The kinetic relativity theory – hiding in plain sight

2.1 The Hafele and Keating (HK) experiment (1972) [1,2]

The theory begins with an experiment report by HK in 1972. They flew four atomic clocks around the world in eastward and westward directions. After each round trip, they compared the clock times to the Naval Observatory atomic clocks.

They measured relativistic time changes for both elevation (gravity) and kinetic motion (velocity). The relativistic velocity was computed by combining the airplane’s ground speed with Earth’s rotational speed. They collected the speed and elevation data from the flight staff for each flight segment. Their atomic clocks were capable of measuring time down to a level of 10⁻¹³–10⁻¹⁴ due to atomic clock stability over time.

They wrote:

… For low coordinate speeds (u ² ≪ c ²), the ratio of times recorded by the moving and reference coordinate clocks reduces to (1 − u ²/2c ²), where c is the speed of light. Because the earth rotates, standard clocks distributed at rest on the surface are not suitable in this case as candidates for coordinate clocks of an inertial space. Nevertheless, the relative timekeeping behavior of terrestrial clocks can be evaluated by reference to hypothetical coordinate clocks of an underlying nonrotating (inertial) space (6).

For this purpose, consider a view of the (rotating) earth as it would be perceived by an inertial observer looking down on the North Pole from a great distance. A clock that is stationary on the surface at the equator has a speed R Ω relative to non-rotating space, and hence runs slow relative to hypothetical coordinate clocks of this space in the ratio 1 – R ² Ω ²/2c ², where R is the earth’s radius and Ω its angular speed. On the other hand, a flying clock circumnavigating the earth near the surface in the equatorial plane with a ground speed v has a coordinate speed R Ω + ν , and hence runs slow with a corresponding time ratio 1 − (RΩ + ν)²/2c ². …

Importantly, both the airplane and laboratory clocks moved relativistically to the hypothetical ECF.

For the airplane clocks, the kinetic time dilation used the small velocity approximation of (u ² ≪ c ²), resulting in a time ratio of

where t m and t f are the respective times recorded by the moving planes and the ECF during flight, respectively, R is the Earth’s radius, Ω is the Earth’s rotation speed, and ν is the airplane ground speed. They explicitly stated that the airplane clocks had a time that “runs slow,” meaning time dilation.

If the small number approximation of

is used in reverse, their calculation becomes

where u is the total velocity of the laboratory or airplane moving relative to the ECF.

HK compared the airplane clock time against the Naval Observatory clock time. They measured a kinematic time gain in the west direction and a kinematic time loss in the east direction. The time dilation due to gravity potential was similar in each direction. Their experiment clearly showed that motion is neither relative between planes nor relative to the Naval Observatory. The East/West directions clarified that time was relativistic to the ECF.

The HK experiment empirically establishes time dilation. Based only on their experiment, one might conclude that every plane, satellite, car, train, walking person, and building moves relative to the ECF.

To clarify this possibility, GPS satellite results are mentioned.

2.2 GPS satellite reports

The GPS description of relativistic time by Ashby and Spilker [3] and Ashby [4,5] agrees with the HK result and calculation method for satellites instead of airplanes. Both the satellites and ground-based clocks move relativistically to the ECF.

In a 2006 paper, the GPS expert Ashby [6] described GPS time dilation by:

“… so the time of the “moving” clock at the top end of the rod, and the “rest” clock at the same location are related by (4):

(4) t ′ = 1 − v 2 c 2 t .

Thus, a clock moving relative to a system of synchronized clocks in an inertial frame beats more slowly. The square root in Eq. (4) can be approximately expanded using the binomial theorem (5):

(5) 1 − v 2 c 2 ≈ 1 − 1 2 v 2 c 2 .

In the GPS, satellite velocities are close to 4,000 m/s, so the order of magnitude of the time dilation effect is (6)

(6) − 1 2 v 2 c 2 ≈ − 8.35 × 10 − 11 .

This is also a huge effect. A reference clock on earth’s equator is also in motion, but with a smaller speed, of order 465 m/s. To obtain the fractional frequency difference between a GPS satellite clock and a reference clock on the equator, we have to compute the difference (7):

(7) Δ f f = − 1 2 v 2 c 2 − − 1 2 ( ω a 1 ) 2 c 2 = − 8.228 × 10 − 11 .

If not accounted for, this would build up to contribute a navigational error of order 2.13 km/day. Because of these frequency offsets, it is best to view the GPS satellite constellation and the reference clocks on the rotating earth from the point of view of the ECI frame.”

Ashby’s math symbols were: t ′ is the moving body time, t is the fixed frame time, v is the moving body velocity, Δf is the frequency change between the satellite and a clock on the equator, and f is the reference frequency on the equator. The term ωa ₁ is the velocity of the reference clock due to Earth’s rotation ω at equator radius a ₁, and ECI is the same as ECF.

The main point of [Eqs. (6) and (7)] is that a laboratory clock also moves relativistically to Earth and the smaller velocity reduces the difference in dilation between the equator clock and the satellite clock.

Ashby attributed the laboratory dilation to the Sagnac effect. However, the GPS report by Denker et al. [7] merely called it centrifugal potential. HK only mentioned it as motion. Regardless of attribution, HK, Ashby, and Denker et al. agree that laboratory clocks also experience time dilation based on Earth’s rotation.

The cited GPS references confirm time dilation when moving relative to the ECF in orbit and when rotating on Earth’s surface.

Next, a relativistic Doppler shifting experiment in a laboratory will be discussed to determine how relativistic length changes.

2.3 The Mandelberg–Witten experiment (1962)

Mandelberg and Witten [8] set up an experiment to improve the accuracy of earlier Ives and Stilwell [9,10] and Otting [11] experiments. Mandelberg–Witten used a similar apparatus to measure the change in the emitted wavelength from moving hydrogen atoms, which was the result of relativistic movement in a laboratory. They used a spectrograph to separate the spectral lines and took pictures of them.

They described their measurement by:

“The experiment consisted in measuring λ _R and λ _B, the wavelengths observed with and opposite to the beam, and taking the average to determine λ _Q and hence λ ₀ β ²/2. By a subtraction,

(8) 2 λ D ≡ λ R − λ B = 2 λ 0 β cos θ .

β was determined by measuring λ R , λ B , λ 0 , and θ . The velocity was thus obtained by direct measurement without assuming a precise knowledge of the accelerating voltage and without making assumptions regarding the collision mechanism which produced the atom. To give an idea of the magnitudes of the parameters involved, a typical run was made with an accelerating voltage of 63.70 kV which produced a beam of excited atoms whose measured velocity corresponded to β = 0.008176. For a wavelength λ ₀ = 6562.793, we measured 2λ _D = 107.317 A and δλ ≡ ¹/₂ λ ₀ β ² = 0.219 A.”

They combined their results with those of Ives–Stilwell and Otting and plotted them in Figure 6 of their paper. Mandelberg–Witten viewed their Doppler shift experiment as being predicted by SR. They reported an overall precision of 5% for velocities up to 2.8 × 10⁸ cm/s.

In addition to linear Doppler shifting, Mandelberg–Witten measured a tiny relativistic wavelength change created by moving hydrogen molecules. The change was measured by laboratory instruments. Photographic plates showed spectral lines shifting to numerically longer wavelengths, which is called “red shift.” The red shift increased for higher particle velocities.

Their red shift calculation, δλ ≡ ¹/₂ λ ₀ β ², where β is the particle velocity as a fraction of c, is an additional amount that is added to the hydrogen spectral line wavelength λ ₀. This is illustrated in Figure 1.

Figure 1

Illustration of relativistic length dilation by a moving light emitter.

In Figure 1, both the laboratory and moving body have rulers that measure the same light beam wavelength. The moving frame ruler and the laboratory frame fixed ruler are aligned at the left, as illustrated. In this case, the moving body ruler is longer due to length dilation.

By only looking at the relativistic effect and ignoring linear Doppler shifting, a light beam with wavelength λ ₀ is created by the moving body source (emitter) at 1 unit length. The wavelength is observed (measured) at λ _Obs. The wavelength measured by the laboratory ruler is longer than one unit. This agrees with a red-shifted beam as measured in the laboratory frame.

In contrast, a change in source is shown in Figure 2, and the two scales remain the same.

Figure 2

Illustration of relativistic blue shift from a fixed source light emitter.

As illustrated, the laboratory source frame creates a shorter beam length (called “blue shift”) as measured by the moving body frame. Regardless of which frame is the source, length dilation is shown for the moving body.

Figures 1 and 2 illustrations of red shift and blue shift agree with well-known relativistic doppler shifting effects.

The Mandelberg–Witten experiment measured length in the direction of motion. This result, along with HK’s time dilation, means that the relativistic effects for both length and time use the same small-number approximation.

The experiments thus far have established relativistic time dilation to an ECF and relativistic length dilation to a laboratory frame.

This raises the question: Is the same relativistic length change measured perpendicular to the direction of motion? That is, does length change in all directions?

2.4 The Hasselkamp, Mondry, and Scharmann experiment (1979)

This question was answered by Hasselkamp et al. [12]. Their experiment also measured light from high-speed hydrogen atoms. The equipment was set up to measure the emitted wavelength perpendicular to the direction of motion. They focused the light on the entrance of a monochromator and measured the position of specific spectral lines by a photomultiplier. They measured red shift, like the Mandelberg–Witten results.

Their experiment used higher particle speeds up to 3.1% of the speed of light, and their error was about 6%. They described their calculations as follows:

“We consider a light source emitting photons of wavelength λ ₀ that is moving with a velocity v with respect to a detector located at an angle θ with respect to the direction of the source. The detector measures a wavelength λ′ which is given by the Doppler formula (9).

(9) λ ′ = λ 0 1 − v c cos θ 1 − v c 2 ,

c is the velocity of light.

Eq. (9) is a consequence of the Lorentz-transformation of time. The experimental confirmation of the validity of (9) is therefore a verification of time dilatation. If v/c ≪ 1, we find for the Doppler-shift Δλ = λ′ − λ ₀ with neglection of higher order terms (10):

(10) Δ λ ≅ − λ 0 v c cos θ + λ 0 2 v c 2 .

Besides the classical Doppler term −λ ₀ v c cos θ (retardation) we have on the right-hand side of (10) a second term as a consequence of the Lorentz transformation of time. This relativistic term is called the “second order Doppler-shift.” It is always positive and independent of the observation angle θ, therefore always causing a red shift of the measured wavelength (or, what is the same, causing a decrease of the frequency of the moving “clock” in the system of the observer, i.e. a time dilatation).”

The wavelength calculation of [Eq. (9)] when the beam is perpendicular to motion (θ = 90°) is similar to the HK Eq. (3) calculation for time. The wavelength detector was in the laboratory frame and therefore measured wavelength λ′ in the fixed laboratory frame. λ ₀ is the spectral wavelength emitted by hydrogen atoms. The similarity to Eq. (3) comes from the square root position of (1 − (v/c)²)^1/2; it is underneath the moving body wavelength λ ₀ which conceptually agrees with the HK calculation.

Hasselkamp et al. also acknowledged SR by:

“The results of the present experiment are therefore in agreement with the theory of SR and especially with the prediction of time dilatation.”

Because the laboratory measured red shift perpendicular to the direction of motion, the experimental conclusion is that red shift occurs in all directions. That is, wavelength dilation occurs in all directions.

The experiments so far have established relativistic time dilation to an ECF and a relativistic wavelength red shift in all directions to a laboratory.

The next question is to clarify that the square root function of Eq. (3) is needed. The experiments so far have used the small number approximation of Eq. (2). A high-speed experiment will confirm it is needed.

2.5 The Bailey et al. muon experiment [13]

A 1977 laboratory time dilation experiment by Bailey et al. used an extremely high speed.

In the CERN Muon Storage Ring, the lifetimes of both positive and negative muons were measured in circular motion using a 14 m diameter ring. The muons moved at a speed of 0.9994 c, and the muon lifetime τ as measured by the laboratory frame was longer than the moving frame τ 0 . The experimental error was 0.2%, with 95% confidence. They noted that the muons experienced a tangential acceleration of 10¹⁸ g. They viewed their results as in accordance with SR.

They described the lifetime of muons (both positive and negative) by:

“… If the muon sample has a velocity v then the lifetime of the sample as measured in the laboratory is given by

τ = τ 0 / [ ( 1 − ( v / c ) 2 ) ½ ] = γ τ 0 ,

where τ₀ is lifetime for the particle at rest.”

Since moving particles do not include a clock instrument, the experimental understanding is that muons experience their lifetime as though they are at rest. Thus, the moving frame time τ₀ has a lower numerical value than the laboratory measurement time of τ. This equates to time dilation by the moving body frame, and their equation is also similar to Eq. (3). However, they used the laboratory as the fixed frame, not the ECF.

If moving/laboratory notation is added to their equation, the agreement with the HK calculation is easier to see.

The experiment broadly confirmed time dilation by moving particles in a laboratory and confirmed the square-root function at high speeds.

The extreme tangential acceleration of 10¹⁸ g is higher than the gravity acceleration at a black hole event horizon. The accurate adherence to Eq. (11) appeared to be only from relative motion in a laboratory. Extreme acceleration had no measurable effect.

The experiments so far have established relativistic time dilation to both an ECF and a laboratory, and a relativistic wavelength red shift in all directions to a laboratory.

The next question becomes, can relativistic doppler shifts in frequency and length be combined with time and length dilation for a consistent view?

2.6 Consistency in relativistic doppler shifting

Relativistic Doppler shifting can be analyzed by isolating the relativistic effect from linear Doppler shifting. This is done by illustrating transverse relativistic Doppler shifting in Figure 3.

Figure 3

Idealized transverse Doppler effect for the source moving (a) and the observer moving (b).

In Figure 3(a), an emitter (source) moves in a perfect circle relative to a detector (observer) at the circle’s center. The emitter creates a beam wavelength based on its relativistic length and time dilation. The detector is positioned in a laboratory frame. Similarly, Figure 3(b) flips the position of the emitter and detector, and the detector moves in a circle around the emitter in the laboratory frame. The detectors are colored to indicate a red shift or a blue shift wavelength measurement.

Since only a relative velocity was used in Doppler shifting experiments, this analysis only uses a relative velocity, not an ECF velocity.

Again, the question is whether both time and length dilation predict the measured relativistic effects. The question is answered by adding time and frequency scales to Figures 1 and 2, resulting in Figures 4 and 5.

Figure 4

Relativistic Doppler shifting (source: moving body frame).

Figure 5

Relativistic Doppler shifting (source: laboratory).

To draw the additional scales for time and frequency, both the source and observer measure the speed of light according to

where c is the speed of light, λ _O, λ _S and f _O, f _S are the wavelength and frequency as measured by the observing and source frames. Eq. (12) is correct regardless of which frame is the source.

The Figure 4 time and length scales are aligned at zero on the left, and infinite frequency is also aligned to the left. The equations on the left indicate how the scales are illustrated. In this case, the illustration assumes an observed laboratory wavelength λ _O and a known dilation factor D _v to create the illustrated scales.

To simplify the illustration, an example of a 3 m long radio beam is created by a moving emitter source, and the moving frame scales use v / c = 0.6, resulting in a dilating factor D _v of

The scales of the moving frame dilate (i.e. become longer) equally for time and length. This results in scales that are numerically smaller than the fixed laboratory scales, according to Eq. (3).

Similarly, Figure 5 shows the results when the laboratory is the light source. The scales are the same as Figure 4.

These two figures are based on the understanding that the light beam does not change in length or frequency after it is emitted. It is the emitter and detector that experience relativistic effects, as mentioned by Cranshaw et al. [14].

An inspection of Figures 4 and 5 results in Table 1.

Table 1 agrees with current and well-known textbook equations [15,16] for relativistic doppler shifting and are posted on numerous web pages.

To summarize, Figures 4 and 5 and Table 1 all agree with the concept of time and length dilation experienced by the moving body relative to the fixed laboratory frame. The moving body time Δt _m and length ΔL _m are calculated according to

where ΔL _f and Δt _f are the fixed laboratory frame length and time, and v is the moving body velocity. The two figures provide a compelling case that time and length dilation in all directions correctly describes transverse relativistic doppler shifting.

2.7 The question of an ECF or laboratory frame

However, another question arises. Both HK and GPS used relative velocities to the ECF. Laboratory experiments used a moving body velocity relative to the laboratory frame. Why is there a difference? Is there a way to reconcile the two?

The answer is simple. Moving body velocities relative to a laboratory frame are an astonishing approximation of moving body velocities relative to the ECF. A moving body in a laboratory only appears to move relativistically to the laboratory frame.

HK’s experiment and GPS operating experience make it clear that all laboratory clocks move relativistically to the ECF. And HK’s airplanes used a combination of ground speed plus Earth’s rotation to determine the total plane velocity to the ECF. The HK and GPS results are the guiding experiments because their atomic clocks have higher resolution.

By using their calculating method, laboratory velocities are mathematically cleared up by adding Earth’s rotational velocity to both the laboratory and the moving body. Then, the Eq. (3) method applies instead of a relative velocity calculation.

Surprisingly, there is only a tiny difference between an ECF velocity calculation and a relative velocity calculation. Figures 6–8 show this.

Figure 6

Time dilation comparison for linear motion East/West.

Figure 7

Time dilation comparison for North/South motion.

Figure 8

Comparison of time dilation for circular motion.

To create the figures, the mathematical comparison of relative motion vs ECF motion will be derived. Both linear and rotational motion in a laboratory will be addressed.

2.8 Theory comparison – linear motion east/west

Figure 6 compares moving body time dilation by using a relative laboratory velocity v m (dashed line) and a moving body velocity v T relative to the ECF (solid line).

At higher velocities, the solid and dashed lines overlap tightly. The dashed-dotted line shows the difference between the two theories.

In Figure 6, the time ratio for a moving body time according to an ECF total velocity v _T is

where Δ t m is the moving body time, Δ t f is the ECF time, v _Lat is the laboratory latitude velocity of Earth’s rotation, v _m is the moving body velocity relative to the laboratory, v _T = v _Lat + v _m, β _Lat = v _Lat/c, and β _m = v _m/c.

And the ratio of laboratory time Δt _Lab to the ECF time Δt _f is

So the moving body time Δt _m ratio to the laboratory time Δt _Lab is the ratio of Eqs. (15) and (16):

Eq. (17) is the solid line in Figure 6 as a fractional dilation by subtracting from 1.

The time ratio for a moving body time Δt _m to the laboratory time Δt _Lab according to the relative velocity β _m is

Eq. (18) is the dashed line in Figure 6 by subtracting from 1. The dashed line represents SR.

The dashed-dotted line is the difference between Eqs. (17) and (18).

Figure 6 also represents length dilation as it is equal to time dilation.

North/south motion is next.

2.9 Theory comparison – linear motion north/south

Similarly, a time comparison for north/south longitudinal motion is shown in Figure 7. The two theories tightly overlap for all velocities. Figure 7 begins when the theory difference is above 10⁻²⁴.

In this case, the total instant velocity is the combination of the Earth’s rotation velocity v _Lat at a laboratory latitude. The moving body has a north/south ground velocity v _m and the total velocity v _T comes from a right triangle sum of the squares:

The ratio of the moving time Δt _m to the ECF time Δt _f is

and using the ratio Δ t _Lab/ Δ t _f of Eq. (16) gives

where β _Lat = v _Lat/c and β _m = v _m/c.

Figure 7 compares Eqs. (18), (21), and their difference. In this case, the agreement between the two theories is higher.

Figure 7 applies to laboratory velocities in the North/South direction and points out the difficulty in seeing Earth’s rotational velocity in a laboratory experiment.

However, an experiment outside of a laboratory, such as a round trip to/from an origin airport to a destination airport, is a different calculation. In this case, the origin clock has a fixed velocity, but the moving body has a variable velocity due to Earth’s variable rotation.

Rotating motion is next.

2.10 Theory comparison – rotating motion

A rotating laboratory velocity is a different mathematical study. The instant velocity is not in a single direction and combines differently with Earth’s rotation.

Figure 8 shows a comparison of the rotating Earth-based velocity v _T and a rotating relative velocity v _m. Similarly, Figure 8 begins when the theory difference is above 10⁻²⁴.

As you can see, Figure 8 is the same as Figure 7. Though the derivation is different, the mathematical result is the same as Eq. (21). Details are in the supplemental material.

Again, length dilation is equal to time dilation, so Figure 8 represents both.

Figures 6–8 are a compelling case that the ECF explains relative velocity laboratory experiments.

The main point of Figures 6–8 is the astonishing agreement between a relative laboratory frame and the ECF. Above the difference line, low-resolution experiments cannot differentiate between the two theories. High-resolution experiments below the dotted line see the difference if designed for it.

Figures 6–8 explain why SR remains a widely accepted theory.

Another question arises from the experimental survey. The cited experiments acknowledge or affirm agreement with SR and often cite the Lorentz factor. However, experimenter math used Eq. (14), which positions the square root function differently. Why would the experiments state support for SR?

2.11 Experimenter comments supporting SR

The answer to this question is uncomplicated. The SR Lorentz factor γ for time change Δ t is written as

where the subscripts m and f refer to the moving and fixed frames, respectively. Because there is “no preferred reference frame,” SR permits an equal interpretation of Eq. (22) as either:

An experimenter can correctly assume that the Lorentz factor position is an option from two.

Surprisingly, all experiments used option (b) for both high and low resolutions. Therefore, option (b) was used in the comparisons of Figures 6–8, and the position of the square root function in Eq. (24).

2.12 Relativistic behavior is orderly

The cited experiments clarify that relativistic effects follow a normal sense of motion relative to the ECF. That is, a faster ECF velocity causes higher relativistic effects.

In a review of 31 experiments, no experiment contradicted this relativistic order. The experiments are listed in the supplemental material.

To provide clarity of what was measured and not measured in the relativistic order, important results of the experiments are:

1. It was always the case that the moving body in a laboratory experiment was the moving frame, and the laboratory was the fixed frame.

2. It was always the case that experimenters in a laboratory experiment calculated relativistic time and length according to Eq. 23(b). It was never the case the experimenters used Eq. 23(a), even if the Lorentz transformation equations or the Lorentz factor was mentioned.

3. It was always the case that the moving body in a laboratory experiment emitted and measured red shift/blue shift according to Table 1.

4. Atomic clocks in the GPS system always experienced time dilation by their velocity to the ECF, whether the clock was in a satellite or in a ground laboratory.

5. It was always the case that similar and later experiments measured the same effects and used the same moving and fixed frames.

6. It was always the case that Earth’s gravity field was an explanation for relativistic effects, even if not measured or considered.

This well-ordered behavior also answers many perplexing questions inherent to a relative velocity theory. These include the twin paradox and the speed at which high-speed particles move past each other. Because a causal frame is clearly identified, there is no twin paradox, and high-speed particles may move past each other up to twice the speed of light. They are only limited by c relative to the ECF.

This well-ordered and predictable behavior supports a view that a common celestial frame for nearby moving bodies is universal.