Question

Does the asymmetric aging of an Earth-bound observer and his twin who travels away and back demand "relativistic speed"? (Overlook the fact that each has a limited life span.)

Short answer

Yes, a high speed, close to the speed of light, or 'relativistic speed', is required for a significant difference in aging to be observed between the Earth-bound observer and the traveling twin due to the effect of time dilation. However, the difference can theoretically occur even at lower speeds, albeit being very small.

Step-by-step solution

Understand Relativity and Time Dilation

Firstly, understand the two postulates of Special Relativity: the laws of physics are the same in all inertial frames of reference, and the speed of light in a vacuum is the same for all observers. These lead to the concept of 'time dilation': to an observer in motion, time seems to pass slower compared to an observer at rest. This predicts that the 'moving' twin will age slower than the 'stationary' twin.

Relate Speed to Time Dilation

Next, understand that the rate of time dilation is influenced by speed. This can be derived from the Lorentz transformation: \( \Delta t' = \gamma \Delta t \), where \(\Delta t'\) is the dilated time, \( \Delta t \) is the proper time, and \( \gamma \) is the Lorentz factor given by: \( \gamma = 1/ \sqrt{1 - v^{2}/c^{2}} \). This equation relates speed \( v \) to time dilation, where \( c \) is the speed of light. Hence, higher speed results in greater time dilation.

Consider Relative Speed of Twins

Lastly, consider the relative speed of the twins in relation to the speed of light. The traveling twin would need to achieve a high percentage of the speed of light to observe significant time dilation. While this speed is theoretically possible, in practical terms it is currently unattainable. Thus, while 'relativistic speed' (comparable to the speed of light) can increase the difference in aging between the twins, it's not necessarily required for the effect to occur, even though it may be minute at lower speeds. It is also essential to note that acceleration and gravitational effects, if considerable, should also be taken into account, as per General Relativity.

Key concepts

Special Relativity

Special Relativity is a groundbreaking theory proposed by Albert Einstein. It transformed our understanding of space and time. An essential part of this theory is the idea that the laws of physics are constant and unchanging in all inertial frames of reference. This means, no matter how fast you're moving, the fundamental rules that govern physics remain the same.
Another significant aspect is the constancy of light speed. In Special Relativity, the speed of light in a vacuum is always the same, regardless of the observer's movement or the light source. This principle has profound implications.
One such implication is time dilation, a phenomenon where time seemingly moves slower for an individual in motion compared to someone at rest. This is famously illustrated through the twin paradox, where a twin traveling at high speeds ages slower than his counterpart who remains on Earth.

Lorentz Transformation

The Lorentz Transformation is a set of equations crucial to understanding Special Relativity. These equations describe how measurements of space and time by two observers are related.
An essential equation from the Lorentz Transformation is the time dilation equation: \[\Delta t' = \gamma \Delta t\]where \(\Delta t'\) is the time experienced by a moving observer, \(\Delta t\) is the time experienced by a stationary observer, and \(\gamma\) is the Lorentz factor.\(\gamma\) is calculated as:\[\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]Here, \(v\) represents the velocity of the moving observer, and \(c\) is the speed of light.
The Lorentz factor becomes significant as \(v\) approaches \(c\), showing that higher speeds lead to more considerable time dilation. This demonstrates how dramatically time slows down at relativistic speeds, making this transformation a cornerstone concept in Special Relativity.

Relativistic Speed

Relativistic Speed refers to velocities that are a substantial fraction of the speed of light. At these speeds, the effects predicted by Special Relativity become pronounced.
For example, time dilation becomes noticeable, affecting aging and perception of events. In the classic twin paradox scenario, the traveling twin moves at relativistic speeds, experiencing less passage of time. It results in the twin aging slower compared to the twin staying on Earth.
Achieving relativistic speeds demands incredible energy, and, in practical terms, isn't within our current technological capabilities. However, even at lesser speeds, time dilation effects can occur but are typically minuscule.
To fully appreciate how time and space transform under these conditions, consider the Lorentz Transformation. These equations reveal the profound nature of motion at such extreme speeds, reshaping our traditional views of time and motion.