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8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics

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8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics 8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics Lectures by: Alan Guth Notes by: Andrew P. Turner May 26, 2017 1 Lecture 1 (Feb. 8, 2017) 1.1 Why general relativity? Why should we be interested in general relativity? (a) General relativity is the uniquely greatest triumph of analytic reasoning in all of science. Simultaneity is not well-defined in special relativity, and so Newton's laws of gravity become Ill-defined. Using only special relativity and the fact that Newton's theory of gravity works terrestrially, Einstein was able to produce what we now know as general relativity. (b) Understanding gravity has now become an important part of most considerations in funda- mental physics. Historically, it was easy to leave gravity out phenomenologically, because it is a factor of 1038 weaker than the other forces. If one tries to build a quantum field theory from general relativity, it fails to be renormalizable, unlike the quantum field theories for the other fundamental forces. Nowadays, gravity has become an integral part of attempts to extend the standard model. Gravity is also important in the field of cosmology, which became more prominent after the discovery of the cosmic microwave background, progress on calculations of big bang nucleosynthesis, and the introduction of inflationary cosmology. 1.2 Review of Special Relativity The basic assumption of special relativity is as follows: All laws of physics, including the statement that light travels at speed c, hold in any inertial coordinate system. Fur- thermore, any coordinate system that is moving at fixed velocity with respect to an inertial coordinate system is also inertial. The statements that all laws of physics hold in any inertial frame and any frame that moves at fixed velocity with respect to an inertial frame is also inertial are often called Galilean relativity. The statement that the speed of light should always be c might seem somewhat counterintuitive. We can avoid the seeming contradictions that this causes by relaxing our assumptions about how observations in two different reference frames are related. That is, everything is consistent if we change our ideas relating the observations of different observers. The kinematic consequences of special relativity can be summarized in three statements: 1 Lecture 1 8.962 General Relativity, Spring 2017 2 • Time dilation: A clock moving relative to an inertial frame will \appear" to run slow by a factor of γ = p 1 . 1−v2=c2 To demonstrate that time dilation is essential, we can consider the thought experiment of the light clock. Consider a clock that, in its rest frame, consists of two stationary parallel mirrors with a light beam bouncing back and forth between them. By measuring the time it takes the light beam to make one full transit, we can use this setup as a clock. Now consider another inertial frame in which the same clock is moving at velocity v, in a direction parallel to the plane of the mirrors. In this frame, the light pulse moves diagonally from one mirror to the other and back. This path is longer than the path in the rest frame of the clock, and so the clock runs slower as viewed from this frame. This uses the assumption that the speed of light is the same for all observers, as well as the assumption that the separation between the two mirrors is the same in both frames. The latter assumption can be argued by consistency. Consider the case that the travelling clock passed through a similar clock in our rest frame. If the separation changed due to the motion, the moving mirrors would either pass between or around the mirrors in the rest frame; either situation violates the symmetry between the two frames, and so the separation between the mirrors cannot change. • Lorentz{Fitzgerald Contraction: Any rod moving along its length at speed v relative to an inertial frame will \appear"contracted by a factor of γ. There is no contraction along directions perpendicular to the direction of movement. There is another simple thought experiment that allows us to derive the necessity of length contraction. We again consider a light clock, and we now assume that we measure the separation between the mirrors exactly using a measuring rod. We then consider going to a frame in which the clock is moving at speed v in a direction perpendicular to the plane of the mirrors. As we have seen, we know this clock must slow down by a factor of γ due to time dilation. We can then compute the separation of the mirrors in this frame necessary to explain this slowing of the clock, and the result is that the separation must have contracted by an factor of γ. • Relativity of simultaneity: If two clocks that are synchronized in their rest frame are viewed in a frame where they are moving along their line of separation at speed v relative to an inertial v`0 frame, the trailing clock will lag by an amount c2 , where `0 is the rest frame separation of the clocks. Imagine two clocks at rest, separated by a measuring rod of rest length `0, that have been synchronized in the rest frame of the system. Now consider a frame in which this system is moving at a speed v along the direction of separation between the clocks. The clocks will not clock be synchronized in this frame. The time t1 read off of the leading clock will be related to clock the time t2 of the trailing clock by v` tclock = tclock + 0 : (1.1) 2 1 c2 How can we synchronize the clocks in their rest frame? If we know the separation `0 between the clocks in the rest frame, we can emit a light pulse at time t1 from the first clock. When the light pulse reaches the second clock, we set the second clock to read time ` t = t + 0 : (1.2) 2 1 c Lecture 1 8.962 General Relativity, Spring 2017 3 Now consider the same process in a frame where the system is moving at speed v along the direction of separation between the two clocks. In this frame, the separation between the 0 clocks is ` = `0/γ. At time t1, the light pulse is emitted at the first clock, when this clock clock 1 0 reads temit . At some later time t2, the light beam reaches the second clock, which is then set to read ` tclock 2 = tclock 1 + 0 : (1.3) rec emit c 0 0 By comparing these times with the times t1 and t2, we can find the amount by which the leading and trailing clocks differ. Here, the word \appear" does not mean what one observer sees with his or her eyes; finding what one observer actually sees requires taking into account the light travel time from the point of emission to the observer's eyes. The word \appears" here means that we have already taken out the effect of this light travel time; we imagine that all of space is filled with observers at rest in the same inertial frame, and the observations are always taken locally and then collected after the fact. 1.3 Notation of Special Relativity We want to be able to describe space and time as a single entity, as they will be mixed by Lorentz transformations. To this end, we introduce four-vector coordinates 0 1 2 3 4 x ≡ (x ; x ; x ; x ) = (ct; x; y; z) 2 R : (1.4) We often denote the components of x with the shorthand xµ, in which µ runs through 0; 1; 2; 3. We will use the convention that Greek letters µ, ν; : : : run over all spacetime coordinates 0; 1; 2; 3, while roman letters i; j; : : : run over only spatial coordinates 1; 2; 3. We will often be dealing with coordinates in multiple frames. The transformations that convert between inertial coordinates are called Lorentz transformations. We will denote such transforma- tions by Λ. The conversion between coordinates is written as µ0 µ0 ν x = Λ νx : (1.5) Here, we are using the Einstein summation notation, in which repeated indices in a single term are always summed over. In particular, 3 µ0 ν X µ0 ν Λ νx ≡ Λ νx : (1.6) ν=0 If we have expressions of the form aµ + bµ, the index µ is not summed over, as it is never repeated in a single term. The symbol x represents a vector in this notation. This means that x has a universal meaning, irrespective of a choice of coordinate system. In our notation, the symbol xµ denotes the coordinates of this vector in the unprimed frame. By comparison, the symbol xµ0 denotes the coordinates of the vector x in the primed frame; note that the prime is on the index µ and not on x, because the components of x change when we change frames but the vector x itself does not. The coordinate basis vectors will be denoted e(µ). Note that µ here is not a Lorentz index; e(0), for example, is itself a four-vector, so we put the index in parentheses to avoid confusion. The vector x can then be written µ x = x e(µ) : (1.7) Lecture 2 8.962 General Relativity, Spring 2017 4 2 Lecture 2 (Feb. 15, 2017) 2.1 Lorentz Transformations In special relativity, we denote coordinate vectors by x. By xµ, we mean the coordinates of the vector x in the unprimed frame, xµ = (x0; x1; x2; x3) ≡ (ct; x; y; z) : (2.1) The same vector can be expressed in another frame, which we can denote with a prime on the index, as 0 0 0 0 0 xµ = (x0 ; x1 ; x2 ; x3 ) ≡ (ct0; x0; y0; z0) : (2.2) µ0 These two inertial frames are related by Lorentz transformations, Λ ν.
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