Tensor manipulations One final thing to learn about tensor manipulation is that the metric tensor is what allows you to raise and lower indices That is for example v g v where again we use the summation convention Similarly v g v where g is the matrix inverse of g g g where is the Kronecker delta 1 if 0 otherwise Another special tensor is the Levi Civita tensor This tensor is defined as being completely antisymmetric In flat spacetime 0123 1 if 0 is the positive time direction and 123 is a right handed set of spacetime basis vectors e g xyz Then 1 if an even permutation of 0123 1 if it is an odd permutation and 0 if any two indices are the same In curved spacetime we define the metric determinant g det g 0 Then the Levi Civita tensor is g 1 2 where is 1 for an even permutation 1 for an odd permutation and zero if two indices are equal as before Note that the Levi Civita tensor may be familiar from cross products A B ijk Ai B j Spacetime and Metrics Now let s get a little more concrete which will eventually allow us to introduce additional concepts Let s concentrate on one particularly important geometry the Schwarzschild geometry aka spacetime It is very useful not least because it is more general than you might think It is the geometry outside of i e in a vacuum any spherically symmetric gravitating body It is not restricted to static objects for example it is the right geometry outside a supernova if that supernova is good enough to be spherically symmetric To understand some of its aspects we ll write down the line element i e the metric in a particular set of coordinates However the coordinates themselves are tricky so let s start with flat space Ask class what is the Minkowski spacetime in spherical coordinates ds2 dt2 dr2 r2 d 2 sin2 d 2 1 Ask class what is the meaning of each of the coordinates this is not a trick question Everything is as you expect and are the usual spherical coordinates r is radius t is time In particular if you have two things at r1 and r2 same t and then the distance between them is r2 r1 No sweat You could also say that the area of a sphere at radius r is 4 r2 Well why all this rigamarole It s because in Schwarzschild spacetime things get trickier Now let s reexamine the Schwarzschild line element ds2 1 2M r dt2 dr2 1 2M r r 2 d 2 sin2 d 2 2 Ask class are the meanings of and changed No they re the same as always This is guaranteed by the assumption of spherical symmetry that comes into the Schwarzschild derivation But what about r Ask class suppose dt d d 0 What is the proper 1 2 distance between r1 and r2 In this case ds grr dr so the distance is D Z r2 r1 1 2 grr dr Z r2 r1 1 2M r 1 2 dr 3 That means that if r1 2M for example the radial distance measured is rather different than in flat space But if you calculate the area of a sphere of radius r you get 4 r 2 as usual and the circumference of a circle is 2 r as usual This is one of the most extreme geometric indicators of the curved spacetime circumference diameter drops like a rock What about time Ask class what is the relation between proper time d at r and the coordinate time dt if dr d d 0 d 2 ds2 gtt dt2 d 1 2M r 1 2 dt Therefore as r 2M the elapsed proper time is tiny compared to the elapsed coordinate time It turns out that t the coordinate time is the time as seen at infinity Therefore to a distant observer it looks like an object falling into the horizon takes an infinite time to do so This is the origin of the term frozen star used by many until the 1970 s for black holes You might think then that if you were to look at a black hole you d see lots of frozen surprised aliens just outside the horizon You actually would not but more on that later Conserved Quantities in Schwarzschild Spacetime Let s take another look at the Schwarzschild spacetime It is spherically symmetric It is also stationary meaning that nothing about the spacetime is time dependent for example the time t does not appear explicitly in the line element Now in general in physics any time you have a symmetry you have a conserved quantity Ask class for a particle or photon moving under just the influence of gravity what are some quantities that will be conserved in the motion of that particle As with any time independent central force energy and angular momentum will be conserved These follow from respectively the symmetry with time and the symmetry with angle In addition the rest mass is conserved more on that later We have three conserved quantities and four components to the motion so if we knew one more we d be set Luckily having spherical symmetry means that we can define a plane of motion for a single particle so we only have three components to the motion in particular we might as well define the plane of motion to be the equatorial plane so that 2 and we don t have to worry about motion in the direction That means we get a great boost in following and checking geodesic motion in the Schwarzschild spacetime from the conserved quantities Test particle Here we pause briefly to define the concept of a test particle This is useful in thinking about the effect of spacetime on the motion of objects A test particle is something that reacts to fields or spacetime or whatever but does not affect them in turn In practice this is an excellent approximation in GR whenever the objects of interest have much less mass than the mass of the system this applies for example to gas in accretion disks We can now think some more about four velocity This is a good time to show a useful technique which is the computation of quantities in a reference frame where they are particularly simple We ll start with the four momentum p From special relativity you know that the time component is E and the space component is the ordinary 3 momentum call it P Let s consider the square of this four momentum p p g p p Assume we have gone into a local Lorentz frame so that g Then p2 p p E 2 P 2 Of course we can boost into another Lorentz frame in which the particle is not moving assuming it s not a photon Ask class what s the square of the four momentum then In that frame P 0 so the square is just E 2 But then the total energy is just the rest mass energy …