For a long time now, I’ve been dissatisfied with the fact that the Lorentz Transformation is taken with respect to velocity. There are two problems which I believe point toward a weakness in using a velocity based transformation.
To describe the first problem, you must first realize that I consider that all phenomena are manifestations of waves. In particular, Mass is a wavelike phenomena. Waves can have rest energy in the form of standing waves, and such standing waves exhibit Dispersion which obeys the energy, mass, momentum relationship. We know that massive objects have a Compton wavelength, which can be show to be identically equal to the wavelength of the corresponding standing wave.
With this fact in mind, consider that a wave-based massive particle has an associated wavelength, and frequency. We can consider these to be the most fundamental forms of length and time measurement, intrinsic to that particle. My longstanding hypothesis has been that when we say that length contracts, and time dilates, it is the wavelength of the wave that shrinks, and it is the period of the wave which stretches.
However, if this hypothesis were correct, two co-moving masses should have different wavelengths and frequencies, given they have different masses. The two co-moving masses should measure space and time differently, based on the difference in mass. The velocity based Lorentz Transformation does not distinguish this difference.
The second problem has to do with Gravity. In treatments of general relativity, where the case is made that we must develop a theory of curved spacetime in order to deal with specific limitations in Special Relativity, one of the primary examples discussed is a photon that is absorbed at a point deeper within the gravity well than it was emitted.
The difficulty which arises when applying Special Relativity to this scenario is usually ascribed to the fact that there are no global reference frames. This statement has always seemed a bit cryptic to me. I believe a better way to describe the problem is like this: The emission and absorption point must be in different reference frames, but Special Relativity, which looks only at differences in velocity, cannot distinguish this fact, when those points aren’t moving relative to each other.
The way Einstein got around the problem of the emission and absorption point being at rest relative to each other (pre-GR), was to introduce an equivalent reference frame that was uniformly accelerated. Now that motion is involved, we actually stand a chance at describing a relative difference in velocity between the emitter and absorber. The difference in velocity, in this case, is introduced by acceleration over the time of flight of the photon.
I decided that I wanted to see if it were possible to come up with an Energy based Lorentz Transform, instead of a Velocity based one, with the belief that such a transform should properly address these two problems.
Let’s look at the canonical transform given a uniform relative velocity along the x axis.
Now, let’s multiply by a wise choice of unity.
We can recognize the coefficients involving gamma to be related to the energy and momentum of a massive particle.
Making these replacements, the transformation looks like this:
So far, so good. Aside from making these equations look just a tad bit simpler (aside from the explicit fraction on y and z), we haven’t changed a thing about the form or properties of a Lorentz transformation.
The first way that we will extend this construction is to multiply both sides by the rest energy.
Again, we haven’t actually changed the relationship between the primed vs. unprimed coordinates. However we now need to think of this as a different kind of transformation. Specifically, the transformation on one side no longer has a determinant of 1 – it no longer belongs to the canonical Lorentz group.
The way I like to think of what’s going on here, is that we are actually transforming both sides of this equation, the primed and unprimed coordinates. So long as the determinant on both sides matches, we will be able to return to something that looks like our original velocity based Lorentz Transformation.
We will now contemplate the meaning of what each side of this two sided transformation actually represents. On the right, the coordinates are those of the lab, which we are initially taking to be at rest. On the left, the coordinates are those of a particle, which we presume to be in motion. On the right, we are transforming by the Energy and Momentum of a particle that is in motion. On the left, we might imagine that we are transforming by the Energy (Rest Energy) and Momentum (Zero) of a particle at rest.
On both sides of the equation, the quantity that we appear to be calculating, is some combination of the Energy and Momentum of a particle, operating on the coordinates for the reference frame under which this Energy and Momentum are observed.
Let’s give this idea a test run. What would the transformation look like if the left hand side were not representing a particle at rest?
The original form of the Lorentz transformation determines how to transform from a frame that has movement into a frame that is at rest. In this new form, we can now talk about the relationship between different energy frames, without requiring one of them to be at rest.
Solving this system of equations yields the following relationships.
The coefficients multiplying the coordinates have a familiar form – namely, they are the Lorentz transformation of the Energy/Momentum of one frame of reference transformed into the other frame. In other words, it is the relative Energy/Momentum between the two energy frames. Applying the Energy form of the Lorentz Transformation to the energy makes this explicit.
We are using double primes now to represent the relative Energy/Momentum between the reference frames. We can now express the transformation in terms of the double primed values
This is exactly what we would expect – namely that we should be able to express the transform between two different energy-frames in terms of a single transformation that encapsulates the relative energy difference.
And now we introduce the second extension. This extension is meant to get at the two problems that were stated at the outset – namely, that objects should be allowed to have a different energy frame, even if they share velocity frames. The two examples listed were 1) co-moving objects in the absence of gravitation with different mass, and 2) co-moving objects with the same mass at distinct points within a gravity well.
Up to this point, our two sided transformation has satisfied the requirement that the determinant be the same on both sides. With this requirement, we are guaranteed to be able to move fluidly between the two-sided Energy form back to the one-sided velocity form.
The second extension is to relax the requirement that the determinant match on both sides of the two sided transformation.
To determine if it makes sense to relax this requirement, we will examine the two examples.
To start, let’s take the example of two co-moving and equal masses, which are separated in a gravity well, introducing an overall difference in potential energy.
We start with the generic two sided energy frames, allow momentum to go to zero. The energy on one side is just the rest mass energy, while the energy on the other side is the rest mass energy plus the gravitational potential energy.
If we factor out the rest energy, the relationship between the coordinates becomes:
Does this result make sense?
Actually, yes it does. It matches the result achieved by Einstein in his 1911 pre-GR attempt at understanding how Gravity influences the propagation of light. Because Einstein was constrained to work with velocity frames, he had to introduce a secondary accelerating reference frame in order to achieve an equivalent velocity distinction between the reference frames. However, using energy frames, this secondary equivalent system is not required, and the result is achieved immediately.
If we try to understand this result in terms of a particle with wave-like properties, whose mechanism for measuring space and time are derived directly from the frequency and wavelength of the wave, then we see that the frequency and wavelength are affected by the gravitational potential. The particle with more energy is waving faster than the particle with less energy, and this affects how that particle may measure space and time.
In the next example, we have two co-moving objects with different masses. Start with the generic two sided energy frame transformation, let momentum go to zero, and let energy equal rest mass energy.
This transformation says that the coordinates of one particle are related to the coordinates of a different particle by scaling those coordinates by the fraction of the mass.
Does this actually make sense? Well, if what we mean by a particle is something that is exhibiting wavelike properties such as a Compton wavelength, and a frequency corresponding to the Zitterbewegung, and if this wavelength and frequency are the primary mechanisms by which this particle may “measure” space and time, then it actually makes a lot of sense.
Say the ratio of masses were 2 to 1. We would expect the more massive object to have a Compton wavelength and a Zitterbewegung period that were half that of the less massive object. We would therefore expect the full coordinate chart of the more massive object to be scaled by 1/2, relative to that of the less massive object.
Imagine the case of a particle/anti-particle pair. They have equal masses, but they have opposite energy. The ratio of masses would merely introduce an overall minus sign. The coordinate chart of the particle would be inverted in both space and time in order to represent the coordinate chart of the anti-particle.
The final case of comparing disparate mass energy frames comes from comparing photon frames with non-photon frames. Technically, this is a different category of analysis, since we are no longer dealing with co-moving objects. However, it’s interesting to consider how one might go about handling a problem that is completely inaccessible to standard SR.
To solve this problem, we will need to address the general case of what is meant by an Energy Frame, which will require the use of Spinor Algebra.
As a brief refresher, in this Algebra, we can represent physical quantities in Scalar/Vector form. For instance, the position 4 vector can be represented as:
In general, both the Scalar and Vector components are allowed to be complex, where imaginary values are associated with pseudo-scalars and pseudo-vectors. Multiplication in this Algebra takes this form
The Algebra contains two involutions: an Algebraic Conjugate which operates by flipping the sign of the vector component, and the Hermitian Conjugate, which operates by flipping the sign of the imaginary components. Each of this conjugates reverses the order of multiplication as well, so that the conjugate of a product is the product, in reverse order, of the conjugates.
Different class of quantities may be represented in this Algebra, for instance, a 4-vector contains a real scalar and real vector component. One way to ensure that a quantity is composed exclusively of a real scalar and vector components is to represent it as the product of a Spinor with its own Hermitian Conjugate. For instance, the momentum 4-vector can be related to a momentum Spinor in the following way:
In the first equation, the momentum 4-vector is related to the product of the Momentum Spinor multiplied by its own Hermitian Conjugate. The second equation describes the Spinor solutions in terms of scalar and vector components. A Spinor might loosely be called the “square root” of a 4-vector, and so we will always expect 2 solutions, represented above with the plus/minus. In the case of the Momentum Spinor, the 2 solutions correspond to the particle and anti-particle solutions.
Having this very brief refresher on Spinor Algebra, let’s dive into how to use the Algebra to do standard SR. A standard velocity based Lorentz transformation can be represented as:
The S in this equation is the Lorentz Spinor, which is defined in terms of Scalar/Vector components as:
The two plus/minus represent the chronous and anti-chronous solutions. If we use the multiplication rule defined above, we will get the following result:
To move toward Energy rather than Velocity frames, we begin by multiplying the Lorentz Spinor by the square root of mc. The result is the conjugate of the Momentum Spinor.
Using this Spinor to Energy Frame the Space Time coordinates gives us:
The result of the two suggested extensions to the Lorentz Transformation is that it is this Energy Framing of an observed quantity which is invariant:
Technically, the observed quantity which we are Energy Framing can likewise be decomposed into a product of Spinors.
Given this decomposition, we can express the Energy Framing invariance of 4-vectors as such:
This equation allows us to relate the Spinors directly to each other, without forming a 4-vector from their product.
The product of Spinors must be related to each other by an overall phase. However, I will make the argument that the Spinor factors themselves already include a phase, and so this overall phase could be seen as arising from the respective phases of each factor in the expression.
This allows us to introduce the third and final extension to the Lorentz Transformation. The invariance of Energy Framing 4-vectors arises due to an underlying invariance of Energy Framing the constituent Spinors which can be decomposed from those 4-vectors.
The Energy Framing of a Space Time Spinor looks like this:
Now, this is a rather complicated expression. Presumably, we would find that equating two such Spinor expression would allow us to return to the canonical Lorentz Transformation, provided we apply the constraints associated with each of our extensions, however, I’ve not done the algebra required to confirm this.
There are 2 interesting elements that I’d like to focus on. The first is the pseudo-vector term, which appears to be related to angular momentum. The second is the fact that the complex phase now plays a direct role in a fundamental symmetry. Aside from these elements, I don’t have a deep level of clarity as to what the ultimate significance of this expression might be. For now, I will set this Spinor expression aside.
Let us turn now to the Energy Framing of photons. Where any attempt at Velocity Framing a photon will fail, we can succeed with Energy Framing. Such a photon has Energy Framed Space Time coordinates which look like this:
We know that the Energy and Momentum of a photon are related to the frequency and wavenumber via a factor of planks constant. We can factor that constant out of the expression.
Let us resume the mental exercise of employing the frequency of the wave as the clock, and the wavelength of the wave as a meter stick, by which this wave can measure space and time. Photons are mathematically represented as plane waves. A plane wave only has a meter stick which can measure a single direction, which means that any measurement of length perpendicular to that direction is undefined. This helps explain why the spatial coordinates play no role outside of the direction of the wave vector.
Let’s imagine that we equate these Energy Framed coordinates with the Energy Framed coordinates of a photon with twice the frequency. The result would be that the spacetime coordinates would need to shrink by an equal factor, in order for an equality to exist. The Extended Lorentz Transformation between two photons amounts to multiplying by the ratio of their frequencies.