Space-Time

The first step in crafting a gravitational theory is to develop a clear picture of space-time. Classical physics and special relativity share the same view of a Euclidean 3D space.  This familiar model of space is a natural starting point to introduce concepts that are used to express the 4D Minkowskian space-time of special relativity.

Euclidean 3D Space

The most intuitive notion of space is as three independent directions that can be described by the Cartesian coordinates x, y and z. A fundamental property of space is that these coordinates are cannot be uniquely defined. A reference frame is here used to denote a particular choice among the possible Cartesian coordinates for a given Euclidean space.

Translations

From any given reference frame one can generate a new frame by select any point and translate the origin to this point. It is also possible to generate a new reference frame by rotating the coordinate system about the origin under yaw, pitch and roll. Finally, a new reference frame can be attained by reflecting our coordinates about any plane.

Any rotation or translation can be produced using two reflections. A single reflection, however, changes the handedness of a coordinate system. Unlike rotations, translations and rotations have a continuous range from the original reference frame to the target reference frame in question. This continuous nature is required for the study of rigid body motion. It is customary to select a right handed coordinate system, where if forward is defined to be along the x direction and left is defined to be along the y direction, then the z direction is up.

If we require our coordinates to have a particular handedness, right handed for example, reflections no longer produce valid reference frames. It turns out that all possible right handed reference frames can be reached from any single right handed reference frame by a rotation and a translation. A particular change from one reference frame to another reference frame is called a transformation.

The process of applying one transformation after another is called composition. A->B means the transformation from reference frame A to reference frame B.

Euclidean Group

Given two transformations, A->B and B->C, this defines a third transformation, A->C. This property is called closure.

If we have a series of transformations, A->B->C->D, it does not matter in which order we compose the transformations. The result will be the transformation from A->D. This is called the associativity property.

The translation consisting of no rotation and no translation, A->A, does not alter the reference frame. Composing this with any other transformation will not change the transformation. This is called the identity property.

For any transformation, A->B, there is a transformation, B->A, such that A->B->A is the identity. This is called the inverse property.

This collection of properties occurs in a broad number of mathematical contexts, so much so that mathematicians have given the name “group” to any situation where these properties apply.

Not surprisingly, the above group is called the Euclidean group. To be pedantic, one should call this the special Euclidean group in three dimensions, or SE(3) for short. This is called special because we did not allow reflections.

In a very real sense, this set of symmetries defines what is meant by space.

Euclidean Transformations

Transformations between Euclidean reference frames can be expressed as functions taking any set of coordinates in one frame, X, to the set of coordinates in another frame, X’, representing the same point.

Translations simply add a constant vector, D,  to all points.
X’ = X + D

Rotations can be expressed using a rotation matrix. To be pedantic, rotation matrices are orthogonal, unitary matrices. This simply means that the transpose of a rotation matrix is its inverse and its determinant is 1.
X’ = R X

Any transformation can be expressed as a rotation followed by a translation, giving a general expression for any translation.
X’ = R X + D

The ability to express these relations using vectors and matrices makes these transformations linear. This makes the Euclidean group a linear group.

Time

For any given reference frame time is independent of space. This means we can represent it mathematically as a fourth coordinate to create a 4D space-time. This time coordinate is unchanged by the rotations and spatial translations defined above. Further, translations are possible in the time direction.

An object will trace a curve in this 4D space time, marking its location at each instant of time. This curve is called the trajectory of the object. The change in space with respect to time is called the velocity of the object, and is constant if the trajectory is a line.

The first clue that space and time are connected actually came when Galileo discovered the law of inertia, also known as Newton’s first law. This law is typically expressed as “an object’s velocity remains constant, unless acted upon by a force.”

Galileo himself noted that this principle implies what we now would call the principle of relativity by observing that if a passenger was asleep when a ship sets sail, they would be unable to determine whether the ship was moving at constant velocity or at rest once they awoke. This can be phrased in the language of reference frames used above as “an object moving at a constant speed in one reference frame is at rest in some other reference frame.”

The transformations involving only a change in velocity are called boosts. When combined with the 3D Euclidean group of our space, this should produce a 4D linear group.

Possible 4D Linear Groups

The possible 4D linear group containing the 3D Euclidean group can be characterized by a single free parameter that expresses how time interacts with space in a boost.

Circular Linear Groups

The first case is that the fourth coordinate could be another spatial direction, scaled by a factor of 1/c. In this case a velocity boost is really just a rotation in 4D by an angle θ.
tanθ=v/c.

Independent Linear Group

The second case is that the fourth coordinate is unchanged by a velocity boost. This results in the Galilean group, which forms the basis of Newtonian mechanics.

Hyperbolic Linear Groups

The final possibility introduces the concepts of a hyperbolic spatial direction, and hyperbolic rotations. The exploration of these concepts will be deferred until the section on Minkowskain Dynamics.

The final possibility is that the fourth coordinate is a hyperbolic spatial direction, scaled by a factor of 1/c. In this case a velocity boost is really just a hyperbolic rotation by a rapidity φ.
tanhφ=v/c

The value c then becomes an invariant speed, any object with this speed in one reference frame will have this speed in any other reference frame. This produces a discontinuity in the transformations between objects traveling less than c and those traveling faster than c.

This results in the Poincaré group, which forms the basis of special relativity. When only rotations and boosts are considered one finds the Lorentz group. Pedantically this is the proper, orthochronous Lorentz group, SO+(1,3). This means that we don’t allow transformations that swap the time-like and space-like component and preserve the direction of time. These groups transform between Minkowskian reference frames.

Observation

In most circumstances the difference between our actual space-time and a Galilean space-time defined by the independent linear group are seen to be negligible. This allows dynamics to be accurately approximated using a Galilean space-time.

These three cases can, however, be distinguished by high precision measurements at high velocities. It was not until electromagnetic waves were studied  in the late 1800’s that we had any ability to reach speeds that were able to distinguish our space-time from the independent linear group.

Light, being massless, travels at the invariant speed. This was the first setting where speeds this high were empirically significant. Physicists, however, were reluctant to discard a Galilean space-time by accepting this as anything more than an optical phenomenon, thus the invariant speed is often called the speed of light.

Since then technologies such as high energy particle accelerators and GPS satellites have established that this is not simply an optical property but indeed the fundamental nature of our physical space-time.

Minkowskian Transformations

Like Euclidean transformations, a Minkowskian transformation can be defined by multiplication by a Lorentz matrix and addition of a translation vector.
X’=L X +D

The Lorentz matrix can be decomposed into a rotation in the Euclidean space and a boost along a particular velocity, where the rotation matrix is orthogonal and the boost matrix is symmetric.
L = R B

Since this is a group, the compositions of two boosts must be a Lorentzian transformation. Unfortunately boosts are not a closed subgroup.
B B‘ = L = R B

R is called the Thomas rotation, and vanishes when B and B‘ are collinear.

We now turn our attention to Minkowskian dynamics.

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