Give an example of a situation where you might have a 4-dimensional function. Give an example of a situation where you might have a 3-dimensional function. What does 4 dimensional geometry have to do with 4-dimensional spacetime?ģ. When a physicist says we live in 4-dimensional spacetime, what dimensions are they talking about?ī. Some questions and answers we didn't make it to this semester:Ģ. I'd need 4D graph paper to graph this equation, you don't get a picture here, but if you knew some 4D math you could find out things like: what different masses an r's give you the same force.Ī: Well, if you have a curved spacetime-like around a gravity well-then there's math that will help you figure out the straightest path a light wave could take going near the gravity well. That's 3 variables: A, H and W, so it's a 3D problem, and gets graphed in 3D:į is the force due to gravity between two objects, M is the mass of one object, m is the mass of the other object, and r is the distance between them. Is the (approximate) surface area of a human body, if the person has height H (in cm) and weight W (in kg). If you want to graph the equation, you need a 3D graph How fast (v=velocity) sound travels through an ideal gass, when it has pressure p and density d Most of the things you are used to are 2D problems, so first I'll give you an example of 3D problems: Sure-anytime anyone has a math equation with 4 variables, it's a 4D problem. So 4D geometry helps theoretical physicist figure out relativity problems in spacetime.ĤD math helps theoretical physicists figure out things like: if a light wave goes near a gravity well (a place where spacetime is curved), what path will it take (it always takes the straightest path possible, but going straight on a curved surface is hard to compute).Ī. (OK-that's simplifying a lot-those physics equations are really pretty complex and different, but they needed the simple 4D math stuff to get it started). Once you have the 4D geometry, then you can apply it to spacetime physics-all you need is the right equations.
(My brother, who is also a mathematician, figured out a formula for the volume of an n-dimensional sphere during church one day using calculus, so going from 3 dimensions to any dimension you want took less than an hour). Geometry is simpler in many ways than physics is, and the math is easy to apply and find analogies for, once you understand it well enough (4th dimensional calculus isn't much harder than 3rd dimensional calculus), so figuring out math tools to help you measure something in a geometric 4th dimension isn't too hard. So what good is doing geometry in 4 dimensions, if you don't have 4 spatial dimensions?Ī. Any time you use 4 variables to describe something, it's a fourth dimensional problem (spacetime: (x, y, z, t)). If I want to specifically locate something, I need to say not only where it is, but also when it is there. Time is rather different than a spatial dimension, because we can't go backwards in time, but it is another degree of freedom. Physicists say that spacetime is four dimensional: 3 spatial dimensions+1time dimension. If you'd be satisfied with a less geometric 4th dimension, then the answer is yes.
You're not going to find a 4th dimension that you can touch and perceive directly, but someone might be able to figure out if there's one out there even though we can't see it or touch it.
There are theoretical physicists who think there is, and others who think there isn't, and so far nobody knows who is right. If you want a 4th spatial dimension, the jury is still out.
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