Bobhawke said:
How can the length of a curve between two points be negative? Sorry I don't know much about any geometry except the good old Euclidean kind
It can't. That was a mistake by me. If we simply replace the scalar product in the standard definition of "length of a curve" with a Lorentzian metric, then the "length" could sometimes be
imaginary, but never negative. However, that's not how these things are usually dealt with. I'll try to explain.
The length (the good old Euclidean kind) of a curve \vec C:(a,b)\rightarrow\mathbb R^n can be defined as
\int_a^b \sqrt{\vec C'(t)\cdot\vec C'(t)}dt
The motivation for this is that the length should be approximately equal to a sum of contributions of the form \sqrt{\Delta x^2+\Delta y^2+\Delta z^2} along the curve, and if we write \Delta x=(\Delta x/\Delta t)\Delta t and similarly for y and z, that square root can be written as
\sqrt{\Big(\frac{\Delta x}{\Delta t}\Big)^2+\big(\frac{\Delta y}{\Delta t}\Big)^2+\Big(\frac{\Delta z}{\Delta t}\Big)^2}\Delta t
which is approximately equal to
\sqrt{v_x^2+v_y^2+v_z^2}\Delta t=\sqrt{\vec v\cdot\vec v}\Delta t
Note that the definition contains a scalar product of two tangent vectors of the curve. In GR, we don't have a scalar product. Instead, we have the metric, which when it acts on two arbitrary tangent vectors at a point in the manifold satisfies all the conditions that a scalar product is required to satisfy,
except the condition \langle v,v\rangle\geq 0 for all vectors v. We can have g(v,v)<0 for some vectors v (for example, any vector in the time direction).
Since the metric is the closest thing to a scalar product we have, the closest thing to a "length" of a curve C:(a,b)\rightarrow M that we can define is this:
\int_a^b\sqrt{\pm g_{C(t)}(\dot C_t,\dot C_t)}dt
where \dot C_t is the tangent vector of the curve C at the point C(t). (Note that there's a dot above C
t. It's barely visible on my screen).
It wouldn't make much sense to use this definition on arbitrary curves. (The "length" defined this way wouldn't be a useful concept). Therefore, we only define the "length" for curves such that g_{C(t)}(\dot C_t,\dot C_t)>0 everywhere on the curve (in which case we use the + sign and call the result of the integration "proper length"), and curves such that g_{C(t)}(\dot C_t,\dot C_t)<0 everywhere on the curve (in which case we use the - sign and call the result of the integration "proper time"). The former type of curve is said to be "spacelike" and the latter type is said to be "timelike".
).
