# Einstein’s Elevator: World Lines, Michelson–Morley Experiment and Relativistic Paradox

## Abstract

**:**

## 1. Introduction

## 2. Uniformly Accelerated Reference Frame

#### 2.1. Coordinate System

#### 2.2. Metric and Change of Coordinates

## 3. Rays of Light

#### 3.1. Lateral Rays

#### 3.2. Any Rays

#### 3.3. Horizontal Synchronization

#### 3.4. Radar Distance

## 4. Non-Zero Mass Particles

#### 4.1. Trajectories and Worldlines

#### 4.2. Velocity and Acceleration for a Particle Released at Rest

#### 4.3. Comparison with the Black Hole

## 5. Experimental Aspects

#### 5.1. Deviation Measurement

#### 5.2. Desynchronization of Clocks

#### 5.3. Michelson Interferometer

#### 5.4. Three Rockets Experiment

## 6. Discussion

## 7. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Einstein, A. A few inferences from the general theory of relativity, and Gaussian Co-ordinates. In Relativity: The Special and the General Theory; Henry Holt and Company: New York, NY, USA, 1920; Available online: https://www.bartleby.com/173/ (accessed on 5 August 2022).
- Desloge, E.A.; Philpott, R.J. Uniformly accelerated reference frames in special relativity. Am. J. Phys.
**1987**, 55, 252–261. [Google Scholar] [CrossRef] - Møller, C. The Theory of Relativity; Oxford University Press: London, UK, 1952; p. 253. Available online: https://archive.org/details/theoryofrelativi029229mbp/page/n7/mode/2up (accessed on 5 August 2022).
- Rindler, W. Relativity: Special, General, and Cosmological; Oxford University Press: Oxford, UK, 2006; Exercise 12.7; pp. 71, 282. [Google Scholar]
- Landau, L.D.; Lifchitz, E.M. The Classical Theory of Fields; Pergamon Press Ltd.: Oxford, UK, 1971; §§ 7, 84, 99. [Google Scholar]
- Desloge, E.A. Spatial geometry in a uniformly accelerating reference frame. Am. J. Phys.
**1989**, 57, 598–602. [Google Scholar] [CrossRef] - Semay, C. Observer with a constant proper acceleration. Eur. J. Phys.
**2006**, 27, 1157–1167. [Google Scholar] [CrossRef] - Rouaud, M. Special Relativity, A Geometric Approach; Lulu Press, Inc.: Durham, UK, 2020; pp. 157, 167, 243, 387. [Google Scholar]
- Born, M. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips [The theory of the rigid electron in the kinematics of the principle of relativity]. Ann. der Phys.
**1909**, 335, 1–56. [Google Scholar] [CrossRef] - Hamilton, J.D. The uniformly accelerated reference frame. Am. J. Phys.
**1978**, 46, 83–89. [Google Scholar] [CrossRef] - Spallicci, A.D.; Ritter, P. A fully relativistic radial fall. Int. J. Geom. Methods Mod. Phys.
**2014**, 11, 1450090. [Google Scholar] [CrossRef] - Desloge, E.A. Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field. Am. J. Phys.
**1989**, 57, 1121–1125. [Google Scholar] [CrossRef] - Muñoz, G.; Jones, P. The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field. Am. J. Phys.
**2010**, 78, 377–383. [Google Scholar] [CrossRef] - Blinnikov, S.I.; Okun’, L.B.; Vysotskiĭ, M.I. Critical velocities c/$\sqrt{3}$ and c/$\sqrt{2}$ in the general theory of relativity. Phys.-Usp.
**2003**, 46, 1099–1103. [Google Scholar] [CrossRef] - Tourrenc, P. Relativity and Gravitation; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Thorne, K.S.; Misner, C.W.; Wheeler, J.A. Gravitation; W.H. Freeman and Company: San Francisco, CA, USA, 1971; p. 848. [Google Scholar]
- Rucker, R.v.B. Geometry, Relativity and the Fourth Dimension; Dover Publications, Inc.: New York, NY, USA, 1977. [Google Scholar]

**Figure 1.**The thought experiment in the Einstein’s elevator with a ray of light. Here, $\overrightarrow{F}$, d, x, and y denote the force exerted, the width of the box, and the vertical and lateral directions, respectively.

**Figure 2.**Minkowski diagram of the rocket in the inertial reference frame ${R}^{\prime}$. The uniformly accelerated rocket has a hyperbolic worldline. In the reference frame $R\prime $, the coordinate lines of $R\prime $ and R are shown. In both cases, the coordinate lines of space and time are orthogonal. In this case, the rocket defines a rigid body [2,3,4], with respect to which one can study the trajectory of the light rays. See Section 2.2 for details.

**Figure 3.**The worldlines of 12 particular rays in a Minkowski diagram $({T}^{\prime},{X}^{\prime},{Y}^{\prime})$ with ${\theta}^{\prime}({t}^{\prime}=0)=\theta (t=0)$ as initial conditions.

**Figure 4.**The circular trajectory of the ray traced on a vertical wall of the elevator. The deviation is greater than that of Newton’s parabolic trajectory. At the level of the horizon the deviation is double.

**Figure 5.**In the $(X,Y)$ plane, the trajectories of non-vertical light rays are portions of circles centered on the horizon. The rays thus arrive perpendicularly to the line of centers which is identified with the horizon. The grayed straight lines correspond to the inertial trajectories, in case the rocket is not accelerated. The gray parabolic lines are the Newtonian trajectories. For $\theta =0\xb0$, a notable difference, the Newtonian rocket will overtake the ray, whereas the relativistic rocket cannot overtake the ray sent forward.

**Figure 6.**The Minkowski diagram in the non-inertial reference frame $(T,X,Y)$. Apart from the limit case $\theta =0$, all the light rays follow a partially helical worldline that asymptotically joins the horizon plane $x=-{d}_{H}$ in an infinite time t. The maximum deviation tends towards the half-turn for a ray emitted in the direction close to $\theta =0\xb0$.

**Figure 7.**The speed of light which decreases and tends towards zero in $x=-{d}_{H}$. In classical theory, there is no horizon and the speed of light increases.

**Figure 8.**Two clocks A and B at the same level are synchronized with the radar method. A light beam is emitted by A, reflected in B by a catadioptric system, and, back in A after $2\Delta {t}_{AB}$.

**Figure 9.**In the $(X,Y)$ plane, the trajectories of the particles are portions of ellipses with the center C on the horizon. The trajectories are perpendicular to the horizon line. The trajectory from $t\to -\infty $ to $t\to +\infty $ is a semi-ellipse, or a semi-circle for a zero-mass particle.

**Figure 10.**The Minkowski diagram in the non-inertial reference frame $(T,X,Y)$. Worldlines for ${v}_{0}=0.50\phantom{\rule{0.166667em}{0ex}}c$. In bold, the worldline for a particle released at rest. The intersection between a horizontal plane and the tubular worldsheet, formed by the set of worldlines for different $\theta $s and a given ${\beta}_{0}$, also forms an ellipse (dotted curve).

**Figure 11.**Falling velocity of a particle, released at rest from O at $T=0$. $T=t/{t}_{H}$ with ${t}_{H}$ the horizon time. In gray, the classical case.

**Figure 13.**The Minkowski diagram in the non-inertial frame $(T,X)$. The cones indicate how the coordinate velocity of light varies with respect to c.

**Figure 14.**Falling velocity of a particle, dropped without velocity from infinity. In dark gray, the curve according to Newton’s laws. Dotted, the speed of light. In light gray, the speed for a local observer.

**Figure 15.**The acceleration of the particle in radial fall towards a black hole. In gray, the classical curve.

**Figure 17.**Minkowski diagram $(T,X)$ for a vertical launch. $\tau /{t}_{H}\left(X\right)$ draws a circle arc. See text for details.

**Figure 18.**Michelson interferometer with two arms $O{M}_{1}$ and $O{M}_{2}$ of same length l: a coherent light source S, such as a laser, emits a light ray which encounters at O a half-silvered mirror, so half of the light is transmitted to the mirror ${M}_{1}$ while the other half is reflected by ${M}_{2}$. Both beams recombine at O and interfere on the detector I. In an inertial reference frame, there is no path length difference. (

**a**) In the non-inertial frame, R, there is a proper acceleration ${\overrightarrow{a}}_{p}$ along the downward direction, then the ray along X come back first and a path length difference $\delta $ appears. (

**b**) The arm $O{M}_{2}$ along the Y-axis rotates to the X-axis to take the place of the arm $O{M}_{1}$. The order of arrival of the rays on I is reversed.

**Figure 19.**Minkowski diagram of the world lines of rays along the two arms of the interferometer for the configuration (a) of Figure 18.

**Figure 20.**In black, light signals: $2\Delta T={T}_{AB}+{T}_{BC}>{T}_{AC}$. In gray, there is an initial velocity ${\beta}_{0e}$ such that: $2\Delta T={T}_{AC}\left({\beta}_{0e}\right)$. Numerical values for $L=2$: ${T}_{AB}+{T}_{BC}=2\Delta T\simeq 3.53>{T}_{AC}\simeq 2.89$, ${D}_{AB}+{D}_{BC}=\sqrt{2}\pi <{D}_{AC}\simeq 1.58\pi $. ${\beta}_{0e}=\sqrt{17/18}\simeq 0.972$, ${\gamma}_{0e}=3\sqrt{2}$, ${D}_{AC}\left({\beta}_{0e}\right)\simeq 1.94\pi $.

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**MDPI and ACS Style**

Rouaud, M. Einstein’s Elevator: World Lines, Michelson–Morley Experiment and Relativistic Paradox. *Physics* **2022**, *4*, 892-911.
https://doi.org/10.3390/physics4030058

**AMA Style**

Rouaud M. Einstein’s Elevator: World Lines, Michelson–Morley Experiment and Relativistic Paradox. *Physics*. 2022; 4(3):892-911.
https://doi.org/10.3390/physics4030058

**Chicago/Turabian Style**

Rouaud, Mathieu. 2022. "Einstein’s Elevator: World Lines, Michelson–Morley Experiment and Relativistic Paradox" *Physics* 4, no. 3: 892-911.
https://doi.org/10.3390/physics4030058