Pre-Einsteinian Gravitation

"Engineering is merely the slow younger brother of physics."
-Sheldon Cooper

Gravitation has been a matter of discussion and research in almost all of human recorded history. From Babylon, Greece and Egypt we have many recordings of how the ancients studied the skies and wondered what was going on up there and what governed the stars. Even though this is a very interesting subject I will begin some 2000 years later, at the end of the dark ages, around 1600, when Johannes Kepler met Tycho Brahe.

Johannes Kepler (Ioannes Keplerus)

Kepler received massive amounts of raw data from Tycho Brahe which he worked with for 20 years before he could make an accurate description of the orbits of the planets around the Sun. His work resulted in three laws:

Kelpers law of elliptical orbits. The planets orbit the Sun in elliptical orbits with the Sun at one focus.
Keplers law of equal-areas. The line connecting a planet with the Sun sweeps out equal areas in equal amounts of time.
Keplers law of periods. A planet's period is proportional to the major axis of the ellipse raised to the power 3/2, \[ T \propto a^{3/2}, \] where the constant of proportionality is the same for all planets.

These three laws were the beginning of the end of the discussion whether our solar system is heliocentric¹ or geocentric².

Galileo Galilei

Galileo Galilei was born in 1564 in Italy, and was thus active at the same time as Kepler. Galileo showed an early interest in science, and already as a young student he formulated a law for swinging pendulums. But Galilei's most important work on gravity was when he dropped balls with different mass to show that a change in mass does not change the acceleration, a result that was later more firmly established by rolling balls down inclined planes. In addition he stated that the distance the ball advanced is given by \[ d \propto t^{2}. \] He also established his Principle of Inertia:

"A body moving on a level surface will continue in the same direction at constant speed unless disturbed."

Newton reinterpreted this law to fit it into his work and in modern times it has led to the following statement:

"All laws of physics are the same in all inertial reference frames."

Here the term `inertial reference frame` is very important, they are reference frames where a freely moving body, i.e. a body that is not influenced by any external forces, continues to move with a constant velocity. These frames are the kinematical foundation for Newton's laws.

Isaac Newton

50 years after Kepler's and Galilei's main discoveries, Sir Isaac Newton came along. In ``Philosophiae Naturalis Principia Mathematica" --- also known as ``Principia" --- from 1687, he formulated the three laws of mechanics and also the law of universal gravitation. Before the laws can be considered one needs to know something about Newton's view of how the world works.

Newton's view on the absoluteness of time and space

"Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration..."
-Isaac Newton

What Newton means by this is that everyone agrees with what `now` is, and it is hence in principle possible for all clocks in the universe to be synchronized, see []. This universal synchronization requires an infinite velocity with which information can be transferred.

"Absolute space, in its own nature, without regard to anything external, remains always similar and immovable..."
-Isaac Newton

Here Newton defines space as an inert arena that never changes, no matter what happens or is done.

Inertial frames of reference

An inertial frame is one in which Newton's laws of motion on standard form are valid. In particular, an object in an inertial frame with no force acting on it remains at rest or it moves in a straight line with constant velocity.

Newton's laws

Newton published his three mechanical laws and the law of universal gravity in ``Principia", where inertial reference frames and absolute time and space serve as a kinematical background³. Here they are with a modern interpretation.

Newton's law of inertia. If a body is not subjected to an external force, it either remains at rest or continues in uniform rectilinear motion.
Newton's law of acceleration. A force $\vec F$ that acts on a body gives it an acceleration $\vec a$ which is in the direction of the force and has a magnitude that is inversely proportional to the mass $m$ of the body; one of many ways to describe this is \begin{equation} \label{Newt-acc} m \frac{d}{dt} \left( \vec v \right ) = m \vec a = \vec F. \end{equation}
Newton's law of action and reaction. When two particles interact, the force on one particle is equal and in opposite direction of the force on the other, see Figure \ref{Fig:NIII}.
Law of universal Gravity. Two point like bodies with masses $M$ and $m$, separated by a distance $\vec r$, exerts a force $\vec F$ upon each other, see Figure \ref{Fig:NIII}, according to \begin{equation} \label{Newt-grav} \vec F = - G \frac{M m}{\vec r^2} \vec {\hat{r}}, \end{equation} where $G$ is the Newtonian constant of gravitation.

$Newtons third law$

Escape velocity and dark stars

From Newton's laws one can derive the so-called escape velocity, the velocity a non-accelerated object needs to have in order to leave the planet's gravitational field, or equivalently the velocity a particle has when it hits the planets surface if it has been accelerated by the planets gravitational field only, from a state of rest at infinity. The escape velocity $v_0$ is given by \[ v_0 = \sqrt{\frac{2GM}{r_0}}, \] when $r_0$ is the radius of the planet. Remarkably, same result is obtained if one uses Einstein's theory of general relativity. Using the above result and setting the escape velocity to the speed of light, Michell a 18$^{th}$ century natural philosopher, obtained \[ r_0 = \frac{2GM}{c^2}. \]

Thus if the Sun had a radius of 2.944 km, or less, then the photons (or corpuscles as they where known as at the time) would not be able to escape but fall back to the star, and thus the star would not be seen from the outside; hence it would be a `dark star`. Today it is well known that Newton's theory breaks down if a star is compacted to these densities, but this was not known at the time.

Even though the results concerning Newtonian dark stars and Einsteinian (Schwarzschild) black holes⁴ are similar, there are some major differences between a dark star and a black hole. In particular, it is possible to escape from a dark star with e.g. a powered vehicle but according to general relativity it is not possible to escape from a black hole.

In Newtonian mechanics gravity is an invisible force that stretches out from all objects with an infinite velocity. This description of gravity was for 228 years the only available one, until Albert Einstein in 1915 presented his general theory of relativity, which soon shall be discussed, but first a look at the special theory of relativity.

1. ^ Sun centered
2. ^ Earth centered
3. ^ It turns out that only absolute time is needed for Newtonian mechanics.
4. ^ We shall discuss these later in \ref{Black-holes}.