point mass creates the potential ${\ displaystyle M}$

${\ displaystyle \ Phi ({\ vec {r}}) = - {\ frac {GM} {r}}}$

and therefore the associated radially symmetrical field with the field strength

${\ displaystyle {\ vec {g}} ({\ vec {r}}) = - {\ frac {GM} {r ^ {2}}} {\ hat {e}} _ {r}}$

This formula also applies to spherically symmetrical bodies if the distance from the center is greater than its radius. It applies approximately to any body of any shape if it is orders of magnitude larger than its extension. If there is a test mass in this gravitational field, the result is ${\ displaystyle r}$${\ displaystyle r}$${\ displaystyle m}$

${\ displaystyle F _ {\ mathrm {G}} = m \, g (r) = m {\ frac {GM} {r ^ {2}}}}$.

This corresponds to Newton's law of gravitation , which specifies the amount of the attractive force acting between the centers of mass of and , which are at a distance . ${\ displaystyle M}$${\ displaystyle m}$${\ displaystyle r}$

Since every arbitrarily extended mass can be broken down into (approximately) point-like partial masses, every gravitational field can also be represented as a sum over many point masses:

${\ displaystyle {\ vec {g}} ({\ vec {r}}) = - G \ sum _ {i} {m_ {i}} {\ frac {{\ vec {r}} - {\ vec { r}} _ {i}} {| {\ vec {r}} - {\ vec {r}} _ {i} | ^ {3}}}}$

where are the locations of the point masses . ${\ displaystyle {\ vec {r}} _ {i}}$${\ displaystyle m_ {i}}$