Geodesics of Spacetime

The purpose of this page is to explain the concept of geodesics in general relativity to general readers. Geodesics are the centerpiece of understanding what UAPs are and what they are doing.

Understanding Geodesics

Geodesics are the trajectories on which a UAP has to move in order to be possible, as discussed on our main UAP page.

A geodesic is a straight line on a curved geometry, which means it has no turns. Such a line may not look straight but it is.

On a flat sheet of paper, a geodesic is a trivial straight line. On a sphere, they are lines of longitude. In both cases, you can move without turns to get from your starting point to your endpoint.

Objects in the universe are embedded in spacetime, which has a curved geometry depending on the masses within it. When not pushed by anything, objects move along geodesics (straight lines) of that curved spacetime geometry. This means geodesics are the trajectories of freely falling objects. The Earth, for instance, curves the spacetime around it and objects simply follow that curvature, which we call gravitation.

Examples of geodesics

The trajectory of a baseball thrown on the moon (so there is no air friction).
The motions of all objects in space, such as orbits of moons, planets, asteroids.
The orbit of space stations, space debris, satellites, etc.

Think of an asteroid that enters the solar system from outside. The sun, planets, moons, and asteroids all create indentations in the spacetime within the solar system. As the asteroid moves through it, it is like an ant walking straight along that curved surface with all those indentations on it. That entire trajectory is the geodesic of the asteroid.

Remember, a geodesic is a curve that has no turns on it.

As the asteroid is moving, it is never turning, even though it looks as though it is moving on a curved trajectory around the solar system. The moon’s orbit also has no turns, even though it moves along an apparent ellipse. This seems like a mistake but is really a misconception because we are intuitively imagining motion in a flat space. But spacetime is curved. The moon is moving straight but along the curved geometry.

This gets clearer if you imagine jumping from an airplane and throwing a baseball (disregard air pressure). As you are falling you see the baseball move away from you in a straight line. This is because you are both in the same freely falling reference frame. You are both moving on a geodesic, your natural trajectory through spacetime. Only an observer on the ground will see both you and the baseball fall along a parabola.

Freely moving objects experience no accelerations because they are never turning. Only if they are pushed away from their geodesic will they experience an acceleration.

You cannot feel gravity. The weight of your body is not due to gravity, even though we tend to say otherwise. What’s really happening is that the Earth is in our way. Our geodesic is pointing inside the Earth but we can’t fall there. So the Earth is pushing us away from our geodesic and that push represents a turn in our motion through spacetime. This constant invisible turn is the weight we are feeling, just like turning in a car produces an acceleration.

Imagine you were an astronaut on the international space station. The station is moving in an orbit around the Earth, so it is on its natural geodesic and thus has no accelerations on it. You and the station are in zero G. But now imagine a rocket booster was to slowly move the space station to the side. As you are floating in the station, you would see one wall come closer to you and the opposite wall move away. Eventually, you would hit the wall and be pushed against it. You would feel your own weight.

In other words, on Earth, we are the ones in an accelerated reference frame. We see a baseball move along a parabola not because it is being accelerated but because we are.

Objects don’t feel a force of gravity. There is no force. They are simply following the natural curvature of spacetime.

This is why a UAP needs to move along geodesics. Their accelerations are so enormous, the only way for them to be possible is for these accelerations not to exist at all. But geodesics around the Earth are orbits and cannonball trajectories. So a UAP has to create its own geodesics by distorting the spacetime in its immediate vicinity, enabling it to fall forward, to any side, or up, as the pilot sees fit. Because they are falling, they have no forces on them. They are not turning, no matter how they are moving.

UAPs are not flying; they are falling.

Here is a very helpful visualization of orbits on a curved surface.

Geodesic Equation

For the more mathematically minded readers, we are deriving the equation of motion for freely moving objects.

We begin by imposing that a freely falling particle experiences no acceleration.

${d^2 X^\mu \over d\tau^2} = 0 ,$

where $\tau = X^0$ is the proper time (time in the particle’s frame of reference), and $X^\mu$ are the coordinates of the freely falling reference frame.

Next, we use the familiar chain rule to obtain

${d X^\mu \over d\tau}={d x^\nu \over d\tau} {\partial X^\mu \over \partial x^\nu} ,$

where $x^\mu$ is any other coordinate system, such as the system of an outside observer. The freely falling coordinates $X^\mu$ are functions of $x^\mu$ . Differentiating with respect to time, we get

${d^2 X^\mu \over d\tau^2}={d^2 x^\nu \over d\tau^2} {\partial X^\mu \over \partial x^\nu} + {d x^\nu \over d\tau} {d x^\alpha \over d\tau} {\partial^2 X^\mu \over \partial x^\nu\partial x^\alpha}$

Thus

${d^2 x^\nu \over d\tau^2} {\partial X^\mu \over \partial x^\nu} =- {d x^\nu \over d\tau} {d x^\alpha \over d\tau} {\partial^2 X^\mu \over \partial x^\nu\partial x^\alpha}$

Multiply the above equation by the following expression:

${\partial x^\lambda \over \partial X^\mu}$

We then get

${d^2 x^\lambda \over d\tau^2} = - {d x^\nu \over d\tau} {d x^\alpha \over d\tau} \left[{\partial^2 X^\mu \over \partial x^\nu\partial x^\alpha} {\partial x^\lambda \over \partial X^\mu}\right] .$

The expression in the bracket is called a Christoffel symbol:

$\Gamma^\lambda {}_{\nu \alpha} = \left[{\partial^2 X^\mu \over \partial x^\nu\partial x^\alpha} {\partial x^\lambda \over \partial X^\mu}\right]$

So we can write

${d^2 x^\lambda \over d\tau^2} = - \Gamma^{\lambda}_{\nu \alpha} {d x^\nu \over d\tau} {d x^\alpha \over d\tau} .$

This is the geodesic equation, the equation of motion of a particle in curved spacetime.

Solving this equation yields the trajectory of a freely moving particle as it moves through spacetime (e.g. planetary orbits).

In the limit of weak, static fields and slow velocities, the geodesic equation turns into Newton’s equation of motion in a gravitational field.

Image Sources

[1] Geodesics on a spheroid