Lighthouse paradox

Last updated December 06, 2022

The lighthouse paradox is a thought experiment in which the speed of light is apparently exceeded. The rotating beam of light from a lighthouse is imagined to be swept from one object to shine on a second object. The farther the two objects are away from the lighthouse, the farther the distance between them crossed by the light beam. If the objects are sufficiently far away from the lighthouse, the places where the beam hits object 2 will traverse the object with an apparent speed faster than light, possibly communicating a signal on object 2 with superluminal velocity, which violates Albert Einstein's theory of special relativity.

The solution to this paradox is that superluminal velocities can be observed because no actual particles or information are traveling from object 1 to object 2. The transverse velocity of the beam along the path in the sky between the objects has an apparent speed greater than light, but this represents separate photons of light. No photons are traveling the path from object 1 to object 2; the photons in the light beam are traveling a radial path outward from the lighthouse, at the speed of light. The theory of relativity says information cannot be transmitted faster than light. This experiment does not actually transmit a signal from object 1 to object 2. The time when the light beam strikes object 2 is controlled by the person at the lighthouse, not anyone on object 1, so no one on object 1 can transmit a message to object 2 by this method. Therefore the theory of relativity is not violated.

Paradox

A lighthouse sends a powerful beam of light which travels significant distances from the point of origin. This light constantly rotates in a circular motion around the lighthouse. This thought experiment proposes that light moving in this situation is actually traveling faster than the speed of light. This presents a paradox because, according to the theory of relativity, the speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source, and nothing can travel faster than this speed.^[1]^[2]

Moon example

Distance observed to be traveled by laser on the moon (from point A to point B) as light source moves laterally on Earth. Moon Diagram.png — Distance observed to be traveled by laser on the moon (from point A to point B) as light source moves laterally on Earth.

A similar example can be explained by the movement of a laser across the face of the Moon.^[3] This paradox arises based on a simple principle: if someone stands a distance of "X" away from an object, and shines a laser from one side (A) of the object to the other side (B) of the object, they would have to rotate their hand by an angle "Y." Thus, as X increases, Y would decrease, as the wrist would have to rotate a smaller angle to move the laser from point A to point B. Also, correlated with a smaller angle would be a decreased time it would take to rotate the wrist (it will take a shorter time to rotate the wrist a smaller angle). With respect to distant objects like the Moon, a paradox arises when one is asked to hypothetically move a laser from one side to the other.^[3] Turning his wrist half a degree, a person can move a laser from one side of the Moon to the other. It would appear that the laser dot is travelling faster than light, as flicking one's wrist at such a large distance would give the illusion that the object was able to cross the diameter of the Moon (6000 km, due to curvature) in milliseconds.^[3] Based on subsequent calculations (distance between points A and B divided by time it takes to move the laser from A to B), it would appear that the dot of light is moving at superluminal velocities, when, in reality, the dots are successive photons being emitted by the source moving across the face of the Moon.^[3]

Resolution of the paradox in special relativity

The paradoxical aspect of each of the described thought experiments arises from Einstein’s theory of special relativity, which proclaims the speed of light (approx. 300,000 km/s) is the upper limit of speed in our universe.^[1]^[4]^[5] The uniformity of the speed of light is so absolute that regardless of the speed of the observer as well as the speed of the source of light the speed of the light ray should remain constant.^[1]^[4]

When considering the movement of an image created by a laser on the Moon, some physical limitations would have to be violated in order to trace out the apparent trajectory at superluminal velocities. To approach the speed of light, and therefore to pass it, an object would have to be accelerated through an infinite potential, an impossibility within the physical universe.^[1]^[4]^[2] The acceleration process would also cause the object to have infinite mass, which is not only logically impossible, but it would also cause severe gravitational effects in the surrounding spacetime.^[1]^[2] However, these effects that have no empirical evidence leading to the conclusion that there is a simple physical explanation.

The fundamental misunderstanding of this paradox is the assumption that the projected image caused by the light ray is a physical object, and therefore must follow physical law. In reality, no physical laws are being broken as there is no physical object travelling faster than light. This paradox uses kinematic processes to explain the motion of this apparent object. However, the projected image on the Moon, or the image created by the lighthouse, is not a real object. The apparent lateral motion across the surface of the Moon is a result of the light source moving through some angular rotation, not superluminal motion across its surface. The angular motion of the source creates a translation of the image projected on the Moon, proportional to the distance between the screen (which in this case is Moon) and the source. Thus, if one was to go close enough to the Moon and rotate the laser through the same angle the image would travel at subluminal speeds even though nothing that should effect its speed has changed. If the image was a physical object, it should be able to travel across the surface of the Moon at the same speed regardless of the distance of the observer. Understanding this, the paradox begins to unravel.^[3]

The movement from point A to point B can be visualized as a collection of photons, each travelling along a different trajectory from Earth to the Moon Moon diagram 2.png — The movement from point A to point B can be visualized as a collection of photons, each travelling along a different trajectory from Earth to the Moon

It is natural to visualize this phenomenon as an abundance of stationary photons within the same ray of light creating the spot on the Moon. To allow the image to move from one end of the Moon to the other, each photon must move laterally with the movement of the projection. In reality, this is not the case: the ray of light is a collection of moving photons and at each instant a different group of photons, detected by the observers eye, are creating the image seen on the surface of the Moon.^[3] The apparent lateral movement is caused by new photons traveling on a different path from the light source to the Moon, caused by the rotation of the source, striking an adjacent position at all instances during the rotation. The movement from point A to point B can be visualized by a collection of photons, each travelling along a different trajectory from Earth to the Moon. The paradox is resolved to be a result of the geometry of the system causing the illusion of superluminal motion, rather than superluminal motion actually occurring.^[3]

A final issue with this explanation is that there appears to be no delay between the flicking of the wrist and the movement of the image on the Moon, a process that is expected if the photon resolution is correct. This does not invalidate the resolution of the paradox. The apparent simultaneity is a result of the large magnitude of the speed of light, and the observers inability to detect changes that fast. In ideal conditions, the expected delay would be noticeable.^[3]

Related Research Articles

In astronomy, aberration is a phenomenon which produces an apparent motion of celestial objects about their true positions, dependent on the velocity of the observer. It causes objects to appear to be displaced towards the direction of motion of the observer compared to when the observer is stationary. The change in angle is of the order of v/c where c is the speed of light and v the velocity of the observer. In the case of "stellar" or "annual" aberration, the apparent position of a star to an observer on Earth varies periodically over the course of a year as the Earth's velocity changes as it revolves around the Sun, by a maximum angle of approximately 20 arcseconds in right ascension or declination.

Faster-than-light travel and communication are the conjectural propagation of matter or information faster than the speed of light. The special theory of relativity implies that only particles with zero rest mass may travel at the speed of light, and that nothing may travel faster.

Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahertz, between the infrared and the ultraviolet.

In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and measuring the change in position of the body relative to that frame with change in time. The branch of physics describing the motion of objects without reference to its cause is called kinematics, while the branch studying forces and their effect on motion is called dynamics.

In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:

The laws of physics are invariant in all inertial frames of reference.
The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or the observer.

The speed of light in vacuum, commonly denoted $c$ , is a universal physical constant that is important in many areas of physics. The speed of light $c$ is exactly equal to 299,792,458 metres per second. According to the special theory of relativity, $c$ is the upper limit for the speed at which conventional matter or energy can travel through space.

In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur.

A tachyon or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such particles did exist they could be used to send signals faster than light. According to the theory of relativity this would violate causality, leading to logical paradoxes such as the grandfather paradox. Tachyons would exhibit the unusual property of increasing in speed as their energy decreases, and would require infinite energy to slow down to the speed of light. No verifiable experimental evidence for the existence of such particles has been found.

Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald contraction and is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.

<span class="mw-page-title-main">Superluminal motion</span> Apparent faster-than-light motion of distant astronomical objects

In astronomy, superluminal motion is the apparently faster-than-light motion seen in some radio galaxies, BL Lac objects, quasars, blazars and recently also in some galactic sources called microquasars. Bursts of energy moving out along the relativistic jets emitted from these objects can have a proper motion that appears greater than the speed of light. All of these sources are thought to contain a black hole, responsible for the ejection of mass at high velocities. Light echoes can also produce apparent superluminal motion.

The relativistic Doppler effect is the change in frequency of light, caused by the relative motion of the source and the observer, when taking into account effects described by the special theory of relativity.

Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. Emission theories obey the principle of relativity by having no preferred frame for light transmission, but say that light is emitted at speed "c" relative to its source instead of applying the invariance postulate. Thus, emitter theory combines electrodynamics and mechanics with a simple Newtonian theory. Although there are still proponents of this theory outside the scientific mainstream, this theory is considered to be conclusively discredited by most scientists.

Relativistic beaming is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of light. In an astronomical context, relativistic beaming commonly occurs in two oppositely-directed relativistic jets of plasma that originate from a central compact object that is accreting matter. Accreting compact objects and relativistic jets are invoked to explain x-ray binaries, gamma-ray bursts, and, on a much larger scale, active galactic nuclei.

An astrophysical jet is an astronomical phenomenon where outflows of ionised matter are emitted as an extended beam along the axis of rotation. When this greatly accelerated matter in the beam approaches the speed of light, astrophysical jets become relativistic jets as they show effects from special relativity.

The Sagnac effect, also called Sagnac interference, named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called a ring interferometer or Sagnac interferometer. A beam of light is split and the two beams are made to follow the same path but in opposite directions. On return to the point of entry the two light beams are allowed to exit the ring and undergo interference. The relative phases of the two exiting beams, and thus the position of the interference fringes, are shifted according to the angular velocity of the apparatus. In other words, when the interferometer is at rest with respect to a nonrotating frame, the light takes the same amount of time to traverse the ring in either direction. However, when the interferometer system is spun, one beam of light has a longer path to travel than the other in order to complete one circuit of the mechanical frame, and so takes longer, resulting in a phase difference between the two beams. Georges Sagnac set up this experiment in an attempt to prove the existence of the aether that Einstein's theory of special relativity had discarded.

The Scharnhorst effect is a hypothetical phenomenon in which light signals travel slightly faster than c between two closely spaced conducting plates. It was first predicted in a 1990 paper by Klaus Scharnhorst of the Humboldt University of Berlin, Germany. He showed using quantum electrodynamics that the effective refractive index n, at low frequencies, in the space between the plates was less than 1. Barton and Scharnhorst in 1993 claimed that either signal velocity can exceed c or that imaginary part of n is negative.

The de Sitter effect was described by Willem de Sitter in 1913 and used to support the special theory of relativity against a competing 1908 emission theory by Walther Ritz that postulated a variable speed of light dependent on the velocity of the emitting object. De Sitter showed what Ritz's theory have predicted that the orbits of binary stars would appear more eccentric than consistent with experiment and with the laws of mechanics, however, the experimental result was negative. This was confirmed by Brecher in 1977 by observing the x-rays spectrum. For other experiments related to special relativity, see tests of special relativity.

<span class="mw-page-title-main">Günter Nimtz</span> German physicist

Günter Nimtz is a German physicist, working at the 2nd Physics Institute at the University of Cologne in Germany. He has investigated narrow-gap semiconductors and liquid crystals. His claims show that particles may travel faster than the speed of light (c) when undergoing quantum tunneling.

A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations.

Cherenkov radiation is electromagnetic radiation emitted when a charged particle passes through a dielectric medium at a speed greater than the phase velocity of light in that medium. A classic example of Cherenkov radiation is the characteristic blue glow of an underwater nuclear reactor. Its cause is similar to the cause of a sonic boom, the sharp sound heard when faster-than-sound movement occurs. The phenomenon is named after Soviet physicist Pavel Cherenkov.

References

1 2 3 4 5 "Maudlin, M. (2011). Quantum non-locality and relativity : metaphysical limitations of modern physics (3rd ed.). Singapore: Blackwell Publishing Ltd
1 2 3 Uzan & Leclercg, J.P. & B. (2010). The Natural Laws of the Universe: Understanding Fundamental Constants. Springer Science & Business Media. pp. 43–4. ISBN 978-0-387-73454-5.
1 2 3 4 5 6 7 8 "How A Laser Appears To Move Faster Than Light (And Why It Really Isn't)". Universe Today. 7 February 2014. Retrieved 2016-04-05.
1 2 3 "", Simonetti, J. Virginia Tech Physics: Frequently Asked Questions About Relativity.
↑ Jorgensen, Palle E. T. (2008-11-13). "The road to reality: a complete guide to the laws of the universe". The Mathematical Intelligencer. 28 (3): 59–61. doi:10.1007/BF02986885. ISSN 0343-6993. S2CID 117975932.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Maudlin-1] 1 2 3 4 5 "Maudlin, M. (2011). Quantum non-locality and relativity : metaphysical limitations of modern physics (3rd ed.). Singapore: Blackwell Publishing Ltd

[:1-2] 1 2 3 Uzan & Leclercg, J.P. & B. (2010). The Natural Laws of the Universe: Understanding Fundamental Constants. Springer Science & Business Media. pp. 43–4. ISBN 978-0-387-73454-5.

[:0-3] 1 2 3 4 5 6 7 8 "How A Laser Appears To Move Faster Than Light (And Why It Really Isn't)". Universe Today. 7 February 2014. Retrieved 2016-04-05.

[Facts-4] 1 2 3 "", Simonetti, J. Virginia Tech Physics: Frequently Asked Questions About Relativity.

[5] Jorgensen, Palle E. T. (2008-11-13). "The road to reality: a complete guide to the laws of the universe". The Mathematical Intelligencer. 28 (3): 59–61. doi:10.1007/BF02986885. ISSN 0343-6993. S2CID 117975932.

[1]

[2]

[3]

[4]

[5]