# physics

## Six-dimensional spacetime

First consider the dual to Minkowski spacetime. Recall that the invariant interval of Minkowski spacetime has one dimension of time with three dimensions of space: (ds)² = (c dt)² – (dx1)² – (dx2)² – (dx3)² = (c dt)² – (dr)² where t is the time coordinate and x1, x2, and x3 are space coordinates of …

## Duals for Galilean and Lorentz transformations

In Newtonian mechanics inertial frames of reference are related by a Galilean transformation and time is absolute. In the special theory of relativity (STR) inertial frames of reference are related by a Lorentz transformation and the speed of light is absolute. By taking account of the three dimensions of time with a single dimension of …

## Geometric vectors in physics

The concept of a vector in physics is similar to that of mathematics: a geometric object with both magnitude and direction. The magnitude is in units that may be any physical units. The direction is in angular units such as radians or degrees. These are called geometric vectors (also known as Euclidian vectors). Note that …

## Speeds and velocities

A common word-problem in arithmetic goes something like this: If someone takes a road trip and for half of the time they go one speed and for the other half they go another speed, how should their average speed be determined? The answer is that the average speed is the arithmetic mean of the two …

## Direction and dimension

What does it mean to say that space has three dimensions? It means that space has directions that have three dimensions, that is, three degrees of freedom. The dimensions are the directions in the space. It’s not that there are some dimensions that are spatial and others are something else but that space is characterized …

## No change in time per length

Speed can be zero, that is, the change in spatial position per unit of duration can be zero. Can the change in temporal position per unit of length be zero, too? Let’s see. First, the denominator cannot be zero. We cannot simply invert a zero speed because that would lead to a zero denominator, which …

## The flow of time and space

Marcus Aurelius wrote, “Time is like a river made up of the events which happen, and a violent stream; for as soon as a thing has been seen, it is carried away, and another comes in its place, and this will be carried away too.” Time flows, and keeps flowing day and night, whether anything …

## Is time three-dimensional?

This post is a companion with the post “Is space one-dimensional?“. As we can compare the alleged one-dimensionality of time with how we think about space, so we can compare the three-dimensionality of space with how we think about time. In both cases the comparison is instructional. Space and time are parallel in both cases …

## Is space one-dimensional?

While the answer is No, space is three-dimensional, it is instructional to compare space with time as people speak about it. The philosopher JME McTaggart made a well-known distinction between an A-series, which is “the series of positions [in time] running from the far past through the near past to the present, and then from the present to the …

## Time for spacetime

Consider a worldline in one dimension of space and one dimension of time that tracks the position of a point that moves from position 20 to 10 to 15. This could represent the movement of a point in the E-W dimension. Another worldline could track the movement of the same point in the N-S dimension. All would agree …