In my last blog, I wrote in detail about zero, one, real numbers, complex numbers and quaternions (or as I now prefer to call them, space-time numbers although I use them interchangeably). For each sort of number, there were rules for addition, rules for multiplication, and a relevant animation. The rules happened to get more complicated going from zero out to the space-time numbers, but they were all of the same form. That makes sense since zero, one, the real numbers, and the complex numbers live inside the tent of space-time numbers.

None of those numbers were used to do any calculations in physics.

In this blog, I will calculate the interval between two events in space-time in spherical coordinates using the typical machinery of differential geometry as best I can. The subject is highly specialized. My only training was four months I spent transcribing Sean Carroll's "Lecture Notes on General Relativity". After that I will start with the same two events written as quaternions and end up at an interval. There is little overlap in between. That happens when one uses different tools.

Intervals Using Differential Geometry

Here is a picture of the roadmap I will follow in this blog:

The set we start with is the real numbers. This is interesting for a few reasons. The real numbers are a totally ordered set, meaning one can say that any member is greater than, less than, or equal to any other member in the set. A group can be formed with the addition operator. The set without zero can form a group with a multiplication operator. Taken together, the real numbers, the addition, and multiplication operators form a mathematical field. Being a mathematical field is essential for doing calculus, the study of change. As structure is added to the real numbers, know that calculus is healthy. By contrast, quantum mechanics starts with the mathematical field of complex numbers. That is necessary to get the interference patterns known to quantum mechanics. Steve Adler, an expert on quaternions, has made this point, referencing a paper by Birkhoff and von Neumann on the foundations of quantum mechanics.

Now that we have the set of real numbers, we need a topology that allows open sets. For any point x in R, for a small positive number epsilon ε, then we can form an open set around x with (x-ε, x+ε). The open set topology is a requirement of calculus which is defined using limit processes.

The next step is to use charts to get to Rⁿ. Then construct an atlas out of a bunch of these charts. By this time, we have an n-dimensional manifold. You will have to read a real math book to understand the details. My eyes glass over on this level of math jargon. I believe at this time I can talk about a coordinate-tuple. One of the dimensions in the 4-dimensional manifold has a label of time, the other three are R, theta, and phi. What can you do with them? Just about nothing.

For each point in this n-dimensional manifold, tangent vectors are created. This is the proverbial 4-vector over the mathematical field of real numbers.

As one moves around the manifold, where every point has its own tangent space, things can change. It is the connection that provides the tools for accounting for this change. There are many choices that can be made about the connection. Einstein worked with a torsion-free, metric compatible connection. Others have studied connections with torsion. Still others work with metric-free connections. I have only a superficial appreciation of the first approach which uses Christoffel symbols of the second kind. Just know if someone brings up such tools, they have made a number of technical choices.

Now one adds the metric. The magic of general relativity is that the metric is a solution of ten non-linear differential equations. It is magic. All of our other theories for physics, be it the well-established quantum electrodynamics (QED), QCD, or out to research involving strings, the metric must be supplied. The metric can be curved, but then that is a solution to a different, unrelated set of field equations.

Consider a space-time curved by a spherically symmetric, non-rotating, uncharged mass. The metric is the Schwarzschild solution of general relativity. There are two events A and A' that are measured using spherical coordinates. Do I know how to subtract them? I know what I am suppose to do: use the process of parallel transport to bring A next to A', then with that transformed 4-vector A as A' neighbor, they can be subtracted. Yet I could get lost in the details for that sort of thing.

Do the easy case first, flat space-time. This allows one to skip the parallel transport step. Subtract A from A' to get dA^μ:

(t', R'/c, θ', ϕ') - (t, R/c, θ, ϕ) = (dt, dR/c, dθ, dϕ) === dA^μ

Notice that two of these coordinates have units of time, while the other two are dimensionless. This is a contravariant 4-vector as indicated by the index as a superscript. Write out the metric tensor in flat space-time in spherical coordinates:

The first two are dimensionless, while the last two have units of time squared. The units may work out yet. I got to pick the signature of the metric. I chose this one because of the overlap with how quaternions work.

To find the interval between the events A and A' in spherical coordinates in flat spacetime, contract two copies of the contra-variant dA^μ.

All inertial observers will agree this is the interval squared between the two events A and A'.

What happens in curved space-time? Locally, the answer remains the same. To someone far away, the interval measured in the curved space-time would be different from the local one.

The difference between a measurement made in flat space-time versus curved space-time is contained in a metric solution of general relativity:

I chose to write the Schwarzschild metric in isotrophic coordinates because these are the ones that an astronomer would put to use. I always have to look it up at the end of Chapter 31 in Minser, Thorne, and Wheeler. I was surprised to see all four terms changing due to the gravitational field. Then I remembered the magic of the Schwarzschild coordinates which pack all the change into the radial component because the Schwarzschild coordinates are non-isotrophic.

The interval in curved space-time is thus:

This is the standard way to write about intervals in curved space-time using isotrophic spherical coordinates.

Intervals Using Space-Time Numbers/Quaternions

In my previous blog, I went into detail about how to add and multiply quaternions. I also introduced the name "space-time numbers" as a synonym for quaternions. Hamilton coined the name quaternions, focusing on the fact that it has four independent real numbers. To my eye, quaternions naturally map to one variable having to do with time, the other three to space. The name space-time number asserts these numbers might work well for space-time physics.

I started from three sets, {0}, {1}, and {R⁺}. None of these sets with the operators of plus and multiplication form mathematical fields. I showed how to use the groups Z₂, Z₄. and Q₈ for the multiplication rules of the real numbers, the complex numbers, and space-time numbers respectively. The addition rules had more basis elements as one progresses from real numbers out to the space-time numbers, but all the rules have the same form and share the same zero. In the long run, it may be productive to think about animations analytically.

My big obsession is with consistency. Once the appropriate structure is built, use it for all calculations, no exceptions. That is the model I am investigating.

Start with one event, A, measured in spherical coordinates:

A = t e + R i + θ j + ϕ k

This is math gibberish, mixing Cartesian basis vectors with spherical variables. At the vary least, I should have used R hat, theta hat and phi hat. In this blog, all previous expressions were 4-tuples. I have taken the liberty to rewrite this blog to use the 4-tuple form so the presentation is consistent]

This is a quaternion. That means as a math object, you can do whatever you please. Take the arctangent if you like. It would be a meaningless calculation for a number of reasons. At any point in this process, one could do a stupid calculation that has no relevance to Nature. The quest is to find a useful calculation out of a great sea of meaningless ones.

The second step is to take the difference between two events, A' and A:

It is understood that any of the deltas could be positive or its additive inverse. The most important thing that comes out of calculating the difference between events is that it removes the role the origin plays in making one observation. With one observation, the values are about the event and the origin. The subtraction of one event and the origin and another event and the origin removes the origin, so the difference is only about the two events, not the origin. I think of this delta as a "poor man's tangent space". It is not a tangent space, but it does eliminate the origin, a necessary thing for two observers to compare data about events.

Can we square this difference? Of course we can, but it would be stupid. The units are still different. There are no metric tensors for quaternions. Look to differential geometry for guidance. It should be simple enough to construct a line element, call it D for the Distance function, so that we not only get the units correct, but also get the relationships between the different spatial parts right:

Square the line element D:

The result of squaring the line element D is another space-time number. Space-time numbers contain the real numbers as a sub-mathematical field. The real number is the first term. Those terms are identical to the differential geometry contraction of a 4-vector using a flat metric tensor in spherical coordinates.

What is going on here? With differential geometry, one needs to pair a 4-vector with a metric tensor to have a way to measure an interval. With space-time numbers, one needs a line element. Square the distance function and the real number sub-field will be identical to the result of differential geometry.

Creating a line element should be straightforward for flat space-time metrics that are diagonal.

A lively debate may happen when discussing rotating gravitational sources. The Kerr metric of general relativity has a dt dθ term that is part of the interval. There is a natural place for that information with a space-time number squared, but it is not in the real part. At this time, I have not investigated rotating gravitational sources.

What happens in curved space-time? If two observers are looking at the same thing from the same distance to the gravitational source, they square the same distance function. If the observers are at different distances to the source, they notice their measurements of time and space are different. That happens in special relativity. In special relativity, both time and space measurements get either larger or smaller. Gravity makes one get larger, the other smaller. The calculated interval is different. 

I wrote about my new proposal for gravity in an earlier blog, Special Relativity and the Road Not Taken. I call the proposal either "Space-times-time invariance as gravity" or "Quaternion Gravity". After a little discussion on math.stackexchange.com, I know how to characterize the proposal mathematically. I am dealing with equivalence classes of space-time numbers. Special relativity uses the following equivalence class:

This says that both the Distance functions on the difference of two pairs of quaternions dq and dq' (one pair for the difference, the other for primed versus unprimed) are quaternions. We say that D(dq) is equivalent to D(dq') when the real part of the square of both are the same. The quaternion gravity proposal has a similar form:

For quaternion gravity, distance functions on the difference of two pairs of quaternions dq and dq' are equivalent if the imaginary part of each squared is the same.

*** New long side-bar: Space-times-time invariant line element

A decent fraction of the readers of this particular blog may not click on above link which provides a road to the line element that is needed below. I will thus provide a sketch of the idea. In GR, one starts with a Lagrangian. Using the Euler-Lagrange equations, the field equations are derived. For a particularly simple situation - a spherically symmetric, uncharged, non-rotating system - the ten equations are reduced to 4 which has the a solution. One then needs algebra to eliminate the constants. One requirement is that the metric approaches the flat Minkowski metric as M/R goes to zero. Another requirement is that to show the results are consistent with Newton's theory of gravity.

By contrast, there is no Lagrange density for the space-times-time invariance proposal. This should sound odd since even Newton's theory for gravity has a Lagrangian. The rules of special relativity apply to all equations written in physics, yet is not based on a Lagrangian. Because the quaternion gravity proposal is just a different equivalence class of the line element squared, it should be like special relativity, not a field theory.

The first goal of the quaternion gravity proposal is to characterize a situation every bit as simple as is done for the Schwarzschild solution of general relativity, so no rotation and no electric charge. There is one additional constraint - that space-times-times is invariant. The most general form of the solution is thus:

If all one needed to do was work for flat space-time, then f=1. To be consistent with Newton's theory of gravity, f=(1 + 2 GM/c^2 R) is enough. Another algebraic constraint is that the proposal should pass weak gravitational field tests. Gravitational systems are periodic, so it is reasonable to propose a function that not only is periodic, but commonly seen in fundamental physics, an exponential. If f=exp(GM/c2R), then all the algebraic constraints are satisfied. It is worth further study.

*** End long sidebar

Here is the curved line element in isotrophic spherical coordinates:

Square it:

All three space-time terms of the square are not changed by their distance to the gravitational source. The interval does depend on location. The interval is different from that predicted by general relativity. Gravity is exceptionally weak, so only the first three terms for dt² and first two for dR² matter in weak field tests. Are those the same? The Taylor series expansion in both cases is:

At this level of accuracy, there is no difference between the field equation approach of general relativity and the algebraic method of space-times-time invariance as gravity.

Calculus With Quaternions

The first few steps in the differential geometry approach are about getting the foundations of calculus correct. Quaternions are a set. One can define open sets with quaternions due to the norm of a quaternion being positive definite. Since both real numbers and complex numbers are sub-mathematical fields, one can do real and complex analysis with quaternions that have three or two zeroes respectively. Unfortunately, the last time I checked, quaternion analysis is broken. I have made an effort on this subject with some marginal progress, but that would be too big a diversion to dive into that subject at this time. I simply want to point out it is a significant area that needs development.

Four-Vectors or Coordinate Tuples?

One Mark G. asked a question:

What is a quaternion representing in your ideas: a four-vector or a coordinate tuple?

This question stumped me for a while. When someone presents themselves as being considerably better trained than myself, I take the comments very seriously. This question as asked has one of two possible answers: a four-vector or a coordinate tuple. Since I don't work in the tangent space, that eliminates the four-vector immediately. A coordinate tuple is a bag of labels that cannot be added together. That is wrong too since quaternions can always be added and multiplied together, even if such an operation has no physical meaning.

Mark G.'s question is well formed English, but not a valid question to ask about this work. A real number can be part of a covariant or contravariant 4-vector, but a real number is not covariant or contravariant. A complex number can be part of a covariant or contravariant 4-vector. The same goes for space-time numbers. It looks like a question based on numerology: there are 4 parts to a quaternion so it must be a 4-tuple or a 4-vector to fit in the differential geometry pipeline.  I had not defined the Distance function, an essential tool to understanding how quaternions handle different choices in coordinates.

The Poynting Vector Test

A more general way to state the space-times-time invariance as gravity proposal is that if some calculation with quaternions ends up with the real term being Lorentz invariant, then the imaginary terms should be invariant if observed at different distances from a gravitational source. One such invariant real number is the difference between the squares of the magnetic and electric fields:

The invariant difference of the fields can be analyzed using the Euler-Lagrange equations. The results are the Maxwell source equations, Gauss's and Ampere's laws. The cross product is known as the Poynting vector. If an electrical system is put at different distances from a gravitational source, general relativity can detail how the difference in the square of the fields changes. But such a calculation is effectively outside of EM. EM needs GR to know what this difference is. The algebraic approach in QG (Quaternion Gravity, not quantum gravity) means that the constraint on the Poynting vector could determine how the difference in the square of the field changes.

The quaternion gravity proposal is thus subject to a new test. Develop a high precision measurement of the Poynting vector.  Lift the machine up a mountain. Drop it slowly down a mineshaft. Quaternion gravity predicts the Poynting vector should remain the same. This is a test of gravity using only EM measurements, nice.

Snarky Puzzle

Mass is Lorentz invariant. Good for mass. That means the thing sitting next to it with space-time number notation - EP, the product of energy and momentum, not a short album - should be invariant if one goes up or down in a gravitational field. Calculate the mass squared using spherical coordinates, but do a few simplifications. Hang out in the plane so you can ditch dθ² and that pesky sine squared. Work with Schwarzschild coordinates so all the changes due to gravity act on dR, not on dϕ. Keep only first order terms in GM/c²R. Mix things around until you get an expression that looks like an equation of motion. Finish the calculation by scratching your head, wondering where the Lagrangian was.