Distance is a physical and mathematical quantity that the measurable non-coincident between the two objects space indicates. This measurable space can exist between concrete and between abstract (eg mathematical) objects. In daily practice, however, a distance to be covered is usually not the same as (a displacement over) a straight line. Think, for example, of a city where a motorist has to travel a distance. The distance traveled will not be a straight line, and the trajectory may also differ in the opposite direction. In this example, the distance traveled by the motorist is the number of times that a given standard size can be measured on the shortest possible connecting road. As an international standard for measuring length is the SI system the meter used.

There are various unit systems for physical quantity, the most commonly used are the International System of Units and the Anglo-Saxon system of units.

Meter

The idea of distance between two points is formalized and generalized by mathematics through the concept of metric. A space where there is a defined distance or metric is called a metric space.

The metric is the rule that defines how to calculate the separation between two events. In the Euclidean space, a space directly associated with classical mechanics and to which we are accustomed in our daily lives, the distance between two points can be calculated, once considered the metric of this space, through the expression:

This expression, widely known among high school students, does not include the greatness of time, and the spatial distance is shown to be completely independent of the temporal distance between the two points (events) considered, which in daily life translates into:

Once the spatial distance and the temporal distance between the events are determined, these are the same for any observers, whether they are in relative motion or not. The length of your street does not depend on whether you are in a stationary or moving car, nor does your wristwatch slow for this reason (this is for relative speeds encountered in everyday life).

Einstein, in publishing his theory of special relativity, brought to light the fact that the universe in which we live is not a Euclidean space, but a hyperbolic space with four dimensions, three spatial and one temporal, mutually inseparable. The metric for calculating the distance between two events in space-time is not Euclidean because in the expression that allows the calculation of this distance *δS*:

Just as the spatial distance between two points in the Euclidean metric does not depend on the observer considered, whether this observer is standing or moving in relation to the points (events) considered, the space-time separation between two events will always be the same regardless of the observer considered, whether in relative motion or not. However, the minus sign that accompanies the time in the expression for *δS* allows considerations that are beyond common sense since, even if the temporal space separation is preserved *δS* for two observers in relative motion, it allows a spatial separation observed by one of the observers to be interpreted by the other observer not as a spatial separation but as a temporal separation, the same goes for the temporal separation in the first case, which can be interpreted by the second observer as a spatial separation. Such a possibility of a relationship between spatial and temporal quantities in the hyperbolic metric of space-time gives rise to the well-known temporal expansion and spatial contraction described in the courses of special relativity.

The universe we live in is governed by a hyperbolic metric, not a Euclidean one.

Yard (yd) | Meter (m) | Kilometer (km) | Mile (mi) | League (lea) | |
---|---|---|---|---|---|

1 Yard (yd) = | 1 | 0.9144 | 0.0009144 | 0.0005681818181818182 | 0.0001893939393939394 |

1 Meter (m) = | 1.0936132983377078 | 1 | 0.001 | 0.0006213711922373339 | 0.000207123730745778 |

1 Kilometer (km) = | 1093.6132983377079 | 1000 | 1 | 0.621371192237334 | 0.20712373074577797 |

1 Mile (mi) = | 1760.0000000000002 | 1609.344 | 1.609344 | 1 | 0.33333333333333337 |

1 League (lea) = | 5280 | 4828.032 | 4.828032 | 2.9999999999999996 | 1 |