Understanding Time and Length Contraction in Special Relativity: A Meterstick Moving at 0.1c

Special relativity, introduced by Einstein, challenges our classical understanding of space and time. Central to this theory are the concepts of time dilation and length contraction. In this article, we explore a particular scenario involving a meterstick moving at 0.1 times the speed of light (0.1c) relative to an observer. We'll analyze how this affects the perception of length and time passing.

Understanding the Problem: Length Contraction

When an object, such as a meterstick, moves at a significant fraction of the speed of light relative to an observer, it appears shorter in the direction of motion. This effect is called length contraction. According to special relativity, the observed length L of the object is given by the formula:

L L_0 √(1 - v^2/c^2)

where L_0 is the proper length (the length of the object at rest), v is the relative velocity of the object, and c is the speed of light.

Applying the Length Contraction Formula

Consider a meterstick with a proper length of 1000 mm (or 1 meter). When moving at 0.1 times the speed of light, we can calculate the contracted length:

Given:

L_0 1000 mm (1 meter) v 0.1c c 3 × 10^8 m/s

Calculation:

L 1000 mm × √(1 - (0.1 × 3 × 10^8 m/s)^2 / (3 × 10^8 m/s)^2)

L 1000 mm × √(1 - 0.01)

L 1000 mm × √0.99

L ≈ 995 mm (or 0.995 meters)

This calculation shows that the meterstick appears to be 995 mm long when observed from the stationary frame of reference.

Calculating the Time to Pass

The time it takes for the meterstick to pass the observer is determined by its contracted length. Given that the contracted length is 995 mm, and assuming the meterstick moves at 0.1c, the time to pass can be calculated as:

t L / v

t 995 mm / (0.1 × 3 × 10^8 m/s)

t 995 × 10^-3 m / (0.3 × 10^8 m/s)

t 3.3167 × 10^-9 seconds ≈ 0.33167 nanoseconds

This calculation provides the time it takes for the observer to see the entire meterstick pass by, considering length contraction.

Length and Time Contraction: A Comparative Analysis

In addition to length contraction, special relativity also affects the perception of time. The contraction of radial lengths and the observed clock frequency inversely relate to the time dilation caused by the ratio of the relative radial speed to the constant vacuum light speed.

For example, if the contraction is 0.5, the observed frequency f is given by:

f √(1 - v/c^2)

Where v is the relative speed. To find v, rearrange the formula:

v/c^2 1 - f^2

v/c √(1 - f^2)

For f 0.5:

v/c √(1 - 0.5^2) 0.866

v 0.866 × 3 × 10^8 m/s ≈ 2.6 × 10^8 m/s

This suggests the relative speed of 2.6 × 10^8 m/s. The time experienced by an observer holding the meterstick (t) when the leading edge of a spacecraft passes 1 meter is calculated as:

t 1/30000000 s × 0.995 0.33167 nanoseconds

Conversely, from the viewpoint of the spacecraft, the time (t') for the 500 mm (0.5 meter) meterstick to pass is:

t' 0.5/30000000 s × 0.995 0.165835 nanoseconds

This example further demonstrates the intricate interplay between time and space in special relativity.