Matters relating to length and duration in physics and transportation
The post Lorentz for space and time derived the standard (spatial) Lorentz transformation and also the temporal Lorentz transformation. It is surprising in this age of relativity how the standard Lorentz transformation is dependent on absolute time. While time is relativized in the sense of Lorentz and Einstein, it remains absolute in the sense of […]
It is not unusual to seek the route in space that minimizes travel time, for example, a drive from point A to point B may go out of the way to include a high-speed facility that reduces travel time even if it increases distance traveled. But what about routes in time? Does it ever make
One surprising result is that an object’s velocity and its inverse, which I’m calling lenticity, may have different directions. Here’s what I mean: Suppose someone drives 30 miles East in 50 minutes, then turns North and drives 40 miles in 50 minutes. Overall, they have driven 70 miles in 100 minutes but as the crow
Here is an updated list of claims about time made in this series of blog posts: Time has 3-dimensions. This is the over-arching claim which is explained and expanded by the other claims. Time is duration with direction. That is, time is a vector variable similar to a space vector (a distance with a direction).
Travel, that is, the movement of something, includes transporting and signalling. To transport means to take something (e.g., people or goods) from one place to another by means of a vehicle or the like (e.g., a car). To signal means to transmit information or instructions from one place to another through a medium or the
Consider again the now-classic scenario in which observer K is at rest and observer K′ is moving in the positive direction of the x axis with constant velocity, v. This time there is a characteristic constant speed, c. The basic problem is that if they both observe a point event E, how should one convert
Consider the now-classic scenario in which observer K is at rest and observer K′ is moving in the positive direction of the x axis with constant velocity v. The basic problem is that if they both observe a point event E, how should one convert the coordinates of E from one reference frame to the
For each mode of travel there are four speeds to consider: the minimum speed, the typical speed, the maximum speed, and the speed of particular objects. The more that impediments to travel are removed (e.g., other objects, the topography, the network), the more that speed reaches free flow. In transportation, the free flow speed is
A physical vector is a physical magnitude with a direction that operates as a mathematical vector. As with all physical quantities, it has units of some kind. Both the magnitude and the direction have units. The directional units are called unit vectors. The units of a magnitude are what it is relative to, for example:
Space has been represented with a three-dimensional geometry since ancient times. Descartes added coordinates, which make these dimensions more explicit. We call this Euclidean space R3 = R × R × R, that is, three dimensions of real-number coordinates. Newton added time but kept it separate since he considered time absolute and space relative. Call this R3
