so how can the photon have a velocity without time?
velocity is measured as as distance over time
No, it isn't. It's measured by the square root of the squared sum of it's components.
I was tutoring an undergrad in physics and covering the basics and recalled this thread. I shouldn't have provided the above answer, or rather I shouldn't have only provided that answer (which is how one calculates the magnitude of any vector (despite there close affinity and how much each has influenced the other math and physics are still different and I always tend to think about the math first; the fact that in modern physics often that's all we have, at least in a certain sense, doesn't help).
In classical physics, you start with a position vector. I'm going to use some notation I rather hate, and call define the position vector parametrically:
r=
x(
t)
i+
y(
t)
j+
z(
t)
k.*
The velocity vector
v is then defined
r'=
x'(
t)
i+
y'(
t)
j+
z'(
t)
k, or the derivative of the position vector. As long as we're talking about motion in 3D space, this holds true. Usually, however, dynamical systems (systems that change over time) are represented mathematically by what is called the phase space of the system, and frequently systems that we think of as existing in 3D space have a 1D phase space or a 6D phase space or an
n-dimensional phase space. Also, in relativistic physics, even the physical interpretation of space changes to 4D spacetime.
*The reason I hate this representation and any like it (any that use the "unit" vectors
i,
j,
k) is because for those of us who took linear algebra and studied vectors in Rn rather than R3 (i.e.,
n-dimensional space, where
n could be 10, 100, 55,456, etc., and R3 is the 3-dimensional space we experience). There is no point in defining vectors of length 1 in any particular dimensional space, and thus unit vectors are represented by one symbol (often
u, and whatever notation- such as an arrow over the u- might be used to indicate it is a vector). This is more compact, and considerably more powerful: it can be equal to any of the 3 unit vectors above, or any unit vector at all. The vectors
i,
j,
k, however, can only be unit vectors in R3 and lose all meaning in any other space (or they have to be re-defined).