Proper length

Proper length, also called rest length, is the length of an object when it is not moving. This idea is important in physics, especially when we study how things change when they move very fast.

In classical mechanics, measuring length is simple because we assume everything happens at the same time. But in the theory of relativity, things are more complicated because what happens at the same time depends on who is looking.

To help solve this, scientists use a term called proper distance. This is a special way to measure distance that gives the same answer for everyone, no matter how they are moving. Proper distance is like proper time, which measures time in a similar way, but proper distance is used for space instead of time.

Proper length or rest length

The proper length or rest length of an object is the length of the object when measured by someone who is not moving relative to it. This measurement can be done using standard measuring tools placed directly on the object.

In situations where an observer is moving relative to the object, the measurement becomes more complex. The moving observer must measure the object's endpoints at the exact same moment, and the result will appear shorter than the proper length. This effect is known as length contraction.

Proper distance between two events in flat space

In special relativity, the proper distance between two events that happen at the same time in a certain view is simply the straight-line distance between them. This distance can be found using a simple formula: Δσ = √ (Δx² + Δy² + Δz²), where Δx, Δy, and Δz are the changes in the x, y, and z directions between the two events.

There is also another way to calculate this distance that works even if the events do not happen at the same time. This formula is Δσ = √ (Δx² + Δy² + Δz² - c²Δt²), where Δt is the time difference between the events and c is the speed of light. Both ways to calculate the distance give the same result because of a special property in space and time called the invariance of spacetime intervals.

Proper distance along a path

The proper distance between two events depends on the shape of space around them. In simple, flat space, the proper distance is just the straight-line distance. But in curved space, like around a massive object, there might be many different straight paths, or geodesics, between two points.

To find the proper distance along any path, we use a special math formula. This formula takes into account the metric tensor, which describes the shape of space, and the small steps along the path. It helps us calculate the distance even when space is curved.