This paper builds upon the existing postulates associated with Relativity as expressed by Einstein and others in order to devise a means to measure the absolute direction and velocity of an object relative to At-Rest affine space.

To much of the Physics community, it has been a longstanding belief that we cannot determine the precise direction and velocity of the earth, or any other body, relative to a state of absolute rest.  This belief is founded on the Principle of Relativity.  Given observers in any number of different inertial frames at motion relative to one another, it has been suggested that there can be no way to determine which if any are at a state of rest and which are in motion.  This is an extension of the equivalence principle.

However, this particular application of the equivalence principle contains an assumption that the only information available to the observers is information from within relative reference frames. While this appears self-evident, the first two postulates that form the basis of Einstein’s theories and of current models of the universe in fact provide a basis for observations that are independent of inertial reference frame.  More specifically, if the following postulates are true as assumed by currently accepted theories, then we have the additional benefit of observations that are relative to affine space at absolute rest.

  1. First postulate (principle of relativity)

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

  1. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body

Note that there is no exclusion in the second postulate for an inertial frame of reference at absolute rest.  Indeed, the existence of an inertial frame of reference at absolute rest should have no bearing on whether these postulates are true.

Examining the second postulate more closely, we note that the velocity of light c, being independent of the state of motion of the emitting body, is an absolute velocity relative to an absolute state of rest in affine space.  As such, every inertial reference frame does possess, in addition to information on motion relative to other reference frames in motion, a key piece of information that is relative to a reference frame that is at absolute rest in affine space.  This recognition that the speed of light being constant and unrelated to the motion of the source as information that is not associated with our reference frame and that therefore breaks the presumed prohibition on deducing our speed and velocity is a realization I have not found elsewhere in my own literature search and represents the first of the ideas in this paper that I contribute. (Note I make no claim to definite knowledge of whether or not this realization has ever previously been published.  But I have found no mention of it myself.)

The challenge, then, is to determine a method by which observations in a reference frame that is in an unknown state of motion (not excluding at rest) can determine the state of motion of the observer’s reference frame relative to the reference frame of the speed of light in an at rest affine space.  For such a method to work, we will choose for experimental reasons to adhere to the following limiting principles:

Limiting principle 1:  All measurements must be taken from a single reference frame.  

This follows from the second postulate above.  If measurements are taken and combined from more than one reference frame, we would then require an ability to relate values absolutely between those reference frames.  But without a-priori knowledge of our direction and velocity of motion, we become dependent on additional postulates and theories that we do not wish to add to our assumptions.  More specifically, we wish to further restrict ourselves by limiting principle 2.

Limiting principle 2:  We may not make any direct measurements of either distance or time.  

Two key extensions of the Theory of Relativity are length contraction and time dilation.  While the Lorentz factor:

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{dt}{d\tau}

has been calculated as the degree of length contraction and time dilation that should be experienced relative to the speed of light, one of the benefits of determining motion relative to an absolute state of rest in affine space is the desire to be able to test whether length contraction and time dilation actually occur, whether the magnitude of length contraction corresponds to the value calculated by the Lorentz factor, or whether length contraction and time dilation are simply optical illusions in the perception of the observer.  We could neither test nor validate these expectations if our method for determining absolute motion relative to affine space at rest were itself dependent on the correctness of those principles.

Limiting principle 3:  Measurements taken in different locations may not be assumed to be synchronized.  

This follows from the relativity of simultaneity.  As with our prior limiting principles, our intent is to minimize those postulates upon which we must depend to establish velocity and direction of motion relative to at rest affine space.  As will be seen in the method proposed, in excluding any assumptions of clock synchronization and simultaneity, we enforce a requirement that time based measurements that are compared or combined must be taken from a single location.

Limiting principle 4: The Cosmic Microwave Background Radiation is not assumed to represent a rest frame in affine space.

Some physicists have theorized that the Cosmic Microwave Background radiation (CMB) itself represents an expanding rest frame of the universe.  If this rest frame is, indeed, an absolute rest frame in an expanding affine space, then the direction and velocity we compute should have as the expected value a velocity and direction of motion consistent with the computed velocity relative to the CMB.  Some may suggest that this entire exercise to determine velocity and direction of motion in a CMB-independent manner is unnecessary.  However, we prefer to exclude the assumption of the CMB constituting a true rest frame as a means of testing that theory. Furthermore, even if the CMB does represent a rest frame relative to all of the known universe, how can we ascertain that the composite totality of all of the known universe is, itself, not in motion, carrying the CMB with it?  Based on the principles of relativity and equivalence, we would not be able to distinguish the difference between a state in which the CMB and our own inertial frame were both at rest versus a state in which we were in the reference frame of the CMB, with that reference frame itself in a state of motion.  In either case, we would observe no dipole shift, but we could not conclude on that information alone that our reference frame at rest relative to the CMB were actually in a state of absolute rest in affine space.

The remainder of this paper will provide an overview of a method that is hypothesized to accomplish this purpose.  The method presented is one that is believed to be able to be achieved to a high level of precision with currently existing technology, albeit at some significant expense.

Method to Determine Velocity and Direction

Holding to the postulates and limiting principles defined for this method, our challenge is to devise a means by which we may take measurements that allow us to calculate our direction of motion and velocity.  Because of postulate 2, we need a mechanism whereby we may take measurements at a defined location that allow us to determine variations in the time for light to traverse a fixed distance in different directions.  Through these measurements, we may determine our direction of motion.  Then, given our direction of motion, we may take measurements to determine our velocity in that direction, again limiting ourselves to measurements of a fixed distance.

But note that in our limiting principles, we precluded ourselves from taking measurements of any actual value for distance.  The key to the solution is to establish equal but unknown distances that we may use to measure traversal time of light.  We propose doing so by establishing a triangle whose vertices must be sufficiently distant to enable the desired degree of precision in our time measurements as well as forming a non-length-contracted equilateral or right isosceles triangle.  A light source and sensors must be positioned at a vertex between two equal length sides and mirrors must be positioned at the other two vertices. Throughout the rest of this article, we will refer to use of a right-isosceles triangle.  Either choice would suffice, but the calculations that will be derived in a later article are simpler using a right-isosceles triangle rather than an equilateral triangle. Given this choice, we will hereafter simply refer to our triangle.

This use of either a right isosceles or equilateral triangle in order to establish non-length-contracted equal measurement distances in differing directions is the second concept in this paper that I have not found elsewhere in my own literature search.

In order to establish such a non-length-contracted triangle, we must establish our vertices in a vacuum or such that the positions themselves are otherwise movable and such that they are not attached to any other body in any manner that the unknown distance between vertices would be influenced by length contraction.  For the purposes of this method, we shall assume we establish the triangle using multiple craft situated in space sufficiently distant from large bodies to maintain relative fixed positions of the vertices and to ignore gravitational effects at our desired level of precision.  We begin by choosing any two sufficiently distant points as our two initial vertices.  To establish a non-length-contracted right isosceles triangle, we will seek the location for a third vertex such that the two new sides formed are of equal length and form right angles to one another.  To do so, we first find the midpoint of our initial side (the hypotenuse of the triangle to be formed.)  We can use light pulses from one of the original vertices to determine the time for light to traverse to the second vertex and return.  Our midpoint on that side will require 1/2 of the round trip time irrespective of our direction and velocity of motion of our reference frame.  Repeated pulses may be used to ensure that the time measurements are constant at our single measurement vertex, assuring us that our two initial vertices and the midpoint are at rest relative to each other.  We then move a fourth craft from our midpoint perpendicular to the first side until such point where the two new sides are perpendicular to one another.

A potential objection would be to the establishment of sufficiently precise right angles to support our later measurements.  While we disallow any specific measurements of distance, we claim that establishment of a true right angle can be done irrespective of length contraction. Implications of this expectation being invalid are discussed towards the end of this paper.  Furthermore, we will not be affected by aberration of light because of the known formula for aberration.  Recall that the angle \theta_o\, perceived by an observer of a light source moving at velocity v and angle  \theta_s\, relative to the vector between observer and source is:

\cos \theta_o=\frac{\cos \theta_s-\frac{v}{c}}{1-\frac{v}{c} \cos \theta_s} \,

In the case of our triangle, we will establish that our various craft are at rest relative to each other before evaluating if we have achieved the desired triangle.  As such, our relative velocity v between any two craft is 0 and our actual angle \theta_s\, we wish to achieve is π/2.  Plugging v=0 and \theta_s\, = π/2, we see that we get cos ( \theta_s\,) = (cos π/2) / 1, so our observed angle   \theta_o\, = π/2 as desired.  In fact, more generally, we see that for relative velocity v=0, the above trivially reduces irrespective of angle to: cos   \theta_o\, = (cos  \theta_s\,) / 1.  So we can be assured than by maintaining a state of relative rest between the vertices of our triangle and the midpoint of the hypotenuse, the triangle constitutes the desired right isosceles triangle and therefore the length of the two sides, while unknown, is fixed to an equal value.  By operating in a vacuum with no connecting body, we know that our triangle is not length contracted.

To determine our direction of travel, we shall form two such right isosceles triangles in planes that are perpendicular to one another.  The base of one side of the two triangles shall be formed by a single pair of vertices shared by these two triangles.  To compute the direction of motion, we simple must measure the ratio of the time for light to travel from the central vertex of the equal sides to each of the other vertices and back, taking all time measurements at the common central vertex.  In a later paper, we will show the mathematical expression derived for the angle θ of our direction of motion in each of our two planes.  The fundamental principle applied is that the ratio between the round trip time on one equal leg and the round trip time on the adjacent equal length leg is described by a sinusoidal function of the angle θ of our direction of motion in the plane of the triangle.  The period of the function will always be π and the phase equal to -π/2. Specifically, the formula for round trip traversal times for equal sides AB and AC of triangle ABC would be:

TACA / TABA = A sin(2 θxy – π / 2) + 1

The amplitude A will be shown in the later posting to be a function of the velocity of motion that is inherently related to the Lorentz factor.  Specifically:

A =  1 – (1-vxy2/c2)1/2

which can also be written in terms of the Lorentz factor γ as:

A = 1 – 1/ γ

This makes intuitive sense when we consider two facts:

  1. The Lorentz factor is traditionally computed to represent the viewpoint of an observer in one reference frame observing effects in another reference frame that is moving at velocity v relative to the observer.  Since observation occurs via light, then all observations must be based on round-trip measurements due to the relativity of simultaneity.  So the fact that our amplitude would itself reflect a function of the Lorentz factor should be expected.
  2. Our use of a right-isosceles triangle, wherein the two sides are therefore orthoganal, should lead to a result, then, that should be a reflection of the ratio of Lorentz’s calculation of length contraction of a body in motion in the directions of the two sides. When orienting our triangle such that one side is in the direction of motion, then we would expect no length contraction of objects orthoganal to the direction of motion and length contraction based on the Lorentz factor in the direction of motion.  Now recalling that length contraction is inverse to time in our non-length-contracted measurements, we would expect the above result wherein the time for non-length-contracted triangles would be higher by the Lorentz factor in the direction of motion.

So now our final equation for the ratio relative to our direction of motion in the xy plane is:

TACA / TABA = (1 – (1-vxy2/c2)1/2) sin(2 θxy – π / 2) + 1

This is the third and final key concept in this paper that I have not found reference to in other writings in my own literature search.  While the Lorentz factor and related transformation are used extensively in physics, I have seen no reference in other writings of a calculation of the sinusoidal function representing the ratio of the round trip travel times of light on each of the equal sides of the triangle.

For motion in the x direction at velocity varying between 0 (at rest), and the speed of light c, we see then that our ratio will range from 1 to 0.5, or alternately the inverse TABA / TACA varies from being equal to the round trip time TABA being double the round trip time TACA.  This also matches our intuitive sense that as our velocity approaches the speed of light in the x direction, the round trip distance traveled in the x axis for round trip TACA will approach half of the round trip distance traveled in the x axis for round trip TABA  while the contribution of the distance traveled in the y axis will become an infinitesimal portion of the total round trip time, effectively approaching 0% of the time required.  Of course, we can never complete the journey at the speed of light and therefore the ratio TABA / TACA will never quite equal 2 just as the portion of the travel time in the y axis will never reach 0% of the total round trip time TACA.

Since we do not know our velocity, we will demonstrate in the later paper that we may compute the angle θ by taking two such sets of measurements using sets of triangles that differ in coordinate axes by π/4, providing two equations in the two unknowns of θ and amplitude A allowing us to solve for θ and A.  Indeed, we could solve directly for θ and vxy, but to maximize the precision of our measurements for velocity in any initial experiments, we would prefer simply to solve for θ at this stage and then later compute velocity when our ratio will have maximum amplitude by repeating the experiment while orienting our x axis in the direction of motion.

When solving for θ, note that the true angle θ for our direction of motion in each of the two planes will be one of the angles θ+k π/2 for any integer value of k, revealing 4 possible values for θ.  We may easily eliminate two values for θ by aligning one side of a triangle in one of the candidate directions and repeating our measurement.  The measurement along the side of our triangle in the direction of motion will yield a higher round trip time than the measurement perpendicular to our direction of motion. We finally are left with determining whether our direction of motion is positive or negative relative to the two diametrically opposed values for θ.  This we may determine through a slightly modified method.  We align our triangle such that one side is on the axis of our direction of positive or negative motion.  We then emit an extremely narrowly focused set of pulses from the third vertex on the side orthoganal to the line of motion.  Assuming our triangle is travelling at a velocity sufficiently greater than 0, we will find that for a sufficiently narrow beam, we will need to angle the direction of the beam in the direction of motion of our triangle in order to receive the maximum intensity of reflection.

With our direction of motion established, we next must determine our actual velocity.  Again we will employ our right isosceles triangle, this time oriented with one equal side along the axis of motion to maximize our ratio.  Solving for v with θ=0, we have:

v = c(1- TACA2 / TABA2)1/2

Note that at this point, we do NOT know for certain if we have calculated our true velocity relative to at rest affine space.  However, in general we can use this velocity for measurements taken for most interesting follow on experiments.  The reason we cannot be certain is because of our limiting principle #2.  We do not wish to assume that length contraction and time dilation are real.  However, if time dilation does actually occur equal to the Lorentz factor, then we know that all of our time measurements have been dilated relative to the measurements of time we would have taken if at rest.  The principle of proportionaltiy assures us that because we relied only on ratios, our direction of motion and velocity measurements will be correct save for our absolute value of our velocity calculation being based on our dilated measurement of time.  In a later paper, we will explore approaches for testing time dilation given our now known direction and calculated velocity in our own reference frame.

As stated above, the step by step mathematical derivations of the above steps of the method will be provided in a subsequent posting.

Checking our Result

There is one step in the above procedure that we rely upon, but that may be checked to ensure a valid result.  We have depended upon the formula for the aberration of light when assuming that right angles were achieved when establishing the positions of the vertices of our triangle.  Once we have found our direction of motion, we should create a new right isosceles triangle in the plane perpendicular to our direction of motion.  Because each of the sides of the triangle are now perpendicular to our direction of motion, we are no longer dependent on issues of length contraction or the formula for the aberration of light to be certain that we have established a non-length-contracted triangle.  In this orientation. our ratio of the round trip times should equal 1.  If we see a variation, we may wish to very slightly adjust the plane of our triangle repeatedly to refine our determination of our direction of travel and repeat the subsequent steps for determining velocity.

Implications and Additional Experiments

If the experiment is successful, the implications of establishing the direction of travel and velocity relative to affine space at rest are many.  First, we may now test the theory of length contraction.  To do so, we may construct a sufficiently long rod for the desired precision and place it perpendicular to our direction of motion.   Using a light pulse and detector at one end and a mirror at the far end, we can measure the length of the rod taking our velocity and direction into account to calculate the distance.  We then rotate the rod to orient it in our direction of motion and repeat the measurement.  Since the measuring location and mirror are affixed to the rod, if length contraction is real, then we should measure a different length.  Our expected result should be the Lorentz factor based on our now known velocity.  Alternately, we can avoid the computation of length and simply measure round trip time.  If measuring time only, then the measurements should be equal, indicating that the length must have contracted an amount equal to the difference in light round trip time caused by our motion.   A significant variation of the contraction from the Lorentz factor would be a serious anomaly suggesting further experiments to further test the theory of length contraction.  Equal length measurements based on our velocity and direction of travel would indicate that length contraction does not actually occur.  Assuming we do confirm the Lorentz factor for length contraction, an additional interesting experiment would be to repeat this procedure using rods composed of different materials to determine if length contraction is solely related to velocity or if the nature of the material involved also affects the degree of length contraction within the limits of our measurement precision.

Revisiting our Postulates

As originally indicated, for this approach to be valid, our initial two postulates must be correct.  Therefore, it should also be possible to consider the potential expectations of certain unexpected results.

The first unexpected result to consider would be if we find that the ratio of the round trip times for light traveling along the two equal legs of our right isosceles triangle is always 1.  Three possible explanations are posited, any of which would have profound implications for the current theoretical underpinnings of relativity:

  1. Could what we perceive as empty space itself experience length contraction?  While this would not, strictly speaking, violate our postulates.  such an implication could potentially resurrect the notion of known space being filled with an aether, that itself experiences length contraction.  This would also raise interesting questions about the effects of such an aether on the speed of light.  In the presence of an aether, could such an aether affect the speed of light and even be necessary for  light to propagate?  Furthermore, if an aether were required for the propagation of light, then would other matter be capable of traveling at velocities exceeding the speed of light through space that did not contain the aether?  Indeed, could our concept of the visible universe itself represent simply the limitations of the extent of expansion of such an aether?
  2. Could the speed of light be affected by the motion of the source?  While we would be highly doubtful of this explanation for a constant ratio of 1, mathematically we might expect such a result if the speed of light was not independent of the motion of the source. Past experiments, and more precise experiments in space using a fixed point and a fast moving point traveling directly towards the fixed point to measure round trip light travel time should be possible to rule out this potential implication.
  3. Our triangle is at absolute rest (or at sufficiently low velocity of motion to be indiscernible at our measurement precision.) This implication would not violate either of our postulates, but would invalidate existing theories that our velocity relative to the CMB represents our true velocity in affine space.  The calculated velocity of earth relative to the CMB is sufficiently high to be well above the velocity that we can measure through the above methods.  This would imply a significant overall rate of motion for the CMB and the known universe.

Subsequent to posting of this article, the currently unpublished paper deriving the mathematics for the calculations was completed.  Any individual wishing to obtain a copy of the completed paper may submit a comment requesting the paper and providing a valid email address in order for us to correspond.  Such comments shall then be removed in moderation rather than cluttering this blog with the requests for the completed paper.  Once the paper has been published, a reference to the publication will be placed here.

 

 

Advertisements