# Length Contraction

## Objects that are moving undergo a length contraction along the dimension of motion; this effect is only significant at relativistic speeds.

#### Key Points

• Length contraction is negligible at everyday speeds and can be ignored for all regular purposes.

• Length contraction becomes noticeable at a substantial fraction of the speed of light with the contraction only in the direction parallel to the direction in which the observed body is travelling.

• An observer at rest viewing an object travelling very close to the speed of light would observe the length of the object in the direction of motion as very near zero.

#### Terms

• the speed of electromagnetic radiation in a perfect vacuum: exactly 299,792,458 meters per second by definition

#### Figures

1. ##### Observed Length of an Object

Observed length of an object at rest and at different speeds

2. ##### Geometry for a Clock at Rest

This illustrates the path that light must traverse when the clock is at rest.

3. ##### Geometry for a Moving Clock

This illustrates the path that light must traverse when the clock is moving from the perspective of a stationary observer.

Length contraction is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer. Length contraction arises due to the fact that the speed of light in a vacuum is constant in any frame of reference. By taking this into account, as well as some geometrical considerations, we will show how perceived time and length are affected.

## Example 1

Let us imagine a simple clock system that consists of two mirrors A and B in a vacuum. A light pulse bounces between the two mirrors. The separation of the mirrors is L, and the clock ticks once each time the light pulse hits a given mirror. Now imagine that the clock is at rest. The time that it will take for the light pulse to go from mirror A to mirror B and then back to mirror A be can be described by:

<equation contenteditable="false">$\Delta t = \frac{2 L}{c}$

where c is the speed of light (Figure 2). Now imagine that the clock is moving in the horizontal direction relative to a stationary observer. The light pulse is emitted from mirror A. To the stationary observer, it appears that the light pulse has a longer path to travel because by the time the light reaches mirror B the clock has already moved somewhat in the horizontal direction. This is the same case for the light pulse on its way back. The stationary observer will perceive that it will take the light a total of:

$\Delta t' = \frac{2 D}{c}$

to traverse its path (Figure 3). We can see that D is longer than L, so that means that .

## Example 2

We have established that in a frame of reference that is moving relative to the clock (the stationary observer is moving in the clock's frame of reference), the clock appears to run more slowly. Now let us imagine that we want to measure the length of a ruler. This time let us imagine that you are moving with velocity v. You can mathematically determine the length of the ruler in your frame of reference (L') by multiplying your velocity (v) by the time that you perceive that it takes you to pass by the ruler (t'). Expressing this in equation form, L' = vt'. Now, if someone in the ruler's rest frame wanted to determine the length of the ruler, they could do the following. They could mathematically determine the length of the ruler in their frame of reference (L) by multiplying your velocity (v) by the time that they perceive that it takes you to pass by the ruler (t). This is expressed in the following equation: L = vt. Just as in the clock explanation, the ruler appears to be moving in your frame of reference, so t will be longer than t' (your time interval). Consequently, the length of the ruler will appear to be shorter in your frame of reference (the phenomenon of length contraction occurred).

The effect of length contraction is negligible at everyday speeds and can be ignored for all regular purposes. Length contraction becomes noticeable at a substantial fraction of the speed of light (as illustrated in Figure 1) with the contraction only in the direction parallel to the direction in which the observed body is travelling.

## Example 3

For example, at a speed of 13,400,000 m/s (30 million mph, .0447c), the length is 99.9 percent of the length at rest; at a speed of 42,300,000 m/s (95 million mph, 0.141c), the length is still 99 percent. As the magnitude of the velocity approaches the speed of light, the effect becomes dominant.The mathematical formula for length contraction is:

$L = \frac{L_{0}}{\gamma (\upsilon )} = L_{0}\sqrt{1-\frac{\upsilon ^{2}}{c^{2}}}$

where L0 is the proper length (the length of the object in its rest frame); L is the length observed by an observer in relative motion with respect to the object; v is the relative velocity between the observer and the moving object; c is the speed of light; and the Lorentz factor is defined as:

$\gamma (\upsilon ) = \frac{1}{\sqrt{1 - v^{^{2}}/c^{2}}}$

In this equation it is assumed that the object is parallel with its line of movement. For the observer in relative movement, the length of the object is measured by subtracting the simultaneously measured distances of both ends of the object. An observer at rest viewing an object traveling very close to the speed of light would observe the length of the object in the direction of motion as very close to zero.

#### Key Term Glossary

contraction
A reversible reduction in size.
##### Appears in these related concepts:
dimension
A measure of spatial extent in a particular direction, such as height, width or breadth, or depth.
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equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
frame of reference
A coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it.
##### Appears in these related concepts:
Length
How far apart objects are physically.
##### Appears in these related concepts:
length contraction
Observers measure a moving object's length as being smaller than it would be if it were stationary.
##### Appears in these related concepts:
light
The natural medium emanating from the sun and other very hot sources (now recognised as electromagnetic radiation with a wavelength of 400-750 nm), within which vision is possible.
##### Appears in these related concepts:
Lorentz factor
The factor, used in special relativity, to calculate the degree of time dilation, length contraction and relativistic mass of an object moving relative to an observer.
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magnitude
A number assigned to a vector indicating its length.
##### Appears in these related concepts:
motion
A change of position with respect to time.
##### Appears in these related concepts:
parallel
An arrangement of electrical components such that a current flows along two or more paths.
##### Appears in these related concepts:
pulse
to emit in discrete quantities
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relative
Expressed in relation to another item, rather than in complete form.
##### Appears in these related concepts:
speed of light
the speed of electromagnetic radiation in a perfect vacuum: exactly 299,792,458 meters per second by definition
##### Appears in these related concepts:
velocity
A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.
##### Appears in these related concepts:
Velocity
The rate of change of displacement with respect to change in time.