by **jinydu** » Thu Mar 30, 2006 8:42 am

Q: What is a dimension?

A: Mathematically, the dimension of a vector space is the number of vectors in a basis for that vector space. That is, the dimension is the minimum number of vectors {v1, v2, v3 ... vn} such that any vector, v, in the space can be written in the form

v = c1v1 + c2v2 + ... + cnvn

where c1 ... cn are scalars

Q: Is it possible to study 4D space (and higher dimensional spaces) without being able to visualize it?

A: Yes; it can be done solely through mathematical reasoning. There are already many theorems about higher dimensional spaces in calculus, geometry and topology.

Q: Why is it not possible to reach the speed of light?

A: According to Einstein's Special Theory of Relativity, the energy of a particle (in a particular reference frame) is:

E = sqrt((mc^2)^2 + (pc)^2)

where m is the particle's rest mass, p is its momentum and c is the speed of light

Furthermore:

p = gamma * m * v = (1-(v/c)^2)^(-1/2) * m * v

As v -----> c, gamma -----> infinity, so p -----> infinity and thus E -----> infinity.

Thus, an infinite amount of energy is needed to accelerate a particle to the speed of light.

Caveat: This argument does not take into account potential energy. But unless you can find a way to make potential energy decrease by an infinite amount, the argument still holds.

Q: Is it true that according to relativity, when your speed approaches the speed of light, time slows down, lengths contract and your mass increases?

A: Special Relativity is mainly concerned with two things:

1) Determining which quantities are the same in all inertial reference frames.

2) Given a description in one inertial reference frame, determining what happens in another reference frame.

The second objective is done by expressing everything in the original frame in terms of four-vectors, and then applying what is called a Lorentz transformation (i.e. multiplication by a 4x4 matrix of a particular form).

Time and length are quantities that vary from one reference frame to another. Suppose we have a clock and a ruler, with no relative motion between the two. Let S be a reference frame where the clock and ruler are stationary. Suppose that in the reference frame S, the period of the clock is T and the length of the ruler is L.

Now suppose we have another reference frame, S', moving parallel to the ruler, with a speed v with respect to S. What is the period of the clock (T') and the length of the ruler (L') according to S'? According to Special Relativity, the answer is:

T' = gamma * T

L' = L/gamma

where gamma = (1-(v/c)^2)^(-1/2) [as implied in the answer to the previous question]

Thus, clocks run slow and rulers contract in a reference frame where the clock and ruler are in motion. Since all inertial reference frames are equally valid, there is no "true time" or "true length".

Also, the effects are totally symmetric. A clock at rest in the S' frame will appear to run slow to an observer in the S frame.

Q: So that means that everything is relative?

A: No. As hinted in the first goal of special relativity, there are quantities which are the same in all inertial reference frames. Examples are

c [the speed of light]

(ct)^2 - x^2 - y^2 - z^2 [called the spacetime distance]

m [rest mass]