The Shape Of Time
A couple of months ago, maybe it was a year before now, can't remember, I came across an article that gives an account of Immanuel Kant's views on space and time. According to it, Kant thought of space as geometry and time as arithmetic and he did that perhaps in order to fit spacetime into a mathematical framework that was inclusive of two fundamental concepts in math viz. arithmetic (numbers) and geometry (shapes).
It's not that difficult to understand where Kant was coming from. There's a very basic link between space and geometry - space is filled with objects of all shapes and sizes. As for time, it's obvious that we use arithmetic (numbers) to make sense of its passing.
Now, it's clear that geometry can be/has been subjected to arithmetization - coordinate geometry, and we measure the dimensions of various geometric shapes with numbers. The converse must be true too as certain relationships between numbers can be geometrized e.g. there are such things as "triangular" numbers.
Space, being about shapes, its own and that of its contents, has also been arithmetitized (translated into numbers so to speak) and that's that.
What I want to discuss is the geometrization of time.
Can we construct figures/shapes using time?
For instance, we can construct a triangle in space with, say, lengths 3 cm, 4 cm, and 5 cm (this is right triangle if you'd like to know). Likewise, can we construct a time-triangle with lengths 3 minutes, 4 minutes, and 5 minutes? The same goes for other spatial geometric figures? Can we construct a time-circle with a radius of 2 hours and a circumference (2*pi*r) of 4pi hours?
It seems that we can. If I fix the speed of my pen at 1 cm/sec, and I draw a triangle with lengths 3 cm, 4 cm, and 5 cm, that would be a time-triangle of 3 seconds, 4 seconds, and 5 seconds. Now, if I double my speed, the space-triangle with lengths 3 cm, 4 cm, and 5 cm remains unchanged but my time triangle is scaled down by a factor of half. Likewise if I halve my speed, the space-triangle (3 cm, 4 cm, 5 cm) stays the same but my time triangle is scaled up by a factor of 2.
Lastly, just like our universe's space is thought to have a specific shape, does time have a shape of its own?
It's not that difficult to understand where Kant was coming from. There's a very basic link between space and geometry - space is filled with objects of all shapes and sizes. As for time, it's obvious that we use arithmetic (numbers) to make sense of its passing.
Now, it's clear that geometry can be/has been subjected to arithmetization - coordinate geometry, and we measure the dimensions of various geometric shapes with numbers. The converse must be true too as certain relationships between numbers can be geometrized e.g. there are such things as "triangular" numbers.
Space, being about shapes, its own and that of its contents, has also been arithmetitized (translated into numbers so to speak) and that's that.
What I want to discuss is the geometrization of time.
Can we construct figures/shapes using time?
For instance, we can construct a triangle in space with, say, lengths 3 cm, 4 cm, and 5 cm (this is right triangle if you'd like to know). Likewise, can we construct a time-triangle with lengths 3 minutes, 4 minutes, and 5 minutes? The same goes for other spatial geometric figures? Can we construct a time-circle with a radius of 2 hours and a circumference (2*pi*r) of 4pi hours?
It seems that we can. If I fix the speed of my pen at 1 cm/sec, and I draw a triangle with lengths 3 cm, 4 cm, and 5 cm, that would be a time-triangle of 3 seconds, 4 seconds, and 5 seconds. Now, if I double my speed, the space-triangle with lengths 3 cm, 4 cm, and 5 cm remains unchanged but my time triangle is scaled down by a factor of half. Likewise if I halve my speed, the space-triangle (3 cm, 4 cm, 5 cm) stays the same but my time triangle is scaled up by a factor of 2.
Lastly, just like our universe's space is thought to have a specific shape, does time have a shape of its own?
Comments (17)
Even if it Kant's position on the matter were misrepresented by me, it doesn't seem to invalidate the general idea that space is geometry and time is arithmetic.
Not even a line can represent time. It's non-dimensional, simply order.
Lorentz Factor
Now that I think of it gravity is proven to affect space, classicaly depicted with massive objects producing dips and dimples in the fabric of space and it's also scientifically proven that mass can cause time dilation and it seems plausible that time dilations can be explicated as mass bending/curving time but that's only a hunch.
Gravity affects spacetime not ""space". When you are talking about things like this within that conceptual framework, it is not appropriate to speak of space and time separately. That is because there is no real method to separate what time is doing, from what space is doing.
That's like saying economic policies affect the nation but not the people.
Thought of that but I'm no mathematician.
Within that conceptual framework which produces these forms, "the dips and dimples" are in the fabric of spacetime, not space. Suppose we have two objects, and assume a straight line between them in a classical 3d representation. It's impossible to measure that distance all at the same time, because measuring takes a period of time. This amount of time is determined by the speed of light, the fastest known way of measuring. But the light, in travelling from the one object to the other will really take a non-straight path due to the influence of gravity. Since light is the fastest thing to travel from the one object to the other, the shortest path between the two, is represented as this non-straight path. That is called the curvature of spacetime.
There is a standard for creating this curvature in models, which is based in the presence of mass, and assumed gravitation. However, some activities will affect the applicability of the standard, so exceptions to the standard curvature must be allowed for. These are things like gravitational waves. What this implies is that light doesn't actually take the shortest route between two objects. We think that it does, because classical representations of 3d space will show light travelling in a straight line. And, since physicists know light as the fastest thing to travel that distance, they equate the 4d fastest route with the 3d shortest route, hence the 3d straight line is equivalent to the 4d curved line. However, since other things are known to alter the shape of this "fastest" route, so there are proven wrinkles in spacetime, we can conclude that it is not really the shortest route.