Please help with 3a and 3b!
Problem 3: Constants of Motion and E= mc2
THE NATURAL MOTION OF N object in Spacetime (i.e., the motion it will follow if not acted upon by outside forces) is along a geodesic, which is a path that maximizes proper time. There are a few different ways to calculate geodesics. In this problem set we use constaufs of motion to describe the paths. Consider first the metric of ordinary flat (Minkowski) spacetime is (considering for simplicity only 1 spatial dimension)
ds2==c2d2+dx2
(3)
Since we will be interested in timelike paths, it makes sense to use the proper time interval dt which will be positive. Recall that by definition d2 = -ds2/c2 so the Minkowski metric can be written
Geodesic Natural motion In fat space
Not a geodesic (acted on by force)
ct
c
(4)
The natural path of an object in flat spacetime is to move at con stant velocity, which is a straight line in the spacetime diagram. As Figure 2 shows, such geodesics have constant slope, and so we can identify constants of motion (i.e., quantities that have the same value at all points on the path)
x
Figure 2: Paths through flat (Minkowski spacetime). The worldline on the left shows motion along a geodesic, which in flat space are straight lines (i.e., moving with constant velocity). The geodesic is seen to have constant slope, such that dx/dr and dt/ have the same value all along the path. The worldine on the right shows motion that is not along a geodesic, and so some external force must have accelerated the object.
dt = constant dr dx constant dr
in flat space)
(5)
in flat space)
(6)
If you want to prove mathematically that these are indeed constants of motion, you can work out the optional problem at the end of this set. Otherwise, we will simply note that from the figure it is apparent that such slopes are constant along the straight line geodesic. Let's get a better feel from the constant of motion given by dt/dr.
3a) )Use the Minkowski metric for dT (Eq. 4) to replace dT and show thaf with some manipulation
dt = dr
(C)
here the Lorentz factor =1//1-/c2 and = dx/dt.
3bhe constant of motion dt/d will be related to the energy of an object. To see this consider the non-relativistic limit c and do a binomial expansion to show that
tp dr
12
(8)