5.8.2 Some Notes on the Physics and the Math (General Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.8.2 Some Notes on the Physics and the Math (General Relativity)
Before we go on to our two examples, I wanted to mention a couple of
points about the mathematics which can be used to develop physics in a
particular space-time.
First, note that for any space-time there is a four dimensional metric
involved. This metric can be used to find the invariant interval between two
space-time points. That interval (recall) can generally be expressed as
(Eq 5:25)
ds^2 = SUM(a & b vary over space and time dimensions) g *da*db
ab
Second, consider a vector in our four dimensional space. Such a vector
(usually called a four-vector) has four components, three relating to space
and one relating to time. Now, in general, the values for these components
will depend on the coordinate system/frame of reference in which you are
considering the vector. However, we can use the metric to act on two
four-vectors to produce an invariant number. In other words, if there are
two four-vectors in a space-time, then two different observers using two
different frames of reference will each find different x, y, z, and t
coordinates which represent those two vectors in their respective frames.
However, when they each act on those two vectors in a specific way using
their own coordinate systems and using their own representation of the
metric, they will each produce the same particular number. The action on the
two vectors is called the dot product of the vectors, and many of you may
have heard of and used it before (though perhaps you didn't realize you were
using the metric--if you have ever had to remember how to produce a dot
product in polar coordinates, then you have seen how the metric in that
coordinate system affects the way you produce the dot product).
So, consider two four vectors, U and V. Remember that these are simply
tensors with either contravariant or covariant components. Now, we can
produce the dot product of U with V as follows.
(Eq 5:26)
a b
U (dot) V = SUM(a,b) g *U *V
ab
This produces a frame invariant number (a scalar), and if U and V have
particular physical properties in space-time, then we can use the dot
product to produce frame invariant physical rules in a particular
space-time.
For our third note in this section, let's discuss the time between two
events. It will be useful for us to find a frame-independent way of
expressing that time. To explore this a bit, consider an observer who is not
being acted on by any forces other than gravity. Because of gravity, he will
simply follow a geodesic through space-time--being at certain points in
space at particular times. Now, consider two events which each occur at the
position of our observer, but which occur at two different times on our
observer's clock. For such events, the time on the observer's clock which
ticks off between the two events is called the "proper time" (T, though it
is usually denoted using the Greek letter "tau") between those two events.
The time this observer reads on his clock does not depend on what any other
observer sees or does, and T is therefore a frame-invariant way of
specifying a time between two such events. Of course, the time as measured
in other frames will be different from T, but every frame will agree that
for the one, unique observer who naturally follows space-time curvature to
be at the position of both events, T is the proper time which he measures on
his clock.
We should note that not all events can be connected by the natural
space-time path of an observer because no observer can travel faster than
light in that space-time. Any two events which can be connected by an
observer's natural space-time path are called "time-like separated", and T
can easily be defined for such events.
Now, consider the invariant interval for some observer's space-time
path between two particular points. Remember that in general the invariant
interval is a function of your position in space-time. Thus, as soon as you
start moving down a path, the invariant interval begins to change. We
discussed this fact briefly in Section 5.5 and decided that we would deal
with it by breaking up the path into small bits and consider the invariant
interval at each bit. Therefore, rather than discuss the entire interval
between the two events, it is better to consider just one point along our
observer's path and look the infinitesimal (ds) at that point. That
infinitesimal in four dimensional space-time is generally made up of an
infinitesimal change in space and an infinitesimal change in time. However,
remember that for the observer and the two events we are considering, both
of the events occur right at the observer's position. So, for him there is
no spatial distance (dx' = 0, dy' = 0, and dz' = 0) between any two points
on the path. Therefore, the invariant interval at any point on his path as
calculated using his coordinates must be made up of only changes in his time
coordinate (dt'). Thus, the value of the invariant interval at some point on
the observer's path is given totally by the infinitesimal change in the
proper time (dT = dt', the infinitesimal change in time on our observer's
watch). We can therefore write the following (taking the spatial components
out of Equation 5:25):
(Eq 5:27)
ds^2 = g *dT^2
t't'
Notice that the component of the metric tensor in the above equation is
expressed in the coordinates of the observer we are considering (i.e. we are
specifically using t' and not t). This must be the case, because it is only
when we measure the infinitesimal invariant interval (ds) using his
coordinates that we can disregard any spatial component and write the
interval totally in terms of dT. However, since this observer is free
falling (only being acted on by gravity), then recall that his local
space-time is flat, regardless of the global geometry of the space-time he
is in. Thus, for small distances in space and time in his coordinate system
(i.e. for infinitesimals like dt') his space-time can be considered to be
that of special relativity (flat space-time). We will find out in the next
section what g_tt is for the flat space-time of SR, and when we plug this
into Equation 5:27 we will find that
(Eq 5:28) dT^2 = -ds^2/c^2.
That equation is true for any space-time, because the space-time of the
observer is locally flat regardless of the global geometry of the space-time
we are considering.
So, how will this help us with the physics? Well, specifically, this
gives us a way to define the momentum of an object in any space-time.
Consider a free-falling object of mass m. In some coordinate system, the
object's position in one coordinate (say "a") can be changing. Note that "a"
could be x in an x-y-z coordinate system, r in polar coordinates (which we
will discuss later), etc. Now, as the object changes spatial coordinates in
this system, it will follow a natural geodesic path through space-time. As
the object's position in "a" changes by some infinitesimal amount (da) its
own "clock" will tick off some small time (dT--note that this is a proper
time because it is measured on the clock of the object itself). In that
case, the "a" component of the momentum for that object in this coordinate
system will be expressed as
(Eq 5:29) a p = m*da/dT
Notice that if we consider the situation where "a" is the time coordinate
itself in our system, then we have a sort of "temporal momentum" who's
significance will be discussed in the next section. Thus, p^a actually has
four dimensions, and is, in fact, a four-vector. Combine this with our
discussion of four-vectors above, and we will find some useful physics, as
we will see in the following examples.
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