General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29

ecture Relativistic Mass The mass of an object moving at a velocity v is (Ch 4.3) Here, m o is the mass you would measure if the object were at rest with respect to you. This is known as the rest mass.

ecture Relativistic Energy Expanding out the expression for relativistic mass we get So in the low velocity limit this consists of a classical kinetic energy, as well as a relativistic rest mass energy. The total energy of the particle is given by

ecture Relativistic Momentum The relativistic momentum is given by It is easy to verify that suggesting that the quantity m o c 2 is a Lorentz invariant.

ecture Energy & Momentum The 4-vector (E,p) transforms the same way as (ct,r) using the same Lorentz transformations.

ecture Photons Photons possess no rest mass, and so the invariant quantity is Suppose a source in S’ emits a photon with energy E o and momentum –p x. The energy as seen in S is Comparing to the Doppler eqn

ecture Minkowski Space-time (Ch 8) We can define new coordinate quantities The Minkowski line element can then be written with

ecture Space-time In a more general coordinate system where g ab is the metric. For spherical polar coordinates

ecture Einstein Summation Einstein decided to drop the summation signs so now The summation rule is “if there are a pair of identical indices, one down and one up, then these are to be summed over”; e.g. but not

ecture Raising & Lowering The metric can lower an index e.g. Note, in general We can also define g ab ; This raises an index

ecture Raising and Lowering So, for the interval The interval is equivalent to an inner product (and in general relativity it is an inner product in a curved space-time). Two vectors are orthogonal if

ecture The Null Cone The norm of a vector is defined by In relation to the null cone (Fig 8.1)

ecture Past & Future We can define a unit time vector For timelike and null vectors It is important to note that the Minkowski coordinates (t,x,y,z) are just “there”, with time having the same properties as spatial points. There is nothing in relativity that compels one time coordinate to advance to the next.

ecture 4-Vectors Remembering the definition of the proper time  we can define several important 4-vectors. where L is the Lagrangian.

ecture General Relativity (Ch 9) Before examining the mathematical properties of General Relativity, it is important to consider the physical principles that led Einstein to its formulation. These are Mach’s Principle The equivalence principle Principle of minimal gravitational coupling The correspondence principle

ecture Mach’s Principle With accelerated motion we see inertial forces which are not present in an inertial system e.g. Coriolis, centrafugal. Newton’s bucket experiment “reveals” these forces. Mach argued that rotation is relative to all other bodies in the universe, and without them we could not tell if a bucket of water was rotating (i.e. the surface would not curve). Mach’s Principle: The mass in the universe causes all inertial forces. In an empty universe, there would be no inertia. (Note, many are not comfortable with Mach’s Principle)

ecture Mass in Newtonian Theory Mass has three different definitions in Newtonian theory; Inertial mass: the resistance to change in motion i.e. F=m I a Passive gravitational mass: the response to a gravitational field through F=-m P r  Active gravitational mass: the source strength for producing a gravitational field with  = -Gm A /r In Newtonian theory (p ) it can be shown m I =m P =m A and we can simply talk about one mass m.

ecture Equivalence Principle Gravitational test particle: responds to a gravitational field, but does not alter it. The equivalence principle can be stated as “the motion of a test particle in a gravitational field is independent of its mass and its composition”. More accurately “there are no local experiments which can distinguish nonrotating freefall in a gravitational field from uniform motion in space in the absence of a gravitational field”.

ecture Equivalence Principle Einstein argued that gravity can be regarded as an inertial force. “A frame linearly accelerating relative to an inertial frame in special relativity is locally identical to a frame at rest in a gravitational field”. Einstein explored this idea in a famous thought experiment involving lifts and rockets (Fig ).

ecture Equivalence Principle Adopting the equivalence principle results in two immediate observational effect; Light should be blue/redshifted in a gravitational field Light paths in a gravitational field should be curved This can be seen when considering the path of a light beam through an accelerating rocket. Note therefore that these two tests of “relativity” are really tests of the equivalence principle!

ecture Covariance & Correspondence Einstein argued that all observers equivalent, including accelerated observers, and should be able to discover the laws of physics. General covariance states that “the equations of physics should be invariant under (tensorial) transformations between coordinate systems.” The correspondence principle states that general relativity should tend to Special relativity for vanishing gravitational fields Newtonian gravity for weak fields and low velocities

ecture Physical Theories Before working with the mathematics of general relativity, we have to be sure we know what we physically want from it. In Newtonian theory, we specify a set of initial conditions (typically positions and velocities) and use the differential equations from the relationships between force and motion to predict where an object will be at a later date. There are a number of ways of deriving these differential equations (Lagrangian mechanics, Hamiltonian mechanics). We basically want the same from relativity.

ecture Geodesics in SR An (affine) geodesic is a curve along which its own tangent vector is propagated parallel to itself (i.e. the straightest possible curve Ch. 7.6). For Minkowski coordinates: for some (affine) parameter u, and the tangent vector satisfying

ecture Geodesics in SR For a non-null path, we can choose the normalization of u appropriately and the geodesic’s tangent vector can have unit length without loss of generality. With this, the (affine) parameter is the proper time  (for a time- like particle). In units with c=1, d  2 = ds 2, while in standard units, d  2 = ds 2 /c 2, so;

ecture Geodesics in SR Consider an observer with a velocity v and so c.f. the clock hypothesis we saw earlier.

ecture Geodesics in SR If we use non-Minkowski (but flat) coordinates (ie spherical polar coordinates) then we can write the metric geodesic equation for a time-like path as where  a bc (the affine connection) is related to the non- Minkowski metric g ab. It is important to remember that the above geodesic equation represents four coupled differential equations.

ecture Axiomatic SR Axiom 1: Space and time are represented by a four dimensional manifold with a metric (tensor) and symmetric affine connection  a bc. The manifold is flat and along any time-like world-line, the parameter  is defined to be d  2 = g ab dx a dx b. Axiom 2: There exists a privileged classes of curves in the manifold: (i) ideal clocks travel along time-like curves and measure , the proper time; (ii) free particles travel along time-like geodesics; (iii) light rays travel along null geodesics.

ecture SR in a nutshell Free particles and light rays travel along geodesics given by with the normalization; For time-like geodesics, = . As the proper time is not defined for a photon, for null geodesics simply parameterizes the path.

ecture The Affine Connection The affine connection (or Christoffel symbol) is determined from the metric and is given by where Note that for Minkowski coordinates,  ab,  a bc =0