Question

An astronaut in a spaceship flying toward Earth's Equator at half the speed of light observes Earth to be an oblong solid, wider and taller than it appears deep, rotating around its long axis. A second astronaut flying toward Earth's North Pole at half the speed of light observes Earth to be a similar shape but rotating about its short axis. Why does this not present a contradiction?

Short answer

Answer: The different observations of Earth's shape and rotation by the two astronauts do not present a contradiction because these differences are due to the relativistic effects caused by their high-speed motions and the differences in their directions of motion relative to Earth. The astronauts are observing Earth from different inertial frames, so their observations will be affected differently by the relativistic effects of length contraction and time dilation. This is consistent with the principles of special relativity, which states that the laws of physics should be the same in all inertial frames.

Step-by-step solution

Understanding the Astronauts' Observations

The first astronaut is moving towards the Earth's Equator while the second astronaut is moving towards the Earth's North Pole, both at half the speed of light. Due to their motions, they observe Earth to be distorted differently: 1. The first astronaut sees Earth as wider and taller than it appears deep, and rotating around its long axis. 2. The second astronaut sees Earth as a similar shape but rotating around its short axis.

Understanding Relativistic Effects

In order to understand why these different observations don't present a contradiction, we need to analyze the situation from the perspective of special relativity. When objects move at high speeds close to the speed of light, they experience relativistic effects such as length contraction and time dilation. Length contraction states that the length of an object moving relative to an observer will appear contracted along the direction of motion, whereas time dilation states that time intervals observed within the moving object appear longer than they would in the observer's rest frame. These relativistic effects are given by the Lorentz transformations, which relate the spacetime coordinates of events in two inertial reference frames moving at a constant velocity relative to each other: Length contraction formula: $$L=L_0 \sqrt{1-\frac{v^2}{c^2}}$$ Where \(L_0\) is the proper length (length measured by an observer at rest relative to the object), \(v\) is the relative velocity, and \(c\) is the speed of light. Time dilation formula: $$\Delta t = \frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}}$$ Where \(\Delta t_0\) is the proper time interval (time interval measured by an observer at rest relative to the object), \(v\) is the relative velocity, and \(c\) is the speed of light.

Analyzing the Two Observations

Since both astronauts are moving at relativistic speeds, they will experience length contraction and time dilation. Due to their different directions of motion, Earth will be distorted differently: 1. For the first astronaut moving towards the Earth's Equator, Earth will appear contracted along the direction of motion (towards the poles) and elongated in the perpendicular directions. This will result in the Earth appearing wider and taller than it appears deep, and rotating around its long axis. 2. For the second astronaut moving towards the Earth's North Pole, Earth will appear contracted along the direction of motion (from pole to pole) and elongated in the perpendicular directions. This will result in the Earth appearing with a similar shape as seen by the first astronaut but rotating around its short axis.

Concluding the Explanation

These different observations of Earth's shape and rotation by the two astronauts are due to the relativistic effects caused by their high-speed motions and the differences in their directions of motion relative to Earth. The fact that the observations of the two astronauts are not in agreement does not present a contradiction. The astronauts are observing Earth from different inertial frames, so their observations will be affected differently by the relativistic effects of length contraction and time dilation. This is consistent with the principles of special relativity, which states that the laws of physics should be the same in all inertial frames.

Key concepts

Length Contraction

Imagine traveling at near light-speeds, looking ahead, and noticing objects appear squished; this is the essence of length contraction in special relativity. It's a counterintuitive phenomenon predicting that an object in motion will shorten along its direction of travel from the perspective of a stationary observer. For instance, if you're flying towards a planet, it will seem narrower along your flight path.

Mathematically, length contraction is described by the formula: \[L = L_0 \sqrt{1-\frac{v^2}{c^2}}\]where:

• \(L_0\) is the proper length, which is the length of the object measured at rest,
• \(v\) is the object's velocity relative to the observer,
• and \(c\) is the speed of light.

To explicate, as the speed of an object approaches the speed of light (\(c\)), the factor \(\sqrt{1-\frac{v^2}{c^2}}\) diminishes, contracting the length (\(L\)) in the observer's frame. This effect is significant at relativistic speeds – that is, speeds which are a substantial fraction of the speed of light.

Back to our astronaut scenario: as the spaceship moves at half the speed of light towards Earth's Equator, the planet's polar diameter contracts. This results in the first astronaut observing an oblong Earth, visually confirming length contraction at play.

Time Dilation

While length contracts, time dilates - stretches out - as velocities ramp up to relativistic speeds. Time dilation is the strange and fascinating prediction that time runs slower on a moving clock when viewed by a stationary observer. If your friend zips past you in a spaceship, their watch ticks more slowly compared to yours, assuming you could peek inside.The formula encapsulating time dilation is:\[\Delta t = \frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}}\]where:

• \(\Delta t_0\) signifies the proper time interval, or the time between events as measured by an observer at rest relative to the events,
• \(v\) is the relative velocity between the observer and the moving clock,
• and \(c\) remains the constant speed of light.

In the context of our astronauts, time on Earth ticks by normally for us, but as they soar towards our planet at high velocities, their perception of time on Earth slows down. Hence, each astronaut, traveling in different directions, perceives Earth's rotation differently because their individual relative motion alters their perception of the planet's temporal progression.

Lorentz Transformations

Lorentz transformations are the mathematical guardians of the speed of light in Einstein's theory of special relativity. They are a set of equations that make it possible to translate physical quantities – like distances and times – from one inertial frame to another moving at a constant velocity relative to the first. Spanning longer than a century of physics, these transformations form the backbone for understanding how space and time interplay.

In their essence, Lorentz transformations reconcile the seemingly incompatible observations of observers in different states of motion. They ensure the constant pull of the cosmic speed limit, the speed of light, across all frames, regardless of how fast you're moving.

With Lorentz transformations, we can better understand why our astronauts, racing towards Earth from different angles, each see a non-contradictory version of distorted Earth. Each astronaut uses these transformations, albeit subconsciously, to decode Earth's shape and rotation from their unique vantage points of high-speed travel. These equations prove that no matter how bizarre the relativistic effects may seem, they're just different sides of the same cosmic dice, tossed by the same immutable laws of physics.