^{1,a)}, Andreas King

^{1}and Daria Adis

^{1}

### Abstract

Special relativity offers the possibility of going on a trip to the center of our galaxy or even to the end of our universe within a lifetime. On the basis of the well known twin paradox, we discuss uniformly acceleratedmotion and emphasize the local perspective of each twin concerning the interchange of light signals between both twins as well as their different views of the stellar sky. For this purpose we developed two Java applets that students can use to explore interactively and understand the topics presented here.

The authors thank Professor Hanns Ruder for the idea of this work and Professor Jörg Frauendiener for many discussions and for carefully reading the manuscript. Thanks also to Professor Jeff Rabin for the suggestion of the book *Tau Zero* (see Ref. 30). This work was supported by the Deutsche Forschungsgesellschaft (DFG), SFB 382, Teilprojekt D4.

I. INTRODUCTION

II. UNIFORM ACCELERATION

III. A SPECIFIC ROUND TRIP

IV. OBSERVATION OF TIME SIGNALS

V. FLIGHT TO VEGA

VI. THE ACCELERATED REFERENCE FRAME

A. Aberration,Doppler shift and length contraction

B. Apparent size of an object

C. Apparent position of an object

VII. VISUALIZATION OF THE STELLAR SKY

A. Aberration and Doppler shift

B. Temperature and brightness

C. Constellations

VIII. A TRIP TO THE END OF THE UNIVERSE

### Key Topics

- Doppler effect
- 19.0
- Optical aberrations
- 19.0
- Kinematics
- 16.0
- Red shift
- 11.0
- Special relativity
- 9.0

## Figures

Tina’s journey is separated into four phases. She starts from point ①, accelerates up to maximum velocity at point ②, and slows down until she reaches the turning point ③. Then she accelerates in the opposite direction and slows down again until she comes home. Signals that were emitted by Tina in the accelerating phase reach the Earth twin Eric in the interval .

Tina’s journey is separated into four phases. She starts from point ①, accelerates up to maximum velocity at point ②, and slows down until she reaches the turning point ③. Then she accelerates in the opposite direction and slows down again until she comes home. Signals that were emitted by Tina in the accelerating phase reach the Earth twin Eric in the interval .

Tina’s current velocity for her flight to Vega and return to Earth with respect to her own proper time. The maximum speed is reached at points ② and ④.

Tina’s current velocity for her flight to Vega and return to Earth with respect to her own proper time. The maximum speed is reached at points ② and ④.

The earth twin Eric sees/receives Tina’s proper time (ordinate) at his proper time (abscissa).

The earth twin Eric sees/receives Tina’s proper time (ordinate) at his proper time (abscissa).

Spacetime diagram for Tina’s flight to Vega and return to Earth with respect to Eric’s frame. At point ③ Tina reaches Vega and immediately returns. At ② and ④ she changes her acceleration direction. The small circles represent the events when Tina sends a time signal to Eric. They are separated by a single year with respect to her proper time . Note that the time units are not equally spaced on Tina’s worldline.

Spacetime diagram for Tina’s flight to Vega and return to Earth with respect to Eric’s frame. At point ③ Tina reaches Vega and immediately returns. At ② and ④ she changes her acceleration direction. The small circles represent the events when Tina sends a time signal to Eric. They are separated by a single year with respect to her proper time . Note that the time units are not equally spaced on Tina’s worldline.

The rocket twin Tina sees/receives Eric’s proper time (ordinate) at her proper time (abscissa).

The rocket twin Tina sees/receives Eric’s proper time (ordinate) at her proper time (abscissa).

The earth twin Eric sends a time signal every year with respect to his proper time . The dashes on Tina’s worldline mark the years of her proper time. Eric’s first signal does not reach Tina until she already decelerates to her destination ③.

The earth twin Eric sends a time signal every year with respect to his proper time . The dashes on Tina’s worldline mark the years of her proper time. Eric’s first signal does not reach Tina until she already decelerates to her destination ③.

Wave vector of an incoming light ray in spherical coordinates with respect to Eric’s rest frame. The twin Tina is currently moving with velocity along the direction.

Wave vector of an incoming light ray in spherical coordinates with respect to Eric’s rest frame. The twin Tina is currently moving with velocity along the direction.

An object a distance from the origin has an apex angle . An observer at rest at the current position would measure an apex angle .

An object a distance from the origin has an apex angle . An observer at rest at the current position would measure an apex angle .

An object is located at position with respect to the initial position of Tina. After some time, Tina’s current position is and the point would have the relative position with respect to an observer at rest at Tina’s current position.

An object is located at position with respect to the initial position of Tina. After some time, Tina’s current position is and the point would have the relative position with respect to an observer at rest at Tina’s current position.

The observation angle is plotted versus the velocity for . In the first instance, an object with fixed distance depending on and arbitrary angle apparently approaches the center of motion. For higher velocities, it recedes again. Because is reached only approximately, an object at seems to “freeze” at .

The observation angle is plotted versus the velocity for . In the first instance, an object with fixed distance depending on and arbitrary angle apparently approaches the center of motion. For higher velocities, it recedes again. Because is reached only approximately, an object at seems to “freeze” at .

The observation angle is plotted versus the velocity for . Note that even objects that are actually behind the observer might apparently “freeze” in front of the observer.

The observation angle is plotted versus the velocity for . Note that even objects that are actually behind the observer might apparently “freeze” in front of the observer.

Stellar sky at in the -representation where is the abscissa and is the ordinate. The center of the image corresponds to the direction of motion. The circles of latitude and the meridians are separated by 5°.

Stellar sky at in the -representation where is the abscissa and is the ordinate. The center of the image corresponds to the direction of motion. The circles of latitude and the meridians are separated by 5°.

Stellar sky at in the representation. Because of aberration, the nodes of the stellar sphere move together.

Stellar sky at in the representation. Because of aberration, the nodes of the stellar sphere move together.

Lines of constant redshift at velocity in the representation. From inside to outside: to , step 0.2; the bold line marks .

Lines of constant redshift at velocity in the representation. From inside to outside: to , step 0.2; the bold line marks .

Lines of constant redshift at velocity in the representation. From inside to outside: to , step 0.2; the bold line marks . Note that most of the sky is redshifted even for directions .

Lines of constant redshift at velocity in the representation. From inside to outside: to , step 0.2; the bold line marks . Note that most of the sky is redshifted even for directions .

The observation angle is plotted versus the velocity . The solid lines are lines of constant Doppler shift according to Eq. (59); the dashed lines represent the aberration of the angle (see Eq. (60)).

The observation angle is plotted versus the velocity . The solid lines are lines of constant Doppler shift according to Eq. (59); the dashed lines represent the aberration of the angle (see Eq. (60)).

Planck spectrum at temperature with a maximum at .

Planck spectrum at temperature with a maximum at .

The stellar sky marked by some constellations as seen at rest. In the representation the right ascension is plotted on the abscissa and the declination is plotted on the ordinate. Abbreviations: (Aql) Aquila, (Cas) Cassiopeia, (Crt) Crater, (Cru) Crux, (Cyg) Cygnus, (Her) Hercules, (Leo) Leo, (Ori) Orion, (Peg) Pegasus, (UMi) Ursa Minor.

The stellar sky marked by some constellations as seen at rest. In the representation the right ascension is plotted on the abscissa and the declination is plotted on the ordinate. Abbreviations: (Aql) Aquila, (Cas) Cassiopeia, (Crt) Crater, (Cru) Crux, (Cyg) Cygnus, (Her) Hercules, (Leo) Leo, (Ori) Orion, (Peg) Pegasus, (UMi) Ursa Minor.

The stellar sky as seen by an observer passing the Earth with 50% of the speed of light. The distortion of the constellation Southern Cross (Cru) is due to the projection and the aberration effect (see Fig. 12).

The stellar sky as seen by an observer passing the Earth with 50% of the speed of light. The distortion of the constellation Southern Cross (Cru) is due to the projection and the aberration effect (see Fig. 12).

The stellar sky as seen by an observer passing the Earth with 90% of the speed of light.

The stellar sky as seen by an observer passing the Earth with 90% of the speed of light.

CIE 1931 color matching functions .

CIE 1931 color matching functions .

## Tables

The coefficients for Eqs. (62) and (64) are taken from Ref. 17.

The coefficients for Eqs. (62) and (64) are taken from Ref. 17.

Star data of some constellations from Figs. 18–20. : right ascension, : declination, : trigonometric parallax (milliarcsec), B-V: Johnson B-V color, : temperature (Kelvin) from Eq. (62), HIP: Hipparcos number.

Star data of some constellations from Figs. 18–20. : right ascension, : declination, : trigonometric parallax (milliarcsec), B-V: Johnson B-V color, : temperature (Kelvin) from Eq. (62), HIP: Hipparcos number.

The stars of Table II have distance (parsec) and temperature (Kelvin) at velocities and in the direction .

The stars of Table II have distance (parsec) and temperature (Kelvin) at velocities and in the direction .

The apparent visual magnitude of the stars of Table II have bolometric magnitudes at velocities , , and in the direction .

The apparent visual magnitude of the stars of Table II have bolometric magnitudes at velocities , , and in the direction .

Distance from Earth, maximum speed and proper time of both twins for several stellar destinations. In the solar system we will reach only a few percent of the speed of light. Thus, time dilation can be neglected. However, in the neighborhood of the solar system time dilation is crucial. The “END” of the universe represents the maximum distance of about 13.7 billion light years that astronomers are able to observe.

Distance from Earth, maximum speed and proper time of both twins for several stellar destinations. In the solar system we will reach only a few percent of the speed of light. Thus, time dilation can be neglected. However, in the neighborhood of the solar system time dilation is crucial. The “END” of the universe represents the maximum distance of about 13.7 billion light years that astronomers are able to observe.

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