# A Heuristic for Interpreting Lorentz Contraction as a Quantum Mechanical Effect

Alex de Visscher

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Alex de Visscher. A Heuristic for Interpreting Lorentz Contraction as a Quantum Mechanical Effect. 2023. [https://hal.science/hal-04200191](https://hal.science/hal-04200191)

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# A Heuristic for Interpreting Lorentz Contraction as a Quantum Mechanical Effect

Alex De Visscher

Department of Chemical and Materials Engineering, Concordia University, 1455 de Maisonneuve Blvd.
W., EV 3.159, Montreal, QC, Canada, H3G 1M8

## Abstract

This paper develops a heuristic that aims to provide an intuitive understanding of Lorentz contraction in terms of molecular forces at the quantum scale. To achieve this, Lorentz' version of special relativity is revisited by applying it to Einstein's thought experiments that led to special relativity. After derivation of the relativistic mass, the Lorentz contraction and time dilation, the relativity principle is used to deduce relativistic effects on the gravitational and electrical forces. Applying the resulting force constants to the calculation of the Bohr radius, it is shown that Lorentz contraction at the molecular scale in the direction of motion but not perpendicular to motion is a natural consequence of relativistic effects. It is concluded that Lorentz contraction can be interpreted as a quantum mechanical effect that must be applicable to all materials.

## 1. Introduction

Einstein's special relativity theory has been very successful for over a century. Starting from the observation that the two-way speed of light is always *c*, regardless of the (inertial) frame of reference, Einstein postulated that the one-way speed of light is also *c*, regardless of the frame of reference, and he reasoned through the consequences, wherever they led to. Based on this framework, Einstein was able to derive the Lorentz transformation, with its Lorentz contraction and time dilation (Einstein, 1905a). Special relativity also provided a law of addition of velocities that was consistent with the experiments of Fizeau on the velocity of light in a moving liquid. In a follow-up paper, Einstein derived the iconic $E = mc^2$ equation (Einstein, 1905b).

Philosophically, one of the main consequences of special relativity is that it is not necessary to assume the existence of an aether in order to describe electromagnetic phenomena. The aether, in other words, is a metaphysical concept and so is the concept of an absolute frame of reference, i.e., a coordinate system in absolute rest.

Einstein's special relativity replaced Lorentz' theory of electrodynamics. Lorentz assumed that the aether was real and that length contraction and time dilation explain the constancy of the speed of light. In other words, the constancy of light is an observational property of light in Lorentz' theory, whereas it is a fundamental property of light in Einstein's theory. It has been argued that the two theories are observationally equivalent (Zahar, 1973a,b). Zahar also argued that the success of general relativity,
rather than the merits of Einstein's version of special relativity, was the main reason why special relativity was generally accepted.

The constancy of the one-way velocity of light is currently accepted as fact. In Einstein's original paper, it was established 'by definition' (Einstein, 1905a), i.e., by convention. This suggests that other conventions could have been made. Paul Dirac (1951), for instance, believed that the aether exists.

There can be merit in maintaining two observationally equivalent theories of the same phenomena. An example can be found in the field of chemical thermodynamics. The study of liquid systems containing more than one compound can be described by liquid-mixture thermodynamics, and by solution thermodynamics. In liquid-mixture theory, none of the compounds is treated differently from the others. Each of the compounds has a reference state, which is the state of the pure compound as a liquid phase. This choice is convenient in most cases, because many compounds have been studied in their pure liquid state. However, this poses problems when the compound does not exist in the liquid state. Solution theory, on the other hand, singles out one compound, typically water, as the solvent, which is treated differently from the other compounds, the solutes. The solvent is treated the same way as a compound in a liquid mixture. For the solutes, the reference state is the compound dissolved in the solvent at infinite dilution. This is more practical when the compound does not exist as a pure liquid. On the downside, the infinite dilution state is generally less studied. Furthermore, the theory breaks down when the solvent is removed entirely from the solution. Two competing theories of the thermodynamics of the liquid phase allow the study of a broader range of systems than would have been possible with either of the two theories separately. In special relativity, there is merit in maintaining both the Lorentzian and the Einsteinian approach. The advantage of the latter is obvious: it has unlocked powerful mathematics to solve a broad range of problems. In this work, it will be argued that there are heuristic benefits of maintaining the Lorentzian approach by considering it in a quantum mechanical context.

The main weakness of the Lorentzian approach in the early development of relativity is that length contraction and time dilation were ad hoc fixes that seemed to have no physical basis. There was no known mechanism that could explain why all materials contracted in exactly the same way as a result of increasing the velocity, regardless of the properties of the material. This allowed Einstein to put forward his version of relativity, which showed that the Lorentz contraction is inescapable when the speed of light is a constant. As a heuristic, Einstein's relativity is more satisfactory than Lorentz' version. However, Einstein's relativity does not address the question of what mechanism causes length contraction. Instead, the problem is avoided altogether.

The purpose of this paper is to establish a new heuristic that explains Lorentz contraction as a quantum mechanical effect intimately linked with relativistic mass. To that effect, a version of special relativity is outlined that is consistent with the existence of an absolute frame of reference. The logic will be similar to Einstein's relativistic thought experiments, but from a point of view similar to Lorentz' version of relativity. Once the repercussions of this approach are discussed in detail, the Bohr radius from quantum mechanics is used to derive Lorentz contraction from purely nonrelativistic quantum mechanics, using only relativistic mass and a relativistic permittivity of space, hence establishing that length contraction can be seen as a universal quantum mechanical effect. This contribution removes the main weakness in Lorentz' version of relativity, as length contraction now has a physical basis. To be clear, this contribution does not aim to invalidate Einstein's theory but rather to complement it with an observationally equivalent alternative.
## 2. Hypothesis and Assumptions

The general approach to the theory development in this study is to follow a heuristic similar to the one used by Einstein (1905a; 1961) in the development of the special theory of relativity, but with a different set of assumptions. The development will rely on the same observational facts underlying Einstein's special relativity. The hypothesis underlying this work is that a theory in line with Hendrik Lorentz' approach to relativity can be developed that is observationally indistinguishable from Einstein's special relativity.

The assumptions or conventions underlying the current work are the following:

* A fixed (absolute) frame of reference can be defined. A possible candidate for this frame of reference is the frame of reference where the cosmic microwave background is isotropic on average, with no relative blueshift or redshift in any direction on average. Observationally, it does not matter which frame of reference is selected. Any inertial frame of reference can be chosen by convention.

* The speed of light, $c$, is constant in the absolute frame of reference. It follows Galilean rules in other frames of reference but time and space measurements in moving frames of reference are such that the two-way (i.e., round-trip average) velocity of light appears to be $c$ in all frames of reference.

* Lengths of moving objects are measured in the same way as measured in Einstein's heuristic.

* Lorentz contraction is a real (physical) effect in objects that move relative to the absolute frame of reference. Time dilation is an observational effect of moving objects. Absolute time is defined as time measured in an absolute frame of reference.

* Clocks that move at the same speed in absolute space, in each other's vicinity, once synchronized, stay synchronized, regardless of their orientation and the mechanism used to measure time. In other words, relativistic effects are the same for all clocks that move together.

* The rest mass, $m_0$, of an object is its mass at rest in an absolute frame of reference. Relativistic mass, $m$, is a real mass increase in moving objects following the equation $E = mc^2$.

* Inertia is represented by the equation $F = d(mv)/dt$, i.e., force equals momentum change per unit time.

* To determine the perceived properties of a moving object and its interactions with the surroundings, we need to consider a measurement device that travels with the object. The measurement is affected by the way the measurement device is affected by the movement.

## 3. Theory

### 3.1. *Derivation of Inertial Mass*

This derivation is not new, but it is included for the sake of completeness.

Consider an object with mass $m_0$ at rest ($v = 0$) in an absolute frame of reference. The mass is pushed forward with a force $F$, causing it to move. The evolution of the velocity of the object is:
$$ \frac{d(mv)}{dt} = F \qquad (1) $$

We only consider the force in the direction of the velocity. The energy change of the object is the work applied to the object, i.e., force times displacement:

$$ dE = Fds \qquad (2) $$

where s is the distance traveled by the object. The distance traveled is velocity times time:

$$ ds = vdt \qquad (3) $$

Combining eqs. (2) and (3) and solving for F leads to:

$$ F = \frac{dE}{vdt} \qquad (4) $$

Substitution into eq. (1) leads to:

$$ \frac{d(mv)}{dt} = \frac{dE}{vdt} \qquad (5) $$

We can eliminate the factor dt and expand the differential on the left-hand side:

$$ mdv + vdm = \frac{dE}{v} \qquad (6) $$

Now we introduce the Einstein equation, $E = mc^2$. In differential form, this becomes:

$$ dE = c^2 dm \qquad (7) $$

Substitution into eq. (6) leads to:
$$mdv + vdm = \frac{c^2 dm}{v} \qquad (8)$$

Rearranging leads to:

$$mdv = \frac{c^2 - v^2}{v} dm \qquad (9)$$

Separation of variables leads to:

$$\frac{v}{c^2 - v^2} dv = \frac{1}{m} dm \qquad (10)$$

Integration from velocity 0 to v and from mass $m_0$ to m leads to:

$$-\frac{1}{2} \ln \frac{c^2 - v^2}{c^2} = \ln \frac{m}{m_0} \qquad (11)$$

which can also be written as:

$$\ln \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \ln \frac{m}{m_0} \qquad (12)$$

Which leads to the relativistic mass equation:

$$m = \gamma m_0 \qquad \text{where} \qquad \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \qquad (13)$$

In this equation, $\gamma$ is known as the Lorentz factor. The expansion of the momentum change in eq. (6) can be interpreted as having two origins: the momentum increase due to velocity increase, and the momentum
increase due to mass increase. To explain why the latter contributes to the force needed to move the object, the newly created mass can be envisioned as being drawn from a source that is at rest.

It is useful to consider the apparent Newtonian inertial mass of a moving object subjected to a force. To that effect, we consider the following equation derived from eq. (1):

$$Fdt = mdv + vdm \quad (14)$$

If the force is perpendicular to the movement, then the motion is circular and the value of the velocity, and hence the mass, does not change. In that case, eq. (14) reduces to the following case:

$$\overline{F}dt = m\overline{dv} = \gamma m_0 \overline{dv} \quad (15)$$

where $\overline{F}$ and $\overline{v}$ refer to the force vector and the velocity vector, respectively. The inertial mass is $\gamma m_0$ (transverse Newtonian inertial mass). Vectors were introduced in eq. (15) because the absolute value of the velocity does not change in that situation.

If the force is in the direction of the movement, we need to consider both terms in eq. (14). We obtain:

$$Fdt = \gamma m_0 dv + vm_0 d\gamma \quad (16)$$

Substituting $\gamma$ leads to:

$$Fdt = \gamma m_0 dv + vm_0 d\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad (17)$$

Solving the differential leads to:

$$Fdt = \gamma m_0 dv + vm_0 \left( -\frac{1}{2}\gamma^3 \right) \left( \frac{-2v}{c^2} \right) dv = \gamma m_0 dv \left( 1 + \gamma^2 \frac{v^2}{c^2} \right) \quad (18)$$

Writing the Lorentz factor in full leads to:
$$Fdt = \gamma m_0 dv \left( 1 + \frac{v^2/c^2}{1 - \frac{v^2}{c^2}} \right) = \gamma m_0 dv \frac{1 - v^2/c^2 + v^2/c^2}{1 - \frac{v^2}{c^2}} = \gamma^3 m_0 dv \quad (19)$$

When the force acts in the direction of the velocity, the longitudinal apparent inertial mass (i.e., from a Newtonian perspective) is the cube of the Lorentz factor, multiplied by the rest mass.

## 3.2. Einstein's Simultaneity in a Lorentzian Framework – Lorentz Contraction, Time Dilation

A train and a platform both have a length $L$ at rest. As the train drives past the platform with velocity $v$, an observer on the train and an observer on the platform conduct length measurements of the platform and the train. In doing so, they use Einstein's concept of simultaneity. This is a key argument in Einstein's seminal paper on special relativity as retold in his general-audience book (Einstein, 1905; 1961). I will follow a similar procedure as laid out by Einstein, but with the assumption that the platform is at rest in an absolute frame of reference. Both observers believe that they are at rest in an absolute frame of reference, and they interpret their observations accordingly. For the theory to be observationally equivalent to Einstein's special relativity, a number of conditions must be met. First, both must observe that the two-way average speed of light is $c$ in their frame of reference. Second, if the observer on the platform observes the train to have a length $L'$ during motion, then the observer on the train must observe the platform to have a length $L'$ as well. This seems contradictory. We know from Einstein that $L' < L$, so, for the theory to work, the observer on the train must observe the platform to be shorter than the train when it is actually longer.

For an approach compatible with Einstein, I will adopt the following approach to measuring the length of a moving object. The observer is located at the middle of the train or the platform (i.e., at distance $L/2$ from both ends) and simultaneously sends a light signal to the edges of the domain (platform or train). The light signals actuate cameras so that photos taken with the cameras can be used to verify the length of the train in reference to the platform and vice versa, by observing the alignment. The flashes of the cameras arrive back at the observer simultaneously. Both observers believe that their own cameras fired simultaneously but they disagree about the other observer's cameras.

First, I will analyze the observations of the person on the platform. The light signals genuinely travel at velocity $c$ in this frame of reference. Choosing the observer as the origin of the frame of reference, the train's nose is at $x = L'/2$ whereas the train's tail is at $x = -L'/2$ when the two photos are taken. Assuming that this occurs at time $t = 0$, the observer must send the light signals at time $t = -L'/(2c)$. The flashes of the cameras return to the observer at time $L'/(2c)$, as expected. This observer sees the train at its real physical size, $L'$. According to this observer, the train has contracted by a factor $L'/L$.

To facilitate the discussion of the observer on the train, I will assume that there are markings on the platform and the space in front of it that will enable a length determination relative to the size of the train. Assume that the observer on the train sends the signal at time 0. The backwards light signal moves at a velocity $c + v$ relative to the train (and not $c$, which is what Einstein assumes) and reaches the back of the train at time $L'/(2(c + v))$. At this time, the back of the train is located at $x = -L'/2 + v \cdot L'/(2(c + v))$. The
light signal returns with a velocity $c - v$ relative to the train and arrives back at the observer at time $L'/((2(c+v)) + L'((2(c-v)))$.

At time 0, the observer on the train also sends a light signal to the front of the train, with a velocity $c - v$ relative to the train, which arrives at time $L'((2(c-v)))$. At this time, the front of the train is located at $L'/2 + v \cdot L'((2(c-v)))$. The light signal returns with a velocity $c + v$ relative to the train and arrives back at the observer at time $L'((2(c+v)) + L'((2(c-v)))$.

The two photos mark the locations $-L'/2 + v \cdot L'((2(c+v)))$ and $L'/2 + v \cdot L'((2(c-v)))$ on the platform, marking off the following apparent train length:

$$L_{\text{apparent}} = \frac{L'}{2} + v \cdot \frac{L'}{2(c-v)} - \left( -\frac{L'}{2} + v \cdot \frac{L'}{2(c+v)} \right) = L' + L'v \cdot \left( \frac{1}{2(c-v)} - \frac{1}{2(c+v)} \right) \quad (20)$$

All terms are written as a single fraction:

$$L_{\text{apparent}} = L' \cdot \frac{2(c-v)(c+v) + v \cdot (c+v) - v \cdot (c-v)}{2(c-v)(c+v)} \quad (21)$$

Expanding the terms between brackets leads to:

$$L_{\text{apparent}} = L' \cdot \frac{2c^2 - 2v^2 + vc + v^2 - vc + v^2}{2(c^2 - v^2)} \quad (22)$$

which simplifies to:

$$L_{\text{apparent}} = L' \cdot \frac{2c^2}{2(c^2 - v^2)} = L' \cdot \frac{c^2}{c^2 - v^2} = \frac{L'}{1 - \frac{v^2}{c^2}} \quad (23)$$

Interestingly, we find:

$$L_{\text{apparent}} = \gamma^2 L' \quad (24)$$
The observer on the train is of the impression that the train remained its length L whereas the platform has contracted so that a length L marks off a contracted length $γ^2L'$. The observer on the train reads the markings as $γ^2L'/L$ times the length of the train, and concludes that the platform has contracted by a factor $L/(γ^2L')$. Because of the principle of relativity, this factor must be equal to the factor seen by the observer on the platform, $L'/L$. Hence, relativity requires that the following relationship holds:

$$ \frac{L}{\gamma^2 L'} = \frac{L'}{L} \quad \Rightarrow \quad L = \gamma L' \qquad (25) $$

Under these circumstances, both observers are of the impression that their own domain (platform or train) remains at a length L, whereas the other domain contracts to a length $L' = L/γ$. This is the same conclusion as Einstein's in his special relativity. So far, we only considered the longitudinal effects (i.e., in the direction of the movement). The transverse effect will be discussed in the next section.

Both observers must observe a velocity of light equal to c for relativity to hold. This allows us to calculate time dilation on the moving train. For the signal sent from the platform, the travel time is $L/c$. The total travel time of the signal sent from the train is: $L'/(2(c+v)) + L'/(2(c-v))$

$$ t_{\text{signal}} = L' \left( \frac{1}{2(c+v)} + \frac{1}{2(c-v)} \right) = L' \frac{c-v+c+v}{2(c^2-v^2)} = L' \frac{c}{c^2-v^2} \qquad (26) $$

We find:

$$ t_{\text{signal}} = \frac{\gamma^2 L'}{c} = \frac{\gamma L}{c} \qquad (27) $$

For the apparent velocity of light to be c for the observer on the train, they must experience an elapsed time of $L/c$. Hence, clocks on the train must run at a rate $1/γ$ that of a stationary clock. This is consistent with time dilation in relativity.

It follows that time dilation, length contraction, and an apparently constant speed of light are consistent with an interpretation of an absolute space and absolute time. Time dilation and length contraction only occur physically in moving objects in this interpretation. The length contraction of an object at rest, observed by a moving observer, on the other hand, is an optical illusion.
### 3.3. Measurement Devices in Motion – Gravitational Constant, Permittivity of Vacuum.

In this section, I will consider the behavior of measurement rods, light clocks, mechanical oscillators, and pendulums at rest and in motion. All these devices, when synchronized, must remain synchronized when they are at rest relative to each other, regardless of whether they are all at absolute rest, or whether they are moving collectively in absolute space.

#### 3.3.1. Light clock

Einstein's main time measurement thought experiment is the light clock: light reflects back and forth between two mirrors. When the clock is at rest and the distance between two mirrors is $L$, then the time for one "tick" of the clock (light going back and forth once) is:

$$t_{\text{tick}} = \frac{2L}{c} \qquad (28)$$

We have established that objects in motion contract by a factor $1/\gamma$ in the direction of the movement. Hence, when the mirrors of a moving light clock are oriented perpendicular to the movement of the clock, the distance between the mirrors becomes $L' = L/\gamma$. The light alternates between moving with the clock with a relative velocity $c-v$ and moving against the clock with velocity $c+v$ so the time of a tick on the clock is:

$$t'_{\text{tick}} = \frac{1}{\gamma c - v} + \frac{1}{\gamma c + v} \qquad (29)$$

writing $\gamma$ in full and writing the fractions on a single numerator leads to:

$$t'_{\text{tick}} = \sqrt{1 - \frac{v^2}{c^2} L} \frac{c+v+c-v}{c^2-v^2} = \sqrt{1 - \frac{v^2}{c^2} L} \frac{2c}{c^2-v^2} \qquad (30)$$

After rearranging, we find:
$$t'_{\text{tick}} = \sqrt{1 - \frac{v^2}{c^2}} \frac{2L}{c} \frac{1}{1 - \frac{v^2}{c^2}} = \frac{2L}{c} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad (31)$$

Which leads to the time dilation equation we already know:

$$t'_{\text{tick}} = \frac{2L}{c} \gamma = \gamma t_{\text{tick}} \quad (32)$$

One tick of the clock takes longer so for the observer moving with the clock, it seems like less time has passed. The time of a tick must be independent of orientation of the clock, so the same result must be obtained when the light moves transversally between two mirrors that are parallel to the movement of the clock. The distance of the light travelled in half a tick is given by Pythagoras' theorem:

$$\left(c \cdot \frac{t'_{\text{tick}}}{2}\right)^2 = L'^2 + \left(v \cdot \frac{t'_{\text{tick}}}{2}\right)^2 \quad (33)$$

where L' is the as yet unknown distance between the mirrors. Solving for L' leads to:

$$L' = \frac{t'_{\text{tick}}}{2} \sqrt{c^2 - v^2} \quad (34)$$

where $t'_{\text{tick}}$ is given by eq. (31). Substitution leads to:

$$L' = \frac{L}{c} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \sqrt{c^2 - v^2} = L \quad (35)$$

We find that there is no transverse length contraction, i.e., no contraction in a direction perpendicular to the velocity. This is consistent with Einstein's special relativity.

The light clock can be used to calibrate the behavior of other clocks.
### 3.3.2. Mechanical Oscillator Clock

In a system at rest, a weight on a spring has a well-defined period of oscillation. This period can be used as the tick of a clock.

At rest, a weight on a spring has a period $T$ given by:

$$T = 2\pi\sqrt{\frac{m_0}{k}} \qquad (36)$$

where $m_0$ is the rest mass of the weight and $k$ is the spring constant, i.e., the force exerted by the spring per unit of length of the spring's stretch.

When a spring in a moving system oscillates in a direction perpendicular to the direction of the motion (in the transverse direction), then the inertial mass of the weight is:

$$m = \gamma m_0 \qquad (37)$$

Assuming that the amplitude of the oscillation is negligibly small, the mass does not change during the oscillation. The period of the oscillator at motion with a velocity $v$ is given by:

$$T' = 2\pi\sqrt{\frac{\gamma m_0}{k'}} \qquad (38)$$

where $k'$ is the as yet unknown spring constant at motion.

Time dilation must apply the same way to all time measurements so $T'$ must be dilated by a factor $\gamma$ as per eq. (32):

$$\gamma T = 2\pi\sqrt{\frac{\gamma m_0}{k'}} \qquad (39)$$

Comparison with eq. (36) leads to:

$$k' = \frac{k}{\gamma} \qquad (40)$$
The spring is looser (i.e., stretches with less force) while in motion. This is in seeming contradiction with relativity because it suggests that motion at relativistic speeds would cause springs to stretch more in a gravitational field in comparison with a mass on a string in a gravitational field at rest. This will be explored later on.

Because the motion of the oscillator clock is perpendicular to the oscillation, the spring is not contracted along its length but it is contracted along its width. The cross-sectional area of the spring material in the direction perpendicular to the oscillation is contracted by a factor $1/\gamma$. It follows that the stress in the string remains the same regardless of whether the system is at rest or in motion. This will be explored further later on.

I will now analyze the oscillation of the weight on the spring in the direction of the motion of the oscillator clock (longitudinal effect).

It was established in Section 3.1 that the inertial mass of a weight in the direction of the motion is:

$$m = \gamma^3 m_0 \qquad (41)$$

Hence, the period of the oscillator becomes:

$$T' = 2\pi\sqrt{\frac{\gamma^3 m_0}{k'}} \qquad (42)$$

Again, eq. (32) must apply, so:

$$\gamma T = 2\pi\sqrt{\frac{\gamma^3 m_0}{k'}} \qquad (43)$$

Comparison with eq. (36) leads to:

$$k' = \gamma k \qquad (44)$$

The spring is stiffer than a spring at rest. This is the expected result because the spring is shorter due to Lorentz contraction. We find that the same relative stretch (i.e., stretch/length in the absence of stress) generates the same force, regardless of whether the spring is at rest or in motion. Because the cross-sectional area perpendicular to the motion and the oscillation stays the same, it is found that the stresses also stay the same.
To reconcile the behavior of a mechanical oscillator clock with the principle of relativity, it is found that we need to assume that stresses (or pressures) remain the same in relativistic conditions whereas forces do not. This requires independent validation, which will be provided in Section 3.4.1.

### 3.3.3. Pendulum Clock

A pendulum is not strictly a harmonic oscillator, but it can be approximated as one when the amplitude of the oscillation is negligibly small. The period of the swing of a pendulum in a system at rest is given by:

$$T = 2\pi\sqrt{\frac{L}{g}} \qquad (45)$$

where L is the length of the pendulum and g is the acceleration due to gravity. The pendulum that is positioned perpendicular to the velocity of motion (transverse effect) is considered first. In this direction, the length is not affected by Lorentz contraction. Hence, the period of a moving pendulum is given by:

$$T' = 2\pi\sqrt{\frac{L}{g'}} \qquad (46)$$

As with the previous clocks, the period of the pendulum must follow eq. (32):

$$\gamma T = 2\pi\sqrt{\frac{L}{g'}} \qquad (47)$$

Comparison of eqs. (45) and (47) leads to:

$$g' = \frac{g}{\gamma^2} \qquad (48)$$

To be consistent with the principle of relativity, a Lorentzian version of relativity must consider an acceleration due to gravity that is less when the gravitating body is moving at relativistic speeds.
To obtain the gravitational force felt by an object in a gravitational field, g is multiplied by the inertial mass of the object. In a direction perpendicular to the velocity of the system, this inertial mass is $\gamma m_0$. Hence:

$$F' = mg' = \gamma m_0 \frac{g}{\gamma^2} = m_0 \frac{g}{\gamma} \qquad (49)$$

Hence:

$$F' = \frac{F}{\gamma} \qquad (50)$$

With regards to forces, the conclusion is the same as with a weight on a spring: the gravitational force must be less in a gravitational field moving at relativistic speeds. This would explain how a spring can be looser in a case of a moving system without showing additional stretch due to gravitation.

Next, the case of a pendulum oriented in the direction of motion (longitudinal effect) is considered. Now the length of the pendulum is contracted. The period of the moving pendulum is now given by:

$$T' = 2\pi\sqrt{\frac{L/\gamma}{g'}} \qquad (51)$$

Again, this must be consistent with eq. (32):

$$\gamma T = 2\pi\sqrt{\frac{L/\gamma}{g'}} \qquad (52)$$

Comparison with eq. (45) leads to:

$$g' = \frac{g}{\gamma^3} \qquad (53)$$

Gravitational acceleration is even weaker in the direction of the motion than in the direction perpendicular of the motion. However, this is not true with the force, which is obtained by multiplying g' by the inertial mass, $\gamma^3 m_0$:
$$F' = \gamma^3 m_0 \frac{g}{\gamma^3} = m_0 g \qquad (54)$$

Hence:

$$F' = F \qquad (55)$$

The gravitational force is not affected by a movement in the direction of the motion of the gravitating body. This was also found for forces in a spring in Section 3.3.2.

The appropriateness of these choices of the gravitational acceleration will be discussed in Section 3.4.2.

## 3.4. Forces at Relativistic Speeds

### 3.4.1. Ideal Gases

In Section 3.3.2, it was found that a Lorentzian view on relativity requires that stresses experienced by materials must not be affected by relativistic velocities, whereas forces can be affected, depending on the direction of the force. The fundamental reason for this will be investigated for the case of the simplest material possible: the ideal gas.

Ideal gases can be described by the kinetic gas theory. The gas is described by a collection of molecules that move around randomly in three dimensions. There is a distribution of velocities but for the purpose of calculating the pressure, it is sufficient to use the root mean square velocity in the calculations.

Assume a cubic box at absolute rest that contains $N$ gas molecules that move with velocity $u_x$ in the $x$ direction. Assume that the cubic box has a length $a$, and therefore a volume $a^3$.

The mean time between two consecutive collisions, $\Delta t$, on the same wall perpendicular to the $x$ direction, is given by:

$$\Delta t = \frac{2a}{u_x} \qquad (56)$$

The momentum change, $\Delta p_x$, of collision is:

$$\Delta p_x = 2m_0 u_x \qquad (57)$$
Unless the temperature of the gas is extremely high, the molecular velocity is nonrelativistic, typically on the order of one millionth of c.

The force exerted by the gas on the wall is the momentum change realized by the wall per unit time, for N molecules. This is given by:

$$F = N \frac{2m_0 u_x}{2a/u_x} = \frac{Nm_0 u_x^2}{a} \qquad (58)$$

The pressure exerted on the wall is the force divided by the wall area, $a^2$:

$$p = \frac{Nm_0 u_x^2}{a^3} = \frac{Nm_0 u_x^2}{V} \qquad (59)$$

For the sake of completeness, I add that the overall root mean square velocity, $u_{rms}$, of the gas molecules is given by its square, which is given by:

$$u_{rms}^2 = u_x^2 + u_y^2 + u_z^2 \qquad (60)$$

In the absence of any motion of the system or external forces, the gas is isotropic, i.e., $u_x = u_y = u_z$. Hence:

$$u_{rms}^2 = 3u_x^2 \qquad (61)$$

Next, the case of the system in motion at relativistic speed is considered. We start with the pressure on a surface that is parallel to the motion of the gas container. In this case we consider a molecule with a velocity component perpendicular to the motion of the gas container (transverse effect). For this paper, I will pursue a naïve approach of assuming that the motion of the gas molecules is negligibly slow in comparison with the motion of the container.

The container does not contract in the direction of the motion of the gas molecule to be considered. Hence, it remains a. The velocity of the molecule is an as yet unknown quantity, $u'_x$. The time between consecutive collisions with the same wall is:

$$\Delta t' = \frac{2a}{u'_x} \qquad (62)$$
Time dilation requires that the time between collisions follows eq. (32). Hence:

$$ \gamma\Delta t = \frac{2a}{u'_{x}} \qquad (63) $$

From comparison with eq. (56), it follows that the velocity $u'_{x}$ is given by:

$$ u'_{x} = \frac{u_{x}}{\gamma} \qquad (64) $$

The Newtonian inertial mass of the molecule in the direction perpendicular to the motion of the container is $\gamma m_0$. Hence, the momentum change upon collision is:

$$ \Delta p'_{x} = 2mu'_{x} = 2\gamma m_{0} \frac{u_{x}}{\gamma} = 2m_{0}u_{x} \qquad (65) $$

It follows that:

$$ \Delta p'_{x} = \Delta p_{x} \qquad (66) $$

The momentum change is independent of a velocity parallel to the surface.

To calculate the force exerted on the surface, the momentum change is multiplied by the number of molecules and divide by the time between consecutive collisions:

$$ F' = \frac{Nmu_{x}^{2}}{a} = \frac{N\gamma m_{0} u_{x}^{2}/\gamma^{2}}{a} = \frac{Nm_{0}u_{x}^{2}}{\gamma a} \qquad (67) $$

Hence:

$$ F' = \frac{F}{\gamma} \qquad (68) $$
The pressure is obtained by dividing the force by the area. The face of a cube is a square in the absence of motion. In motion, the square contracts in one direction, leading to an area of $a^2/\gamma$. Hence, the pressure is:

$$p' = \frac{Nm_0u_x^2}{\gamma a \cdot a^2/\gamma} = \frac{Nm_0u_x^2}{a^3} \qquad (69)$$

It follows that:

$$p' = p \qquad (70)$$

Forces perpendicular to the velocity of the system decline at relativistic speeds, but pressures (i.e., stresses) stay the same. This is consistent with the observations made with the mechanical oscillator clock.

Next, a system moving in the direction of the motion of the molecule, perpendicular to the surface, will be considered (longitudinal effect). In this case, the container contracts in the direction of the movement of the molecule in consideration, whereas the surface that the molecule collides with does not experience contraction. The distance between the walls in the direction of the movement of the molecule is $a/\gamma$, whereas the area of the wall of interest is $a^2$. The time between two consecutive collisions with the same wall is:

$$\Delta t' = \frac{2a/\gamma}{u_x'} \qquad (71)$$

where $u_x'$ is not necessarily the same as in eq. (62). Time dilation requires the time between collisions to follow eq. (32). Hence:

$$\gamma\Delta t = \frac{2a/\gamma}{u_x'} \qquad (72)$$

Comparing with eq. (56), this leads to:

$$u_x' = \frac{u_x}{\gamma^2} \qquad (73)$$
The inertial mass of the molecule in the direction of the velocity of the container is $\gamma^3 m_0$. The momentum change per collision is calculated as usual:

$$ \Delta p'_{x} = 2mu'_{x} = 2\gamma^3 m_0 \frac{u_x}{\gamma^2} = 2\gamma m_0 u_x \quad (74) $$

Hence:

$$ \Delta p'_{x} = \gamma \Delta p_{x} \quad (75) $$

The force acting on the wall is the momentum change of the collisions of N molecules per unit of average time between consecutive collisions:

$$ F' = \frac{Nmu_x'^2}{a/\gamma} = \frac{N\gamma^3 m_0 u_x^2 / \gamma^4}{a/\gamma} = \frac{Nm_0 u_x^2}{a} \quad (76) $$

Hence:

$$ F' = F \quad (77) $$

The pressure is the force per unit area:

$$ p' = \frac{Nm_0 u_x^2}{a \cdot a^2} = \frac{Nm_0 u_x^2}{a^3} \quad (78) $$

Hence:

$$ p' = p \quad (79) $$

Forces and pressures (stresses) in this direction are the same as in the absence of relativistic motion. This is consistent as what had to be assumed in the case of a mechanical oscillator clock.

We find that the requirements of the principle of relativity, when applied to an ideal gas, are the same as the ones required in a mechanical oscillation clock.
### 3.4.2. Gravitational Acceleration

The purpose of the current section is to calculate for the pendulum clock what was calculated for the mechanical oscillator clock in Section 3.4.1. In particular, I will calculate how the universal gravitational constant should evolve at relativistic speeds in order to preserve the principle of relativity. The outcome of this calculation will be compared with the findings in Section 3.3.3.

Imagine a person on a planet that is stationary in space, and a person on an identical planet that is moving in space at a velocity $v$. If the two persons measure the gravitational acceleration, $g$, they should get the same result. This will allow to calculate the real value of $g$ on the moving planet, and from the value of $g$, the value of the universal gravitational constant, $G$, can be calculated.

I will start with a person standing on the moving planet so that the line connecting the person with the center of the planet is perpendicular to the direction of the motion of the planet (transverse effect). A falling object from the perspective of the person moves in a direction where there is no length contraction. Hence, the apparent (measured) path length in the direction of the fall equals the actual path length, as well as the path length at rest:

$$x_{\text{meas}} = x' = x \qquad (80)$$

The measured time is expressed in number of ticks of a clock that occur during the process. The real time is the number of ticks of a stationary clock. The number of ticks of a stationary clock per unit time is:

$$n_{\text{ticks}} = \frac{t}{t_{\text{tick}}} \qquad (81)$$

For a process on the moving planet, the number of ticks is:

$$n'_{\text{ticks}} = \frac{t'}{t'_{\text{tick}}} \qquad (82)$$

Substituting eq. (32) leads to:

$$n'_{\text{ticks}} = \frac{t'}{\gamma t_{\text{tick}}} \qquad (83)$$
The gravitational acceleration, g, can be measured by measuring the time needed for an object to fall a certain distance x. The relationship is:

$$x = \frac{gt^2}{2} \qquad (84)$$

For the moving planet, the (real) relationship is as follows:

$$x' = \frac{g't'^2}{2} \qquad (85)$$

We write eqs. (84) and (85) in terms of the number of clock ticks, with eqs. (81) and (83):

$$x = \frac{g(n_{\text{ticks}} \cdot t_{\text{tick}})^2}{2} \qquad (86)$$

$$x' = \frac{g'(n'_{\text{ticks}} \cdot \gamma t_{\text{tick}})^2}{2} \qquad (87)$$

It was established that $x = x'$. In addition, the measured times, i.e., the number of ticks of the clock, must be the same for the measurements on both planets. Hence, setting eq. (86) equal to eq. (87):

$$\frac{g(n_{\text{ticks}} \cdot t_{\text{tick}})^2}{2} = \frac{g'(n'_{\text{ticks}} \cdot \gamma t_{\text{tick}})^2}{2} = \frac{g'(n_{\text{ticks}} \cdot \gamma t_{\text{tick}})^2}{2} \qquad (88)$$

This equation simplifies to:

$$g = g'\gamma^2 \qquad (89)$$

This is identical to eq. (48), showing that time dilation in a moving system explains the lower gravitational acceleration needed to synchronize a pendulum clock with other types of clocks.

Next, the gravitational constant consistent with eq. (89) is calculated. The relationship between the gravitational constant and the gravitational acceleration on a planet at rest is:
$$gm = \frac{GmM}{R^2} \qquad (90)$$

For the planet in motion, a choice needs to be made whether Newton's law of gravitation is used with rest masses or with relativistic masses. The former choice is made. Hence, in the direction perpendicular to the motion of a moving planet, eq. (90) becomes:

$$g'\gamma m = \frac{G'mM}{R^2} \qquad (91)$$

Substituting eq. (89) leads to:

$$\frac{gm}{\gamma} = \frac{G'mM}{R^2} \qquad (92)$$

Comparing eq. (90) to eq. (92) leads to:

$$G' = \frac{G}{\gamma} \qquad (93)$$

Next, gravitation in the direction of the motion of the planet is considered (longitudinal effect). The relationship between g and g' is calculated first. In the direction of the planet's motion, the trajectory of the falling object relative to the planet is contracted by a factor 1/γ. However, the measured trajectory is the same as the trajectory on a stationary planet because measurement rods are also contracted. Hence:

$$x_{\text{meas}} = \gamma x' = x \qquad (94)$$

Time dilation is the same as in the transversal case. Once again, the measurement equation is:

$$x' = \frac{g't'^2}{2} \qquad (85)$$

Now, eq. (94) is substituted, as well as eq. (83):
$$ \frac{x}{\gamma} = \frac{g'(n'_{\text{ticks}} \cdot \gamma t_{\text{tick}})^2}{2} \qquad (95) $$

Comparison with eq. (86) leads to:

$$ \frac{g(n'_{\text{ticks}} \cdot t_{\text{tick}})^2}{2\gamma} = \frac{g'(n'_{\text{ticks}} \cdot \gamma t_{\text{tick}})^2}{2} \qquad (96) $$

And because the number of ticks is the same in both cases, this equation can be reduced to:

$$ \frac{g}{\gamma^3} = g' \qquad (97) $$

which was also found with the pendulum clock.

Because of the Lorentz contraction, the gravitational acceleration is more strongly reduced by relativistic effects in the longitudinal case than in the transversal case.

Next, the gravitational constant is calculated from the gravitational acceleration. Now, Newton's law of gravitation is as follows:

$$ g' \gamma^3 m = \frac{G' m M}{R'^2} \qquad (98) $$

Because of Lorentz contraction of the planet, this leads to:

$$ g' \gamma^3 m = \frac{G' m M}{\left(\frac{R}{\gamma}\right)^2} \qquad (99) $$

Eq. (97) is used to substitute $g'$:

$$ \frac{g}{\gamma^3} \gamma^3 m = \frac{G' m M}{\left(\frac{R}{\gamma}\right)^2} \qquad (100) $$
Comparison with eq. (90) leads to:

$$G' = \frac{G}{\gamma^2} \qquad (101)$$

It is interesting to speculate about the origin of the factor $1/\gamma$ in eq. (93) and $1/\gamma^2$ in eq. (101). If we envision gravity as the exchange of gravitons traveling at the speed of light between two moving objects a distance $R$ apart, then the two-way average distance traveled by the gravitons is $\gamma^{1/2}R$ when the line connecting the two objects is perpendicular to the direction of the motion, and $\gamma R$ when the objects are aligned in the direction of the motion. Substitution of these distances into Newton's gravitation law leads to exactly the gravitational constants derived in this section.

### 3.4.3. Electrostatics

Coulomb's law is an inverse square law, just like Newton's law of gravitation. It follows that the effect of motion should be exactly the same as the effect of motion on gravity. Coulomb's law can be expressed as follows:

$$F = \frac{q_1 q_2}{4 \pi \varepsilon_0 R^2} \qquad (102)$$

where $q_1$ and $q_2$ are the charges of the two objects, and $\varepsilon_0$ is the permittivity of space. The latter is inversely proportional to the strength of the electric force so the effects of motion on $\varepsilon_0$ is the inverse of the effect on the gravitational constant $G$. Thus follows, for two charges that are aligned perpendicularly to the direction of the motion (transverse effect):

$$\varepsilon'_{0} = \gamma \varepsilon_{0} \qquad (103)$$

When the charges are aligned in the direction of the motion, then the relationship is (longitudinal effect):

$$\varepsilon'_{0} = \gamma^{2} \varepsilon_{0} \qquad (104)$$
The same comment can be made about these relationships as were made about the effect of motion on the gravitational constant. If it is assumed that the electric force is mediated by photons, then the two-way average path lengths of photons moving from one charge to the other are such that eqs. (103) and (104) are reproduced.

## 4. Quantum Mechanical Explanation of Lorentz Contraction

Now that we have relationships for the effect of relativistic velocities on some of the fundamental constants of nature, we can use these to investigate the physical basis of the phenomenon of Lorentz contraction. To that effect, we focus on the so-called Bohr radius, which is a measure of the size of the hydrogen atom. It can be assumed that all molecular sizes can ultimately be traced back to a multiple of the Bohr radius. Hence, if Lorentz contraction is real, it should be expressed in the effect of relativistic velocities on the Bohr radius. It is calculated as follows:

$$a_0 = \frac{4\pi\varepsilon_0\hbar^2}{e^2m_e} \qquad (105)$$

where $\hbar$ is the reduced Planck constant, $e$ is the charge of the electron, and $m_e$ is the mass of the electron. For the Bohr radius of an atom at relativistic speeds, this becomes:

$$a'_{0} = \frac{4\pi\varepsilon'_{0}\hbar'^{2}}{e^{2}m'_{e}} \qquad (106)$$

It can reasonably be assumed that charge is a conserved quantity. In fact, that assumption is implicit in the reasoning in Section 3.4.3. In the case of the Bohr radius perpendicular to the direction of the motion (transverse Bohr radius), the following equation is obtained:

$$a'_{0} = \frac{4\pi\gamma\varepsilon_{0}\hbar'^{2}}{e^{2}\gamma m_{e}} \qquad (107)$$

If we assume that the Planck constant is independent of motion, the following is obtained:

$$a'_{0} = a_{0} \qquad (108)$$
which is consistent with Lorentz contraction, which is non-existent in the directions perpendicular to the motion.

In the direction of the motion, the Bohr radius is as follows:

$$a'_{0} = \frac{4\pi\gamma^2\epsilon_0\hbar'^2}{e^2\gamma^3m_c} \qquad (109)$$

Again, we assume that the Planck constant is independent of velocity. Hence:

$$a'_{0} = \frac{a_{0}}{\gamma} \qquad (110)$$

Again, the Bohr radius correctly predicts Lorentz contraction.

The Bohr radius is derived from the Schrodinger equation. In fact, its derivation by Bohr even predates the Schrodinger equation, by assuming that electrons circling a nucleus must have an angular momentum that is a multiple of Planck's constant. It is important to realize that neither the derivation of Bohr, nor the derivation of the Schrodinger equation has any dependence on the theory of relativity. As such, eqs. (108) and (110) can be seen as independent predictions of Lorentz contraction. It is a prediction in the sense that the result is unexpected because nothing preceding Section 4 of this paper has any relationship with quantum mechanics.

The only objection that could be made against the notion of an independent prediction is that an assumption had to be made: the constancy of the Planck constant. However, this is not an *ad hoc* assumption. The Planck constant has the units of energy times time, and of momentum times distance. This is obvious from the Heisenberg uncertainty relations. It is worth checking these properties in Section 3.4.1 about ideal gases. Both in the direction of the motion of the system and in the direction perpendicular to the motion, both kinetic energy times time and momentum times distance are independent of the velocity of the motion, indicating that these products are conserved at a very fundamental level upon bringing a system to a relativistic velocity. If the Planck constant were dependent on the velocity of a system, then velocity would have a measurable effect on the quantum properties of the system, in violation of the relativity principle. Therefore, Planck's constant must be independent of velocity.

This result has an unexpected, appealing interpretation at a philosophical level. The result suggests that **Lorentz contraction is a quantum mechanical effect of relativistic velocities.** It provides an answer to the main criticism faced by Lorentz about his theory: it was felt by many that Lorentz contraction was an *ad hoc* assumption needed to make the theory work, whereas in Einstein's version of relativity, it followed naturally from the assumption of a constant one-way velocity of light, along with Einstein's definition of simultaneity. It can be argued that Lorentz would have had stronger support for his version of special relativity if quantum mechanics had been developed prior to the theory of relativity.
5. Conclusions

A heuristic was proposed that allows the interpretation of Lorentz contraction as a quantum mechanical
effect. First, it was shown that the relativistic mass of a moving object can be derived by revising
Newton’s second law to “force equals momentum change per unit time” and combining it with $E = mc^2$.

Next, Lorentz contraction was derived in an absolute frame of reference by requiring that the relativity
principle applies, i.e., neither observer has a way of determining absolute motion, only relative motion.
Both an observer at rest and a moving observer must see the same length contraction in the system of the
other observer, but in the case of the observer at rest, the contraction is real, whereas for the moving
observer it is an optical illusion. Time dilation was also derived in this argument.

Next, Lorentz contraction in an absolute frame of reference was applied to a light clock, leading to a
second derivation of time dilation. In this framework, time dilation is an apparent slowing down of the
phenomenon of light bouncing between two parallel mirrors.

By requiring that the same time dilation applies to a mechanical oscillator clock, it was shown that springs
become looser by a factor equal to the Lorentz factor ($\gamma$) in the direction perpendicular to the linear
motion of the clock, whereas it stiffens by the same fraction in the direction of the movement. In both
cases, the stresses in the material stay unaffected by the velocity.

By requiring that the same time dilation applies to a pendulum clock in a gravitational field, gravitational
acceleration was found to decrease by a factor $\gamma$ in the direction perpendicular to the motion of the clock
and by a factor $\gamma^3$ in the direction of the motion.

Forces and movements of molecules as point masses (kinetic gas theory) were calculated in a system
moving at relativistic speeds. The system showed pressures unaffected by the velocity of the system,
consistent with the observations in a mechanical oscillator.

Requiring that an observer on a planet moving at relativistic velocities experiences the same apparent
gravitational accelerations as on a planet at rest confirmed the gravitational acceleration effects deduced
from a pendulum clock. This led to the observation that the gravitational constant decreases by a factor $\gamma$
in a direction perpendicular to the motion of the planet, and by a factor $\gamma^2$ in the direction of the motion of
the planet. Likewise, because electrostatic effects should be subjected to the same effects, it was shown
that the permittivity of space increases by a factor $\gamma$ in the direction perpendicular to the motion of the
charges, and by a factor $\gamma^2$ in the direction of the motion. The effect is speculatively explained as the
result of a path length increase of the carriers of the force (gravitons and photons).

When these effects were incorporated into the calculation of the Bohr radius, it was found that the Bohr
radius predicts Lorentz contraction correctly when it is assumed that the Planck constant is not affected by
relativistic velocities. However, because location times momentum, and energy times time, are not
affected by relativistic velocities, it is expected that the Planck constant should not be affected by
relativistic velocities. It follows that Lorentz contraction can be interpreted as a quantum mechanical
effect operating at the atomic scale, that affects all materials independently of their properties.

The Lorentzian view of relativity is a theory that complements the Einsteinian view. They are the result of
a different choice of conventions. Neither set of conventions is provably right or wrong. Hence, they are
both valid approaches.
References

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Einstein, A. (1905b). Is die Trägheit eines Körpers von seinem Energiegehalt abhängig? *Ann. Phys.* **18**, 639-641.

Fizeau, H. (1851). The hypotheses relating to the luminous aether, and an experiment which appears to demonstrate that the motion of bodies alters the velocity with which light propagates itself in their interior. *Phil. Mag.* **4**(2), 568-573.

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