In this blog, I will again define the space-times-time invariance proposal using simple graphics, an explanation intended for a wider audience, videos, and information for nerds. It is taken directly from my own web site that has nearly exactly the same information. In Part B, the equations of motion will be derived, something technical people would reasonably ask for.
The deepest problem physicist have struggled with since the 1930s is how to unify general relativity with quantum mechanics. Perhaps this was the wrong goal. Special relativity imposes algebraic constraints on physics equations that allows us to make many predictions about intervals. General relativity is a field theory that allows us to make many predictions about intervals. Maybe the mathematical physics of intervals should be about algebra or field theory but not both.
General relativity is the simplest field theory for gravity that is consistent with experimental tests of weak and strong fields. It also applies to photons. It is hoped that this next door neighbor of special relativity which uses quaternions is the simplest algebraic approach to gravity consistent with tests. Like special relativity, it should apply to photons. This work is a direct challenge to the accounting system provided by the scaffolding of differential geometry: tensors, metrics, connections, and all that. This effort asserts the rules learned in third grade about addition, subtraction, multiplication, and division need to apply to all expressions used by physicists.
Measure the difference between space-time events.
Two stars go supernova while four kids watch.
There is a difference in time (dt).
There is a difference in space (dR).
Together they make a difference in space-time.
Each kid measures different values for time and space.
Yet the walkers agree on something they can calculate called the interval. This is the standard physics of Special Relativity, the physics of moving.
This blog promotes a new proposal for gravity called Space-times-time Invariance for Gravity (or Quaternion Gravity, QG) where the kids above or below agree on a different value they calculate, space-times-time. In this standard approach to gravity known as General Relativity, this is almost, but not quite true.
A Page-Cast for Measurements (4' 15")
Special relativity is special because it is restricted to inertial observers. To cover more cases requires the machinery of differential geometry. A tensor can be added to another tensor or multiplied by a scalar. An interval is formed by contracting two rank 1 contra-variant tensors with a symmetric, rank-2 metric covariant tensor. A connection is needed to describe how the metric changes in space-time. There are many technical choices one makes along the way to calculating an interval in curved space-time.
This blog issues a formal challenge to the algebraic standards of differential geometry used today by physics. In place of tensors, metrics, and connections, only quaternions will be used. For those trained in the craft of differential geometry, that should sound wildly inadequate. It is a great challenge to do more with less.
Measurements while walking - special relativity
Walking changes how one measures deadly supernovae.
As long as the kids move at a constant rate, special relativity comes into play.
Relativity is not an arbitrary change, but incredibly precise change.
And there is that interval that is exactly the same size.
Note: the numbers are far too big (off by 16 orders of magnitude, I just didn't want to write lots of zeros).
Page-cast for special relativity (3' 40")
Take measurements made by two observers written as quaternions and square them.
If the first terms are the same, then the two observers are in reference frames that are moving in a steady way relative to each other. Note that the observers can be in a gravity field which is a non-inertial reference frame, but that does not cause an issue here.
You can use whatever coordinate system you like, just DBD (Don't Be Dumb, making sure the square of a quaternion has the same form as a 4-vector contraction with the appropriate metric tensor in the standard differential geometry approach).
Measurement While Up or Down - Quaternion gravity
Looking down or looking up from below changes time and space measurements in opposite ways.
If one kid is at a different height in a gravity field to another, then time measurements get smaller while spatial ones get bigger. That is standard physics (general relativity, GR).
The Quaternion Gravity (QG) proposal says the space-times-time values are precisely the same. With general relativity, this space-times-time is not mentioned, but it is almost - but not quite - the same.
Note: the numbers are far too big.
Page-cast for Quaternion Gravity (4' 15")
Special relativity could have been called "special invariance" because it is the invariant interval that all inertial observers agree upon. Invariance principles are deep insights into how Nature works. They are truths that do not change.
The quaternion gravity proposal postulates a new invariance principle: that different observers making measurements in different locations in a gravitational field will agree on an invariant value for space-times-time. Some care is required to say this in a coordinate-independent way. All observers are free to pick their coordinate system. There then exists a norm-preserving rotation in space such that the space-times-time measurement of one observer is exactly equal to another.
But how precisely does the interval change? For a spherically symmetric, non-rotating, uncharged source, only one dimensionless ratio comes into play: GM/c2R. Orbital systems are harmonic systems, suggesting that one use exponential of the dimensionless ratio. The requirement that the space-times-time term is invariant means the time term is the inverse of the exponential experienced by the space term, like so:
The resulting exponential interval has the same form as the Rosen metric. The Rosen metric makes the same experimental predictions as the Schwarzschild metric for all weak field gravity tests to first order Parameterize Post-Newtonian (PPN) accuracy. The Rosen metric also allows for a dipole moment for gravity waves so is not consistent with energy loss from binary pulsars. At second order PPN accuracy, the exponential interval predicts 6% more bending of light around the Sun. We have yet to achieve the precision to decide this issue on experimental data.
Special relativity is not a field theory. It is a constraint on all field theories. The quaternion gravity proposal is also not a field theory. Quaternion gravity is a constraint on all field theories. As such, there is no need for a graviton. Quaternion gravity makes the search for quantum gravity moot.
SR + QG - Special Relativity and Quaternion Gravity
This graphic says a lot.
Start with the reference square which has an interval of 16 and a space-times time of 30.
Compare the reference square with the walkers. They all have the same interval of 16 because that is what is invariant for inertial observers, folks moving at a constant speed compared to the reference.
Compare the reference square with the girl above and boy below. Because they are in a gravitational field, they are not inertial observers. The quaternion gravity proposal says the space-times-time value are exactly the same at 30. The interval will be of different sizes.
If one compares a walker to one of the kids above or below, there is no overlap between the squares they calculate. One must compare with the reference observer.
Page-cast For All Measurements (3' 5")
Since there is a gravitational field everywhere, there are no inertial observers. Working with the squares of quaternions, things are a little easier. Just compare the reference square with any other square. For the walkers, since they travel at a constant speed and are at the same location in the gravitational field as the reference square, they will have the same interval.
The kids above and below are not moving compared to the reference square. By the quaternion gravity proposal, the space-times-time is an invariant. All agree on the value of 30. What then has to be different is the interval. But how different, and how does that depend on the gravitational source mass?
Fortunately, there is no choice in answering the question if one is to be consistent with current experimental tests of gravity. For a spherically symmetric, non-rotating, uncharged source, gravity depends on the ratio of the gravitational source mass over the distance to the center of that mass. Whatever function is used to make the time measurement smaller must be the exact inverse of the one that makes a spatial measurement larger. Since gravitational systems follow simple harmonic patterns for billions of years, an exponential and its inverse that depends on the M/R ratio is an obvious thing to propose.
The interval looks just like the Rosen bi-metric proposal, even though quaternion gravity uses no metrics. The Rosen metric is known to be consistent with current tests of weak field gravity up to first-order Parametrized Post-Newtonian accuracy. The extra metric creates a problem for Rosen's proposal since gravity waves would have a dipole moment and lose energy faster than observed. The simplicity of the quaternion gravity proposal would require for an isolated mass in space that the lowest mode of emission is a quadrapole, consistent with what is seen. Yet there is no graviton with quaternion gravity.