The Dark Forest Hypothesis and Relativistic Missiles
In order to understand the dark forest hypothesis, it is crucial to examine relativistic missiles. These weapons, capable of traveling at near light speed, pose a significant threat and understanding their properties is key to grasping the dangers within the dark forest scenario.
Understanding Relativistic Missiles
Contrary to conventional missiles, relativistic missiles travel through space at nearly the speed of light. This extreme velocity introduces relativistic effects that make interception extremely difficult.
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The most alarming aspect of these weapons is that they travel at almost the same speed as the signals they emit and reflect.
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This makes their detection practically impossible. By the time a signal is detected, it's too late to react.
The Problem of Detection
The difference in detection time between conventional and relativistic missiles highlights the danger.
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Conventional Missiles: Signals emitted by conventional missiles are observed well in advance of their arrival, allowing ample reaction time.
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Relativistic Missiles: For an external observer, a relativistic missile and its signal travel at nearly the same speed. The missile is only nanoseconds behind the signal.
Due to wavefront compression, the missile may even appear to be moving faster than the speed of light to the target, a phenomenon known as apparent superluminal motion. This does not mean the missile is breaking the laws of physics.
Apparent Superluminal Motion Explained
The negligible time between detection and impact renders relativistic weapons nearly impossible to defend against. This begs the question: Can objects with mass travel faster than light, and if not, why do some objects appear to exhibit superluminal motion?
The Projection Effect: Explaining Apparent Superluminal Motion
The effect responsible for the apparent faster-than-light motion is known as the projection effect. This is best understood by first using sound as an example.
Sound Wave Analogy
Imagine a sphere playing a one-minute song.
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The first note is represented by a blue wave, and the last note by a red wave.
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The sphere is 20 km away (the distance sound travels in one minute).
If the sphere is stationary, the song will end exactly one minute after we start hearing it. The duration is the same for both the sphere and the observer.
However, if the sphere moves toward us at half the speed of sound:
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The first note remains unaffected, but the last note reaches us in half the time.
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From our perspective, the song plays twice as fast. The 1 minute duration is compressed into 30 seconds. This is due to the Doppler effect and the decreasing distance between successive sound waves.
If the sphere moves at 80% of the speed of sound, the 1-minute song will be compressed into only 12 seconds of perceived duration. If the sphere travels faster than the speed of sound, the song will play in reverse from our perspective.
Light Wave Application
Since light waves also propagate at a finite speed, there is an inherent delay in the transmission of information. While nothing can travel faster than light, relativistic objects can exhibit superluminal motion under specific conditions.
Let us now consider a similar situation but on cosmic scales, using light.
Cosmic Scale Example: Sphere Near Saturn
Imagine a sphere near Saturn and a telescope on Earth observing it.
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The sphere emits light for 60 minutes.
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Light takes 60 minutes to reach Earth from Saturn.
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If the sphere is stationary, the final light will reach our telescope 60 minutes after the first.
If the sphere moves toward us at 1,000 km/s, it will have traveled 3,600,000 km in 60 minutes, consistent with its velocity.
Relativistic Speeds and Apparent Motion
If the sphere moves towards us at half the speed of light:
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The sphere will reach the vicinity of Jupiter after 60 minutes.
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The final light wave will reach us in half the time, because it is emitted from a point halfway closer to us.
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From our perspective, the sphere appears to have traveled from Saturn to Jupiter in just 30 minutes.
At 90% of the speed of light, the initial wave reaches us as expected, but the final wave (emitted 33.3 minutes later) arrives only 3.3 minutes after the first.
This means, from our perspective, the sphere has covered 540 million km in 3.3 minutes, which vastly exceeds the speed of light, showcasing the difference between relative speed and apparent speed.
Line of Sight Impact
Any object moving collinearly or almost collinearly with the observer's line of sight will appear to move faster than it actually does, even at non-relativistic speeds.
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For example, a car moving at 70 km/h would appear to move at 70.00004 km/h, a negligible difference.
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As the object's velocity approaches the speed of light, this discrepancy increases exponentially.
While apparent velocity can exceed the speed of light, this is purely a perceptual effect. No information or object is actually traveling faster than the speed of light. It is simply an optical phenomenon resulting from the delay of light we use to perceive motion.