Definitions

# Vertical direction

In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force (per unit mass) at that point.

## Discussion

Although the word vertical is very commonly used in daily life and language (see below), it is subject to many misconceptions. The precise definition above and the following discussion points will hopefully clarify these issues.

• The concept of verticality only makes sense in the context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are too small or too distant from the point of interest), the notion of being vertical loses its meaning.
• In the presence of a simple, time-invariant, rotationally symmetric gravity field, the vertical directions passing by different points in space (and not belonging to the same vertical direction) intersect at the center of mass of that gravity field. This implies that no two different vertical directions are ever parallel to each other.
• In general, a vertical direction will only be perpendicular to a horizontal plane if both are specifically defined with respect to the same point: a plane is only horizontal at the point of reference. Thus both verticality and horizontality are strictly speaking local concepts, and it is always necessary to state to which location the direction or the plane refers to.
• In reality, the gravity field of a heterogeneous planet such as Earth is deformed due to the inhomogeneous spatial distribution of materials with different densities. Actual vertical directions are thus neither straight lines nor even convergent.
• At any given location, the total gravitational force is a function of time, because the objects that generate the reference gravity field move relative to each other. For instance, on Earth, the local vertical direction at a given point (as materialized by a plumb line) changes with the relative position of the Moon (air, sea and land tides).
• Furthermore, on a rotating planet such as Earth, there is a difference between the strictly gravitational pull of the planet (and possibly other celestial objects such as the Moon, the Sun, etc), and the apparent net force applied (e.g., on a free-falling object) that can be measured in the laboratory or in the field. This difference is due to the centrifugal force associated with the planet's rotation. This is a fictitious force: it only arises when calculations or experiments are conducted in non-inertial frames of reference.

## Practical use in daily life

The concept of a vertical line is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life.

This dichotomy between the apparent simplicity of a usual concept and an actual complexity of defining (and measuring) it in scientific terms is because the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the latter appears to be flat locally, and vertical directions in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy.

In graphical contexts, such as drawing and drafting on rectangular paper, it is very common to associate one of the dimensions of the paper with a vertical, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the vertical direction is typically from the side of the paper closest to the user to the opposite side (farthest away). This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context.