PoS Notes - Physics Notes PDF

PoS Notes Bikram Phookun and Anugraha Arun January 2024 1 Week 1 1.1 Introduction The course will look at how scientists attempt to answer questions that arise in the field of astronomy and examine th...

PoS Notes Bikram Phookun and Anugraha Arun January 2024 1 Week 1 1.1 Introduction The course will look at how scientists attempt to answer questions that arise in the field of astronomy and examine the principles of science that we encounter through this. We primarily address three concepts: gravity, space and time. We will go through four questions dealing with exoplanet detection, the special theory of relativity, the evolution of the universe as a whole and black holes. You will be introduced to the scientific way of thinking - making observations, detecting patterns, developing an understanding and corroborating said understanding with the observations. 1.2 Motion Motion is a change in position of an object relative to some reference points – a frame of reference. When these reference points are markings on some ruler, they are called coordinates and we use them to describe position. The dimension of a space (or object) is the number of coordinates needed to fix a position within it. The changing coordinates of an object w.r.t the chosen observational frame describe motion. To describe motion, we need the time ’coordinates’ along with the position coordinates. This can be illustrated in a nice way using graphs. 1.2.1 Uniform motion in 1D Uniform motion in 1D means moving in a straight line covering equal distances in equal intervals of time, for example when you walk from one end of the classroom to another at a uniform speed. Take a look at the position v. time graph for this motion. Figure 1: Position v. Time This graph also gives you information on the velocity of the object – the slope tells you the rate at which the object is moving. Calculating velocity: x2 − x1 v= t2 − t1 (1) ∆x = ∆t 1 here (x2 , t2 ) and (x1 , t1 ) are any two points on the graph and ∆ is used to indicate a change in the variable. In this case the velocity is constant. We cannot use the v-t graph to obtain the x-t graph as v-t doesn’t specify a unique position at some given time. For a given v-t graph, we can plot infinite corresponding x-t graphs that are parallel to each other, since the v-t graph gives us the slope of the graph, but not the position at a given point in time. We can fix an x-t graph by assuming a position at some time and choosing the line that passes through it. We can use the v-t graph to obtain acceleration - the rate of change of velocity, similar to how we used the x-t graph to obtain velocity. Here the acceleration is zero as velocity does not change. Figure 2: Velocity v. Time Figure 3: Acceleration v. Time Try this: Draw the x-t lines for a stationary object at the origin, to the left of the origin and the right of the origin. Draw these lines on the same graph. What do their v-t lines look like? What about the a-t lines? For motion in 1D, what will the v-t, x-t graphs look like given a constant non-zero acceleration? Since the relationship between acceleration and velocity is analogous to the relationship between velocity and dis- tance, use what you already know about the x-t graph for a constant non-zero velocity to find the possible v-t graphs. After fixing a v-t graph, break it down into increasing little pieces of uniform velocity and plot the corresponding x-t curve. The resulting curve should look like half of a parabola. Draw the x-t, v-t and a-t graphs for a ball dropped to the floor that keeps bouncing up to the same height. Assume that the starting height is the origin and the floor is in the negative direction. 2 1.2.2 Uniform motion in 2D Now we need two coordinates to describe the position but the motion is still in a straight line at a constant speed, for example if you walk diagonally from one corner to the opposite corner of the classroom (here the coordinate system is defined such that x and y axes are parallel to the walls of the classroom). We look at the motion in the two dimensions separately, i.e., there is now an additional set of graphs for motion along the y coordinate, see Figure 5. Figure 4: Position of the object as it moves 3  Figure 5: Position, velocity and acceleration graphs for x and y corresponding to motion described by Figure 4. In Fig 4 we looked at the object in position space (or just space). We can also make a vx -vy plot, this now describes the motion of the object using velocity space. Such plots are called hodographs. In the scenario we described above, the hodograph will be a single point. Now try this: Draw x-t, y-t, x-y, vx -t, vy -t, and vx -vy graphs for: object is moving only along y axis object is moving along x axis with a constant velocity, while it has a constant acceleration along the y axis. Again, use relevant bits from what we know so far. The graphs describing x and y motion separately will look like some of the graphs we worked out before. For the y-x graph - we know that x moves linearly with time, and we know how y changes with time. This translates to x being a stand in for time! So the y-x graph will have a similar shape as the y-t graph, though the curvature maybe different depending on vx. 4 1.3 Newton’s laws Newton laid down fundamental laws of motion in his Philosophiae Naturalis Principia Mathematica. These would serve as the foundation for classical mechanics as we know it now. Using his three laws, Newton described the rela- tionship between the motion of an object and the forces acting on it. The genius of Newton was that while his laws of motion would have been obvious upon observation of a body in deep space where it would not have interactions of any kind, they are not so evident on Earth. 1.3.1 First Law An object isolated from any interaction - for example, in deep space - can only move at a constant velocity or be at rest. This constant velocity cannot be ’felt’, we can only measure it with respect to some reference point. This means when no force is acting on a body, the magnitude of the speed and the direction of the motion remains unchanged. When this does not happen, there must be a reason. This leads us to the second law. 1.3.2 Second Law For a body to accelerate, there must be a force acting on it and the enthusiasm with which it responds is dependent on its mass. 1 ⃗a = F⃗ (2) m So when a table is pulled by a spring, it moves when the force exerted by the spring exceeds the friction exerted by the ground on the table. Physicists try to model such kinds of forces in various scenarios to understand the motion of objects with some acceleration. 1.3.3 Third Law The force that arises between two objects is because of an interaction between them and acts on both objects with the same magnitude but in the opposite direction. This means when you drop a pen, as the pen falls down to meet the earth due to the gravitational force between the pen and the earth, the earth also rises up to meet the pen. But because the earth is much, much, much more massive than a pen, the earth’s acceleration due to this gravitational force is almost negligible. 1.4 Uniform circular motion Now we consider motion in a circle at constant speed and we ask ourselves – what is the acceleration of this motion? What does it depend on (separate the motion from the cause/origin of the acceleration for now)? Since the direction of the velocity changes, the acceleration is obviously non-zero. And we can guess that the acceleration depends on the radius of the circle and the velocity of the motion. We can use dimensional analysis to arrive at a probable form 2 for the acceleration: a ∝ vr. This will not tell us if there are any constants in the relationship. Physicists often use this method to build an expectation of the correct answer before embarking on longer, more drawn out ways to get to the final relationship between quantities. Before we embark on this longer argument, we have to establish some definitions. Vectors When we use a vector to define position, it can be visualised as an arrow originating from the origin of the coordinate system, with the object at the tip of the arrow. Here velocity is given by, ∆⃗r ⃗v = (3) ∆t In this language, the arrow for the velocity vector will shift at each point in the object’s path depending on where the motion leads to and its size is dependent on the magnitude of velocity. Acceleration is given by, ∆⃗v ⃗a = (4) ∆t 5  Figure 6: Vector representation and change in position Radians For a circle, radians is a unit that defines angles as fraction of a circle, 2π rad = 360◦ The length of an arc x subtended by an angle θ for a circle of radius r (see Figure 7) can be given by, θ x = 2π 2πr x = rθ In a total solar eclipse the angular size (θ) of the Moon and the Sun are the same. For motion along a circular path with radius r, the magnitude of the position vector at any given time is the same but its direction is different. Thus while |r⃗1 | = |r⃗2 | = r, r⃗1 ̸= r⃗2 and change in position is, |∆⃗r| = r∆θ (5) To find velocity, we start by calculating the magnitude. |∆⃗r| |⃗v | = ∆t r∆θ (6) = ∆t = rω The direction of this vector |⃗v | will always be perpendicular to |⃗r|. The magnitude of velocity is constant |v⃗1 | = |v⃗2 | = v, since the object is moving at a uniform speed. 6  Figure 7: (a) Radians and arc length (b) Calculating ∆⃗r Now try this: Map the motion in velocity space. Use points that are obvious first - where the circle intersects x and y axes, then points where the x and y velocities are equal (45◦ ). Recall that the magnitude of velocity is constant, which means all the points on the hodograph should be at the same distance from the origin. You will end up with a circle with a radius equal to the magnitude of velocity. 7 2 Week 2 2.1 Uniform circular motion in velocity space To arrive at the acceleration for uniform circular motion, we now consider this motion in velocity space. Here the y and x axes stand for vy and vx respectively. When we map the motion in this space, we will get a circle in velocity space – this circle has a radius of magnitude |⃗v | = rω. Figure 8: Motion in velocity space The rate at which the angle changes in velocity space and in position space is the same for both these circles, and ⃗a ∝ ⃗v (analogous to the relationship we saw between ⃗v and ⃗r). The magnitude of this acceleration is, |⃗a| = vω = rω 2  v 2 =r (7) r v2 = r 2.2 Newton’s law of gravitation We learned that when a body moves at a constant speed at a some fixed distance around a central point, the body 2 possesses an acceleration of the magnitude, a = vr pointing radially inward. From Newton’s laws we can conclude that this acceleration must be due to its interaction with something else, and since the acceleration is inward, the other body must be in the centre and the force that arises due to this interaction is probably dependent on the distance between them. We take Newton’s law of gravitation for granted, m1 m2 Fg = G 2 (8) r12 where G is the gravitational constant. The law of gravity is a fundamental law of physics. The fundamental laws of physics cannot be derived, they must be arrived at by making informed guesses based on observational data. Newton could derive Kepler’s laws using his laws of motion and law of gravity. Kepler set the stage for Newton by performing rigorous data analysis on the extensive naked eye observational data collected by Brahe. Science progresses through guesses like this. We start by observing systems and recording their behaviour, then this data is analysed. We attempt to make models that best fit our observations - these models are made using 8  Figure 9: Earth and Sun revolving around a common centre mathematical frameworks built on intelligent and profound guesses about the nature of the system using the pat- terns that arise from data analysis. If the real world observations corroborate the predictions of the model, then we presume that the model accurately describes the real world and we make further predictions using this model. As more predictions are corroborated by real observations, we become more confident in our model. 2.3 Orbital motion 2.3.1 Centre of mass Now we will refer back to Newton’s laws of motion along with what we just learned to ascertain relationships between measurable quantities in orbiting pairs of bodies. From Newton’s third law, a force acting on one body must mean an equivalent force acting in the opposite direction on the other body involved in the force generating interaction. Thus, 1 ⃗ a⃗1 = F12 m1 1 ⃗ a⃗2 = F21 m2 This leads to the inevitable conclusion that the central mass must also be accelerating. Since the system as a whole is not accelerating, the Sun must be revolving in the opposite direction as the Earth around a common central point. This point is called the centre of mass of the system. The centre of mass is the singular point where all the mass appears to be concentrated, it is a representative point that describes how a complex mass moves in reality and is defined as, mp rp = ms rs Considering the Sun and the Earth, the total distance between them is rsp = rs + rp , here rs and rp are the distances of the Sun and Earth from the c.o.m. We can also conclude that the time taken by each body to complete its respec- tive orbit must be the same, which means the angle covered by each body in the same amount of time will be the same. ωs = ωe vs vp = rs rp From Newton’s second law and 7, vp2 ms mp v2 mp = G 2 = ms s (9) rp rsp rs For a planetary system, rs ≪ rp , from c.o.m definition, since ms ≫ mp. Thus rsp = rs + rp ≈ rp and the equation (on the left) reduces to, 9  vp2 ms mp mp =G 2 (10) rp rp s Gms vp = (11) rp Thus if we can observe radius and velocity of orbiting bodies, we can use that as a weighing scale to find the mass of a central body. When there are multiple planets, we can constrain the problem better. As expected from the equation, vp2 is inversely proportional to the distance from the Sun. We can often find Figure 10: Orbital velocity vs. Semi-major axis planets in our solar system mass distributions in the universe by studying the motion of rotating objects and working backwards (the actual calculations will be more complex but the basic principle remains the same, we merely dissected the simplest possible scenario.) This is how dark matter was discovered! 2.4 Detecting other planets Detecting exoplanets directly has been difficult as planets are very dim compared to the very bright stars they orbit around and this makes them invisible even to very powerful telescopes. Currently very few of the thousand exoplanets detected are done by direct imaging (used from mid 2000s) - it uses infrared radiation and only cer- tain kinds of planets (huge, very bright) can be detected using this method. And directly tracking the position of the central star is difficult as mp ≪ ms =⇒ rs ≪ rp. The velocity of the central star will also be small, by the same logic we just used. But this shift is easier to measure as we can use the Doppler shift of light to get this velocity. Doppler shift Doppler shift is the change in measured wavelength/frequency of a signal when the emitting source is moving relative to the observer, along the line of detection. 10  Figure 11: Doppler shift 2.4.1 Radial Velocity method This method uses the motion of the star to detect exoplanets. From what we studied so far, we know that the presence of a planet around a star will cause some periodic change in its velocity (wiggle) while a star without any bodies in orbit will have a uniform velocity. We can measure the component of this velocity directed along the line of sight by translating the Doppler shift of spectral measurements of the star into radial velocity of the star. When a single planet orbits the star, the radial velocity measurements look like a sinusoidal curve. Figure 12: Radial velocity measurements for a star with one planet in orbit from Gorrini et al. (2023) Planetary companions orbiting the M dwarfs GJ 724 and GJ 3988 - A CARMENES and IRD collaboration. The radial velocity method works best for orbits that are more edge-on (plane is parallel to the line of vision i.e., has an inclination of 90°) as this is when the velocity directed away from/towards the observer is maximum. For a face-on orbit, there is no radial velocity, and for arbitrary inclinations in between there is an overall diminution of radial velocity. It is one of the more commonly used methods to detect exoplanets. 11 3 Week 3 3.1 Radial velocity method During the orbital motion of a star, if there is a component of velocity directed along our line of sight we can use the radial velocity method to find the mass of the exoplanet(s) associated with the star. We try to solve for the mass of the planet: Using the second part of Equation 9, ms mp vs2 G = ms (rs + rp )2 rs Taking rs as common factor from denominator on LHS and cancelling out the common factors on both sides, mp vs2 G rp 2 = rs2 (1 + rs ) rs rp From definition of centre of mass, we can rewrite rs , mp vs2 G m = rs2 (1 + mps )2 rs  2 ms From our assumption that ms ≫ mp , the denominator reduces to mp and cancelling out common factors, Gm3p rs vs2 = m2s Using v = 2πrT , where T is the time period of the orbit, we substitute for rs , since we cannot measure the radius of the star’s orbit and we get,   13 T 2 mp = · ms3 · vs 2πG Now our observations only capture the component of velocity along the line of sight, so anything that we derive using this velocity will have an overall dimunition associated with it for any configuration other than the edge-on orbit. This is introduced into our equation by rewriting it using vs obs = vs sin i to get,   13 T 2 mp sin i = · ms3 · vs obs (12) 2πG 3.2 Transit method For planet-star systems which have edge-on or almost edge-on orbits - the planet passes between the star and the detector in the course of the orbit and this causes a dimming in the light detected. These dips in brightness when observed over time tell us about the length of the orbit and size of the planet. The time taken for full planet to be in the star’s light emitting region is captured in the transit curve and this can help us find the size of the planet, and thus its density when used in conjunction with the radial velocity method. The time periods using these two methods should also match. According to the NASA exoplanet archive, majority of the verified exoplanets were detected using transit method. 12 Figure 13: Relative brightness with multiple transits; zoomed in to one transit from https://avanderburg.github.io/tutorial/tutorial2.html We now attempt to introduce additional layers of complexity and see how the radial velocity curve will change. Remember that all these orbits are edge-on since we detect transits. 3.2.1 Elliptical orbit For an elliptical orbit, the center of mass is now at one of the two foci of the ellipse – so this is the fixed point. The star will move faster when it closer to the c.o.m. and slower when it is further away from the c.o.m. 13 Figure 14: Ellipse with major axis perpendicular to line of sight, ellipse with arbitrary orientation. (Note that for both these orientations, we have assumed an edge-on orbit) Assume the ellipse is oriented such that the major axis is perpendicular to our line of sight, and the star is moving away from you when it is nearest to the centre of mass. The radial velocity curve will no longer have an equal crest and trough. Crests will be higher than troughs and it will spend less time near the crests and more time around the troughs (assuming star is moving away from you when it nearest to the centre of mass). There are two asymmetries - the distance to the peak vs trough AND the rate of turnaround from highest point and the lowest point. This behaviour will be more exaggerated as the orbit becomes more elliptical (eccentricity increases). 14  Figure 15: Radial velocity curve for an ellipse with major axis perpendicular to line of sight. In this orientation, the curve will still be symmetric about the line drawn from the peak (or trough) to the x- axis. Figure 16: Radial velocity curve as eccentricity increases (major axis is perpendicular to line of sight) In the real world, the major axis is rarely perpendicular to the line of sight. In any other orientation, the radial velocity curve loses the symmetry mentioned before. 15  Figure 17: Radial velocity for an elliptical orbit with some other orientation 3.2.2 Multiple planets In a system with multiple planets, the radial velocity curve of the star is a more complicated squiggle as it is now the sum of the motions the star is executing in reaction to each of the planets’ motion. Figure 18: Radial velocity data for a two planet star Take a look at Fig 19 to understand how superimposing individual sinusoids gives you a complex squiggle. With a some complicated math (which we will not attempt to do) we can separate out the the individual sinusoids given the complicated curve. 16 Figure 19: Sum of individual sine curves 17 4 Week 4 In this section of the course we will come up with a way to relate measurements made by different observers moving at some relative velocity to each other. We start by stating that one event in one frame of reference is one event in another frame of reference. Here event is a localization in space and time - anything that happens at a specific position at a specific time - and has a unique set of (x, y, z, t) coordinates. Henceforth we only consider motion along the x – axis. We will derive the relationships that allow us to ’translate’ between an event in different frames of reference moving at some constant relative velocity in a common sense framework. 4.1 Galilean transformations Figure 20: S’ moves with a velocity v relative to S At t = 0 and t′ = 0, the origins of the two frames of reference in Figure 22 coincide, i.e., (x, y) = (x′ , y ′ ) = (0, 0). At some other time t = t, t′ = t′ = t (by common sense, time is the same for all observers), the separation between two frames is vt. We arrive at a transformation between (x, t) and (x′ , t′ ): x′ = x − vt t′ = t For an object in frame S’, if the length along x’ is L0 (measured as the difference between its two endpoints), its length along x will be L0 only if both endpoints are measured simultaneously from S. If the endpoints are measured at different times, the difference between endpoints as measured from S will not be the length (L0 in the common sense transformations). Our common sense understanding tells us that time intervals between events are conserved for all observers and length is absolute for all observers. Try working this out for yourself by using the coordinate transformations given above. Now we consider the particle moving with some velocity u as measured from S. We know, ∆x u= ∆t x2 − x1 = t2 − t1 18 The particle’s velocity is u′ as measured from S’. We try to derive its relation to u, ∆x′ u′ = ∆t′ x′ − x′1 = 2′ t2 − t′1 Using the transformations defined previously, x2 − vt2 − (x1 − vt1 ) = t2 − t1 x2 − x1 t2 − t1 Rearrange and take v as common factor, = −v t2 − t1 t2 − t1 ∆x ∆t = −v ∆t ∆t Substituting u and cancelling common factors, =u−v Galilean (or common sense) transformations: x′ = x − vt (13) ′ t =t (14) u′ = u − v (15) Within this framework, when two events are simultaneous for one observer, they must be simultaneous for another observer – simultaneity is absolute. Now let us consider a room of length L0 moving with the frame S’. If a signal is sent out from the middle of the room when the origins of S and S’ coincide, an observer in S’ would witness this signal hitting the right wall and left wall at the same time. From our understanding so far, this necessarily implies that an observer in S should also witness the signal hitting the walls at the same time. (a) Emission (t = t′ = 0) (b) Impact Figure 21: Simultaneity in S (black) and S ′ (red) 19 Let us consider three events: Event 0: Signal is emitted from the centre of the room and the origins coincide Event L: Signal hits the left wall Event R: Signal hits the right wall 4.1.1 Event 0 When the signal is emitted, an observer in S’ sees: x′0 = 0 t′0 = 0 Similarly an observer in S sees: x0 = 0 t0 = 0 4.1.2 Event L When the signal hits the left wall, an observer in S’ sees: u′L = −u′ L0 x′L = − 2 ′ L0 tL = ′ 2u An observer in S sees: uL = −u′ + v L0 xL = vtL − 2 vtL − L20 tL = uL We can manipulate tL and rewrite it, L0 uL tL = vtL − 2 L0 tL (v − uL ) = 2 L0 2 tL = (v − uL ) 4.1.3 Event R When the signal hits the right wall, an observer in S’ sees: u′R = u′ L0 x′R = 2 L 0 t′R = ′ 2u An observer in S sees: uR = u′ + v L0 xR = + vtR 2 L0 + vtR tR = 2 uR 20 We can manipulate tR and rewrite it, L0 uR tR = + vtR 2 L0 tR (uR − v) = 2 L0 2 tR = (uR − v) We know that t′L = t′R = t′ , therefore from absoluteness of simultaneity, tL = tR = t′. Using this information, and our previous manipulation of tR and tL , we know that for this to be true, uR − v = v − uL = u′ has to hold. Luckily for us, this is exactly the relationship that the velocity addition laws give us, uR = u′ + v and uL = −u′ + v. This means that any signal will have a velocity addition associated with it when observed from different frames. But... Our observations tell us that there exists a signal that travels at the same speed for all observers – light. 4.2 Properties of space and time Any phenomena that occur in space-time manifest the properties of space-time. Physics is the description of some of these phenomena, and the properties of space-time are incorporated into the experimental results and theoretical laws that describe these phenomena. Humans construct these laws to describe the world as we know it. These fundamental laws of physics are well defined relationships between physical quantities. Since mathematics is the only language that can concisely and precisely describe these relationships, they are expressed as well defined mathematical equations. We list out the properties of space-time implicit in the physical relationships we just derived – Space is homogeneous – All points are the same. For example, if you were restricted to walking on an infinite plane with no markers, such a plane would be homogeneous. But if you could fly the same space is no longer homogeneous. So an infinite plane is homogeneous in 2D but not in 3D. The same experiment conducted at any point in a homogeneous space will give you the same answer, i.e., no experiment can be designed to differentiate between different points in space. Space is isotropic – All directions are the same. The same experiment conducted in any direction will have the same result, i.e., no experiment can be designed to differentiate between different directions. Time is homogeneous – All moments in time are the same. The same experiment conducted at different times will have the same answer, i.e., no experiment can be designed to differentiate between moments of time. Another property is that absolute velocity is meaningless – all velocities are described in relation to some reference, i.e., they are relative. We cannot ’feel’ motion at constant velocities, we perceive it by using some reference points. We can only ’feel’ when the velocity is changing – acceleration. 4.3 Deriving a new set of transformations Since we ran into a inconsistency with real world observations using our previous set of equations, we must try to come up with a new framework that incorporates this anomalous observation. Einstein applied himself to this task (within the confines of electrodynamics) and came up with the new transformation laws. We will attempt to derive these transformations in our own way. We start by taking the properties of space-time as given, and we assume that one event is one event for all observers. This means the new equations should be linear transformations with the general form of the equations given as, x′ = a11 x + a12 t (16) ′ t = a21 x + a22 t (17) Here the terms aij describe the transformation and we will try to find out what they are. This mapping is unique – every (x, t) in S will be mapped to a unique (x′ , t′ ) in S’. It is also invertible – given (x′ , t′ ) you can find (x, t) using the inverse of these transformations. And since aij cannot vary with location and time, they cannot depend on x or t, but they can depend on v and universal constants. The transformation should allow same speed to be measured by all observers for atleast one speed – u′ = u. And finally these transformations should reduce to the Galilean transformations for the speeds that we usually encounter. 21 5 Week 5 5.1 Deriving the Lorentz transformations We start by stating that one event in one frame of reference is one event in another frame of reference. Here event is a localization in space and time - anything that happens at a specific position at a specific time - and has a unique set of (x, y, z, t) coordinates. Henceforth we only consider motion along the x – axis. The origins of the two frames of reference in Figure 22 coincide at t = 0 and t′ = 0, Figure 22: S’ moves with a velocity v relative to S There must exist a transformation that allows us to ’translate’ between this event in different frames of reference. Using Figure 22, this means, x′ = a11 x + a12 t (18) t′ = a21 x + a22 t (19) Here the terms aij describe the transformation and we will try to find out what they are. This mapping is unique – every (x, t) in S will be mapped to a unique (x′ , t′ ) in S’. It is also invertible – given (x′ , t′ ) you can find (x, t) using the inverse of these transformations. And since aij cannot vary with location and time, they cannot depend on x or t, but they can depend on v. The transformation should allow same speed to be measured by all observers for atleast one speed – u′ = u. And finally these transformations should reduce to the Galilean transformations for the speeds that we usually encounter. We can also find the general form of the inverse transformations: Multiply Equation 18 by a22 , Equation 19 by a12 , a22 x′ = a22 a11 x + a22 a12 t a12 t′ = a12 a21 x + a22 a12 t Subtracting a22 x′ − a12 t′ = x(a22 a11 − a12 a21 ) 1 =⇒ x = (a22 x′ − a12 t′ ) a22 a11 − a12 a21 Similarly, multiply Equation 18 by a21 , Equation 19 by a11 , 22  a11 t′ = a21 a11 x + a11 a22 t a21 x′ = a21 a11 x + a21 a12 t Subtracting a11 t′ − a21 x′ = t(a22 a11 − a21 a12 ) 1 =⇒ t = (−a21 x′ + a11 t′ ) a22 a11 − a12 a21 Thus the inverse transformations are: 1 x= (a22 x′ − a12 t′ ) (20) D 1 t = (−a21 x′ + a11 t′ ) (21) D where D = a22 a11 − a12 a21 (22) Newton assumed that time flows uniformly for all observers – but perhaps this is not true! This was a revolutionary paradigm shift in our understanding of the world, and all because our common sense transformations were proven to be incorrect. We try to derive the terms aij by conducting some thought experiments. Use Figure 22 for the two frames and the associated velocities. Thought experiment 1: We measure the origin of S’ from S. The origin of S’ is x′ = 0, and when measured from S, this lies at x = vt. Substituting this in Equation 18, 0 = a11 vt + a12 t = (a11 v + a12 )t =⇒ a11 v + a12 = 0, since t ̸= 0 =⇒ a12 = −a11 v And Equation 18 becomes, x′ = a11 (x − vt) Setting a11 ≈ 1 for everyday velocities, this will become the Galilean transformation. Thought experiment 2: Flip your point of view – look at the origin of S from S’. The origin of S is x = 0, and when measured from S’, this lies at x′ = −vt′. Substituting this in Eqn 20 1 0= (a22 x′ − a12 t′ ) D −t′ 0= (a22 v + a12 ), =⇒ a22 v + a12 = 0, since t ̸= 0 D =⇒ a12 = −a22 v Comparing to the relationship we got in thought experiment 1, −a22 v = −a11 v = a12 =⇒ a11 = a22 This means Equation 19 becomes, t′ = a21 x + a11 t a21 = a11 (t + x) a11 23 Setting aa21 11 ≈ 0 for everyday velocities, this also reduces to the Galilean transformation. The general transformations now look like, x′ = a11 (x − vt) a21 t′ = a11 (t + x) a11 And, ∆x′ = a11 (∆x − v∆t) a21 ∆t′ = a11 (∆t + ∆x) a11 Upon dividing, ∆x′ (∆x − v∆t) = a21 ∆t′ (∆t + ∆x) a11 1 Multiplying numerator and denominator of RHS by ∆t , u−v u′ = a21 1+ u a11 Under the conditions we set for aa21 11 under everyday velocities, this reduces to the relative velocity under the Galilean transformations. To further progress we need to find aa21 11. For this we do a mathematical manipulation, a21 a21 a12 = × a11 a11 a12 a21 × −a11 v = , using previously obtained relationship a11 × a12 −a21 v = a12 We substitute this back into the equation for u′ to get, u−v u′ = 1 − aa12 21 uv Let’s set aa12 21 = k12 , where k has dimensions of velocity so that the equation is dimensionally correct. When k is very large, this reduces to the Galilean transformation. Forcing u′ = u = c, c−v =c 1 − k12 cv cv c − v = c(1 − ) k2 c2 v c−v =c− 2 k c2 v = 2v k =⇒ k 2 = c2 =⇒ k = c 1 1 a21 This happens when u = c, so we set k2 as c2. Now we can substitute for a11 using the above calculations and our transformations look like, x′ = a11 (x − vt) v t′ = a11 (t − 2 x) c Rewrite the inverse transformations also using the relationships we derived between the coefficients. 24 5.2 New velocity addition law We now have a new velocity transformation: u−v u′ = uv (23) 1− 2 c The inverse of this is: u′ + v u= (24) u′ v 1+ 2 c Try this: Show this mathematically. This is obvious because for an observer in S’, S moves at a velocity of −v. 5.3 Finding a11 To solve for a11 , we go back to the general form of the transformations. Using the relationships between coefficients a21 v we derived last week (a11 = a22 , a11 v = −a12 , − = 2 ), the forward and inverse transformations can be rewritten a11 c as, x′ = a11 (x − vt) v t′ = a11 (t − 2 x) c a11 ′ x= (x + vt′ ) D a11 ′ v t= (t + 2 x′ ) D c Once again, when we map it to the physical assertion the only change should be the measured velocity (of S) for the observer (now in S’) going from v to −v. The inverse relations reflect this expectation, but they have an additional 1 D. Our expectations based on the physical scenario means that, D=1 a22 a11 − a12 a21 = 1   2 a12 a21 a11 1 − =1 a11 a11 v2   a211 1 − 2 = 1 c 1 =⇒ a211 = 2 1 − vc2 1 =⇒ a11 =r v2 1− c2 a12 a21 v Here we once again use the previously derived relationships, a11 = a22 , = −v, = − 2. a11 a11 c We now get the final form of the Lorentz transformations: ∆x′ = γ(∆x − v∆t) (25) v ∆t′ = γ(∆t − 2 ∆x) (26) c where γ = v 1 v2 u u 1− t c2 25 And the inverse Lorentz transformations: ∆x = γ(∆x′ + v∆t′ ) (27) v ∆t = γ(∆t′ + 2 ∆x′ ) (28) c 5.4 Length contraction Consider an object that is at rest in S’ with a length Lo. What will the length, L as measured from S be? Figure 23: Length contraction From our definition of length, when measured from any frame except the rest frame of the object, the endpoint measurements need to be taken simultaneously. Thus if we consider Lo to be ∆x′ and L as ∆x, then ∆t has to be zero. Substituting in Equation 25 that contains these three quantities, Lo = γ(L − v × 0) Lo = γL Lo L= (29) γ Since gamma is always greater than 1, L will always be less than Lo. 5.5 Time Dilation Consider a clock at rest in S’, measures some time interval τo. What will this time interval, τ as measured from S be? Figure 24: Time dilation 26 Since the clock is at rest in S’, ∆x′ = 0. If we further consider τo to be ∆t′ , τ to be ∆t and substitute these three quantities in Equation 28, v τ = γ(τo + 2 × 0) c τ = γτo (30) Since gamma is always greater than 1, τ will always be greater than τo (proper time). We have now derived a more general set of transformations – the Lorentz transformations that reduce to the common sense (Galilean) transformations at in the domain of everyday experience. And we have seen the implications of these new transformations on the measurements we make – any observer not at rest w.r.t to a body will measure its length to be shorter than its proper length (length in rest frame), and any observer who moves between two events, will measure the time interval between these events to be longer than its proper time (time measured by an observer for whom these events occur at the same location). This also means that simultaneity is no longer absolute – two events that occurred at the same time for one observer, need not have occurred at the same time for another observer (δt = 0 ̸=⇒ δt′ = 0.) The other implication of these new transformations is the existence of a limiting velocity in the universe – the velocity of light. 27 6 Week 7, 8 and 9 6.1 Scientific progress Science often translates to either a series of tedious equations and manipulations or a romantic field of magical discoveries and inventions, depending on who you’re asking. In the principles of science course, we attempt to converge these perspectives to some degree. Any scientific problem arises in a specific context - this informs what problems are important and relevant to study. To begin to study the problems, we lay out an appropriate appropriate foundational framework. Within this framework, we make a model specific to the phenomenon we are trying to explain. This model makes predictions that we can use to verify our theory. When our understanding, model and mathematics work in a loop to supplement each other to refine experimental results and draw conclusions, our understanding of the world is furthered. When a model doesn’t align with observations, as a god scientist we should first check our procedure and experimental set up for errors. If everything has been done carefully, we try to edit our model. If all else fails, we address our foundational framework. Such a paradigm shift has enormous ramifications. This is Kuhn’s scientific revolution. We saw this in the first half of the semester when we studied orbital motion in the Newtonian framework. When the observation of the speed of light didn’t fit into that framework, we moved to Einstein’s relativistic framework. This led to a new understanding of spacetime and energy and transformed all of physics. But this doesn’t mean Newton’s laws are abandoned forever – while incomplete description of the world, they still work very well in a lot of situations that we encounter and are easier to work with than the relativistic equations (context matters!). 6.2 The Great Debate A French comet hunter Charles Messier made a catalogue of objects in the sky that frustrated him in his search for comets. This included fuzzy objects later termed spiral nebulae. It took us a long time to identify the objects in this catalogue, and many of these objects are still referred to by their Messier numbers. Theories about space and the nature of the universe have depended on our abilities to observe it. As these abilities got more refined, we draw conclusions about the universe that are consistent with new observations. This led to the central questions of the Shapley-Curtis debate – what is the nature of our galaxy? and what to make of the spiral nebulae in the sky? In 1920, Heber Curtis and Howard Shapley met to deliver lectures supporting their respective positions on the matter. Shapley presented his estimates of the size of the Milky Way and placed our solar system a couple of thousand light years away from the centre of the galaxy, where it was previously believed to have been. He also argued that given the Milky Way (henceforth MW) was much larger than previously assumed, the spiral nebulae were inside our galaxy. He made three major arguments to support this claim – Relative size of spiral nebulae: If we assume the nebulae are objects like the MW, for them to the size we observe in the sky, they must be some millions of light years away, and this was unlikely. Relative brightness of nova: Shapley observed a nova in Andromeda (Messier 31/M31) that briefly outshone the nebula. This would mean a ridiculous, even impossible amount of energy output if the nebula was a distant galaxy. Maanen’s observation of a spiral nebula rotating over a few years: which implied matter with velocities faster than that of light if the nebula was outside the bounds of the MW. Curtis represented the philosophical and scientific school of thought surrounding the idea of ’island universes’ (coined by Immanuel Kant). He used scientific publications to argue that – Dark paths: Photographs of spiral nebulae showed dark paths like those seen in our own MW, thus these nebulae could be distant galaxies. Nova occurrences: He brought up the rate of nova occurences in the MW and in M31 – M31 had greater nova occurrences. If this was part of the MW, why was this tiny portion of the galaxy so special? These led Curtis to the conclusion that the spiral nebulae are not part of the MW, but distant galaxies like the MW. Both of them made important contributions to our current understanding of the universe – Shapley moved the solar system from the centre of the universe and Curtis told us that our galaxy was one among many others in a much larger Universe. 6.3 Cosmic distance ladder To further our understanding of the nature of the universe, it becomes imperative to be able to measure the distances to the things we see in the sky. These distance being on a large range of scales, no single method can be consistently used throughout. Astronomers have built a series of methods referred to as the cosmic distance ladder to address 28 the distance measurement problem. Each rung of this ladder helps us see farther out and builds on the methods that came before it – none of the higher order methods would be useful if they weren’t initially calibrated using the previous methods. Some basic ideas that recur in the ladder include – Standard candle – A class of objects whose intrinsic luminosity is known, which can then be used to measure distance from the apparent brightness. Standard ruler – A class of objects whose size is known, which can then be used to measure distance using apparent size. Both of these involve knowing some intrinsic property of the object being observed, which is difficult when we are looking at things in space. So we need to start with fundamental distance measurement methods – that do not rely on the nature of the object being observed to calculate the distance. In the succeeding sections we will investigate some distance measurements. 6.3.1 Parallax method We can determine distances directly using the parallax method. In this method we use the change in apparent position of a star when viewed from different points – each measurement being made six months apart when the earth is at diametrically opposite points in its orbit for the largest observed shift. In this method stars that are closer to us move more as compared to stars that are extremely faraway (these form the fixed background relative to which we measure the parallax). [Can you explain why?] Look at Figure 25, and connect it back to what we studied about radians and arc lengths in the first section. Using that, if we consider the radius of the Earth’s orbit r as the arc subtended by the distance to star, d sweeping an angle θ (angle A in figure), then we have the following relationship when θ is measured in radians, r θ= d r =⇒ d = θ When θ = 1”, then the distance to the star is 1 parsec. [Calculate this in metres.] Figure 25: Parallax method Here, the parallax method has been demonstrated using the earth’s orbit as a baseline. We can also use other baselines, using the motion of the Sun, and taking measurements separated by many years or even decades to look 29 further out. Since measurements are not made at the same moment in time, we also have to account for the motion of stars themselves. This becomes even more important when you make measurements separated by longer periods of time. The parallax is also limited by the angular resolution of the detector, it can only be used to measure distances to objects that have an apparent shift that is a couple of times the minimum resolution. As our telescopes get better, we are able to use parallax to greater distances. The distance errors that arise as a result of symmetric errors in angle measurement are asymmetric in one direction (systematic error) and more often than not make it appear that that stars are closer than they actually are. 6.3.2 Cepheid Variable stars Cepheids are a class of variable stars that pulsate periodically. The light curve of such a star is shown in Fig 26. Henrietta Leavitt observed that the period of dimming and brightening of Cepheid stars is related to their intrinsic luminosity as shown in Fig 27. The calculations for intrinsic luminosity were done by looking at Cepheid variables in the parallax distance range. But once this relationship was determined, we could now look at Cepheid variables that were further out. And by observing the period (and correctly identifying the class) of these Cepheids, we can determine their intrinsic luminosity and compare that with the observed brightness to determine the distance to them. These were the standard candles used by Hubble in his observations that make up the famous Hubble Law, which states that objects in the Universe are moving away from each other, with things farther away moving away faster. This method is dependent on the distribution of Cepheids – you need to find a Cepheid variable star near/in the object you are calculating distance to. Figure 26: Light curve for a Cepheid variable star (NASA, ESA and Z. Levay (STScI)) 30  Figure 27: Period-luminosity relationship for Cepheids (ATNF) 6.3.3 Type 1a Supernovae Cepheid variables are still stars - so as they get further away they become too dim for us to measure pulsation and use as standard candles. Now we need to look for a standard candle that releases more energy than a star. Type 1a supernovae are explosions in the sky that are about 5 billion times brighter than our sun. Figure 28: Light curves of Type 1a supernovae (Garcia-Bellido, Juan. (2004). Modern Cosmology. ) Though the light curves are slightly different, the peak luminosity is strongly correlated with the time period over which they occur and multiplying with the appropriate stretch factor we can calibrate the differences in the light curves. This means we can use the period of the supernova to find its peak absolute luminosity and use these supernovae as standard candles. Since they are so bright, they can be used as to measure distances much further away than we can measure using Cepheid variables. But they are transient phenomena, so we need to optimise our sky observation to spot a supernova when it occurs and then track it over time. 31 6.4 Hubble’s diagram In the 1920s, Edwin Hubble calculated distances to galaxies using Cepheid variables and their velocities using redshift (measured by Slipher) and plotted the famous Hubble diagram. This linear increase in a galaxy’s velocity with its distance from us shows us that the universe is expanding. Figure 29: Hubble diagram (Edwin Hubble, Proceedings of the National Academy of Sciences, vol. 15 no. 3, pp.168- 173) Fig 29 shows that vrecession ∝ d and vrecession = H0 d. Two major competing theories came up to explain this expanding universe – the Big Bang Theory and the Steady State Theory. Working within the currently accepted Big Bang theory, if the universe was expanding, we could trace back this motion and find out how long the universe has been expanding. The slope of Fig 29 is vd and has dimensions of time−1. To calculate time the universe has been expanding (assuming everything has been moving at a constant velocity since the Big Bang), we need to find d −1 v. This is the inverse of the Hubble constant. Hubble got 500 kms Mpc−1 as the slope of his graph. This would translate to an age of about 2 billion years. However even at that time geologists and biologists knew the Earth was about 4.5 billion years old – so how could the Universe be younger than the Earth? The error lay in Hubble using the wrong class of Cepheid variables to calculate distance. Current measurements give a Hubble constant of about 70 kms−1 Mpc−1. [Can you calculate the age of the universe using this value for the Hubble constant?] Lemaitre, Robertson, Walker and Friedmann’s solutions for Einstein’s theory of gravity assuming a homogenous and isotropic universe also showed an expanding universe. And when traced back in time led to singularity that we now call the Big Bang. 6.5 Big Bang theory The Big Bang postulates the universe expanded out from an extremely high density state with extremely high tem- peratures to reach its current form. Most naysayers fell into the school supporting the Steady State theory that postulated that the universe has always been, and will always be as it is in the present moment. They explained the constant density using multiple points of creation that introduced new matter to ’fill in the gaps’ as older matter continuously expanded outwards. The current, almost universal support for the Big Bang theory arose a result of some experimental observations that resulted in the falsification of the Steady State model. The following section will discuss a major observation that solidified the Big Bang model. 32 6.5.1 Cosmic microwave background radiation In 1964, Penzias and Wilson working at Bell Lab’s Holmdel Horn Antennae found a persistent noise in their receiver, even after all their attempts at removing said noise. At one point, they even hypothesized that it could be because of all the pigeon excreta from the pigeons nesting in the antennae, but the noise still remained after cleaning this out. Around this time, theorists had said that if the Big Bang really occurred, there would be a background radiation field as a result of this that could be characterized by a specific temperature. The significance of this ’noise’ was finally understood – the Cosmic Microwave Background was discovered. After the Big Bang, matter and radiation were dense plasma and they ’talked’ to each other and behaved in the same way and were at the same temperature. This is because at speeds comparable to c, matter behaves like light. This communication was more effective the hotter it was. As the universe expanded, density and temperature went down. And a last scattering happened where a photon after its last collision would move freely. So in some sense we can ’see’ up to the last scattering, but beyond that the past is opaque to us. (Similar to how the inside of the sun is not visible, but the surface is) And this happened as temperatures became low enough that matter could clump together and hydrogen atoms started forming from electrons orbiting a proton nucleus. After this the communication reduced drastically, as atoms only absorb and emit at certain wavelengths. Matter and radiation stopped talking! After this the expansion of each took place independently. Radiation didn’t change much after this. And still contains a signature of what matter and radiation in the universe looked like at that time. The radiation from that time that has travelled an undetected path to our detectors is what we call the Cosmic Microwave Backround radiation, and it shows us a picture of the universe as it was at the time of the last scattering. The temperature of the radiation signature calculated as a consequence of this evolution of the universe over time matched the temperature of the radiation ’noise’ observed by Penzias and Wilson. Thus the Cosmic Microwave Background radiation characterized by a temperature of 2.7K (as measured presently, Penzias and Wilson had gotten a slightly different value) was finally discovered. Figure 30: Cosmic Microwave Background radiation with the Milky Way subtracted (ESA/Planck Collaboration) Fig 30 is what the universe looked like when matter and radiation stopped communicating. The different colours show different temperatures, but these variations were very small (about 0.001K). These small fluctuations are very important as they mean that matter density was not uniform at that time. Which is why every massive thing that we know now exists. Small regions of greater density led to the formation of larger clumps of mass and these clumps of mass grew to form galaxies. In this framework there was a time in the early universe when neutrons and protons could come together for the lightest elements to be formed. The density of lighter elements in the universe as expected from this formation matches the density we observe. This gives us more reason to believe our description of the evolution of the universe. 33 Every scientific story has to be corroborated with many different lines of observations. The above observations along with some others sounded the death knell of the steady state theory. This story also led us to an homogeneous distribution of matter – but if matter was perfectly homogeneously distributed, how did the structures that we see now arise? It turns out that there are actually small fluctuations in the CMB, and these are the seeds of current structures that we observe and these fluctuations could be explained by a period of exponential ’inflation’ at the beginning of the universe. These regions of matter homogeneity resulted in areas where the gravitational collapse outweighed the outward pressure and slowly formed the structures that we now see. 6.6 Dark matter The fluctuations in CMB could not be fully explained by luminous matter. And when Vera Rubin was studying the rotational motion of galaxies she noticed something strange. She analysed the spectrum of an edge on spiral galaxy and plotted the relationship between orbital velocity and distance from the centre of the galaxy. (Just as we can plot velocities of planets in the solar system as a function of their distance from the sun. Try this!) The observed curve did not match the curve she expected based on the luminous matter distribution. A spiral galaxy with a central bulge (where most of the mass is seemingly concentrated) should have a rotation curve where velocities fall off as we go further out. Instead, most of the stars in the galaxy had more or less similar speeds. This led to the conclusion that there must be some matter (it could also be that our understanding of gravity is incorrect, but luckily for us introducing dark matter seems to work) that we cannot see that is responsible for the observed rotation curve, and this mass forms a halo around the galaxy. Without this extra mass the stars at the outer edges should be flung outwards at the observed velocities. Thus we concluded that the universe is more massive than we know – and this matter we could not see was named dark matter, which is calculated to be about five times the amount of luminous matter. Dakr matter still hasn’t been directly detected by telescopes as it does not interact with radiation, atleast at levels we can detect. But we know of it by studying its gravitational interactions - rotation curves, gravitational lensing, CMBR anomalies, etc. Figure 31: Rotation curve of a spiral galaxy (Wikipedia) This minimal interaction also means in the early stages of the universe, it decoupled from radiation before normal matter and formed the little pools of matter that luminous matter fell into. With the help of dark matter, we could explain the observed large scale structure of the universe. We still do not know the nature of this matter - it could be anything from exotic, unknown matter to brown dwarves to tiny black holes, etc. 34 7 Week 10 7.1 The fate of the Universe In a universe governed by matter, gravity is the force that controls large scale interactions. Depending on the initial outward ’thrust’ a universe controlled by gravity will either slow down, reach zero velocity and collapse (Big Crunch) or slow down and settle into a constant velocity. Both of these cases imply a decelerating expansion. But observations show that the expansion is accelerating! We call the cause of this outward push dark energy, and have added a constant to our equations describing the Universe to account for it. But we don’t really know much else about it. So let us first try to understand the fate of the universe using the tools within our current ability. To do this we will have separately study three new(ish) concepts and put them together to reach some equations that can describe the evolution of the universe in terms of things we can measure. Keep in mind that while physicists do this using Einstein’s theory of gravity (general theory of relativity), we aren’t equipped for that level of mathematics and we will try to derive a version of the equations using Newtonian gravity. This means some assumptions may not be strictly true, and we might resort to some tricks but they will nevertheless help us form a sufficient understanding of the cosmos. 7.2 Escape velocity When we throw a ball up, it comes back down but a (successful) rocket launch sends a rocket outwards in such a way that it doesn’t fall back. You will also notice that the harder you throw something, the higher it goes before coming back – and rockets take off in a massive expulsion of exhaust. This upward thrust (be it your flinging or the exhaust expulsion) imparts energy to the body that is launched upward. So when does a launch result in breaking free from the gravitational pull and when do things come back? To understand this, we introduce the concept of energy. Any object has energy associated with its motion - kinetic energy, KE = 21 mv 2 , and energy associated with configuration (arrangement with respect to other interacting bod- ies) - potential energy, P E = − GMr m. Since we are interested in the interaction of masses, we will only discuss gravitational potential energy, and henceforth PE will mean only gravitational PE. If you carefully observe an object that takes off, you will notice that it is fastest at the moment of take off and gets slower as it goes higher up. This is the KE reducing and as the P E increases so that the total energy remains the same. When an object falls back, it has hit zero KE and all of the KE that it started with has become P E. If we want something, say a rocket, to escape from the Earth’s gravitational pull, we would want to it to launch with enough velocity, that KE never runs out and therefore it never falls back. Let us observe such an object at two points – at the moment of take-off and at infinity. At take-off, 1 GM m E = mv12 − (31) 2 r1 And at infinity, 1 GM m E= mv22 − (32) 2 r2 Here, r2 = ∞ and therefore the P E term in Equation 32 goes to 0. And we require our KE be positive so that the object never hits zero velocity and starts falling back down. Thus for a body escaping from the gravitational pull of the earth, the total energy needs to be greater than 0 (E > 0). Substituting this in Equation 31 you get, r 2GMearth v> (33) rearth 7.3 Force of gravity due to spherical mass distribution We know that the gravitational force exerted between two masses is as given by Newton’s law of gravitation. Imagine yourself suspended in a hollow spherical shell of uniform thickness and density. The total force on you due to gravity will be zero. (You can derive this yourself by solving an integral to get the sum of the forces, but you can also just believe me.) We can think of the universe as an infinite sphere of uniform density, which means any (and every) point can be taken as the centre of the sphere. From one such arbitrary centre, we can look out to another point and wonder what the gravitational force is at such a point. To answer this we can divide the universe into a two pieces - a sphere with us at the centre and the point we are observing on the surface and a thick spherical shell starting from the surface and extending outwards to infinity. From our previous statements we know that the shell does not exert 35 any gravitational force on the point of obervation. Therefore the total mass that affects the point we are observing will be M (r) = 43 πr3 ρ. Figure 32: Force acting on a point in the universe 7.4 Expanding universe In an expanding universe, the distances between objects changes. We can measure the distance between us and a distant galaxy to be r0 at the present moment, as time changes the distance, r will change due to expansion. When we consider a homogeneous, uniformly expanding universe (where no point is special and the rate of expansion looks the same from every point), shapes that you draw using some points remain preserved. This uniform expansion can be factored out into a scale factor, a such that r(t) = a(t)r0 , where r(t) is called the proper distance and varies with time, a(t) is the scale factor and varies with time and r0 is the comoving distance and remains constant with time. At the present moment, comoving distance and proper distance are equal and a = 1. If we think about Hubble’s law using this insight, we will notice that Hubble’s constant is only constant at a given moment of time. More generally it is called the Hubble’s parameter and we can find its relationship to the scale factor. v = Hr v = Har0 ∆r and by definition, v = ∆t r0 ∆a v= ∆t r0 ∆a =⇒ Har0 = = r0 ȧ ∆t ȧ =⇒ H = a 7.5 Friedmann equation We will now use the previous three sections to reach an equation describing the universe in time and then use that to draw conclusions about the fate of the universe. In a matter dominated universe, the interaction that decides how the universe will evolve is the gravitational force. And we saw how the different ways gravitational interactions play out in Section 7.2. The qualitative difference between an eventual collapse or an indefinite expansion being the total energy of the system. So let us write down the equation for energy of a test mass m at some point in an infinite spherical universe of uniform density ρ, 1 GM m E = mv 2 − 2 r 36 Rewriting M from Section 7.3 in terms of r , 1 4 E= mv 2 − mπr2 ρG 2 3 Therefore the energy per unit mass U for the universe is, E 1 4 U= = v 2 − πr2 ρG m 2 3 Rewriting this in terms of the Hubble parameter by substituting for v we get,  2 ȧ 8 2U = πGρ + 2 a 3 r If we explicitly show the time dependent variables,  2 ȧ 8 2U = πGρ(t) + 2 (34) a 3 r0 a(t)2 This is the Newtonian version of the Friedmann equation. The relativistic form of this equation is,  2 ȧ 8 κc2 = 2 πGϵ(t) − 2 a 3c R0 a(t)2 Here ϵ is the energy density and κ is the curvature of the Universe. If you carefully examine the Newtonian equation you can see that to determine the fate of the universe, we need to figure out the sign of U in terms of observable quantities. Positive U (κ = −1, negative curvature) means a universe that expands forever and negative U (κ = +1, positive curvature) leads to an eventual collapse. In between this lies U = 0 (κ = 0, flat universe), the critical state when the universe stops expanding after an infinite amount of time. Starting from Equation 34, for a critical universe the density would be,  2 ȧ 8 H2 = = πGρc (t) a 3 3H 2 ρc = 8πG Notice how the critical density can be calculated for the present moment using the Hubble constant and other known quantities. Let us see if this can help us determine the sign of U. Substituting for Hubble’s parameter in Equation 34, 8 8 2U πGρc (t) = πGρ(t) + 2 3 3 r0 a(t)2 2U 8 8 = πGρc − πGρ r02 a2 3 3 4 U = πGa2 r02 (ρc − ρ) 3 4 ρ U = πGa2 r02 ρc (1 − ) 3 ρc a2 r02 H 2 U= (1 − Ω) 2 where Ω is the ratio of density of matter in the universe to critical density. Notice that the sign of U depends on the terms in the bracket, everything else is positive. The terms in the bracket are ρc and ρ, both of which we can measure and calculate at a given moment in time. If the actual density exceeds critical density the universe will collapse and if it is less than the critical density the universe will expand indefinitely. But as we mentioned before, ρ and ρc are time dependent – how can we be sure that the sign of U will always be the same as what we measure in the present time? Think back to escape velocity, remember that E and hence U always remains the same! How the scale factor changes with time captures how our universe is expanding and depends on the relative density of matter and energy in the universe. 37  ρ Figure 33: Scale factor changing with time for different cases, here Ωm = ρc 7.6 Large scale structure of the Universe Dark matter’s earlier decoupling from radiation and pooling together caused normal matter to fall into these pools resulting in the large scale structures that we see today. 38  Figure 34: Large scale structure of the Universe from SDSS 7.7 Accelerating universe and dark energy No matter what density we get for the universe, we expect the expansion to decelerate – since the force governing the fate of the universe is gravity and a attractive force like gravity will cause an eventual slowing down of the expansion. However, while drawing the Hubble diagram using SN1a data which allowed to look further out than ever before, something unexpected happened. The line fell above what we expected – the universe is accelerating outwards! You can alter the Friedmann equation to account for this new observation by introducing a term with what we call the cosmological constant. As of now we do not have a physical cause for the outward acceleration, but we have incorporated it into our equations describing the universe’s evolution as ’dark energy’ and successfully matched observations. 39  Figure 35: Hubble diagram showing an accelerating universe Another line of observation also led to dark energy – the curvature of the universe and the expected energy density. We can calculate the curvature of the universe using the fluctuations in CMBR. And our calculations lead us to a Universe that is almost flat – which implies a density that is equal to critical density. When we measure the density of the universe, we see that luminous matter contributes to about 5% of the expected density and dark matter about 25%. The remaining 70% is unaccounted for. While it is not necessary that this ’missing’ density would cause an expansion, it is convenient that accounting for this with dark energy that pushes the universe outwards would also explain the acceleration. We now believe that about 4 billion years ago, the dark energy started dominating the universe and the expansion started accelerating. In fact, an observer of the universe from 5 billion years ago would encounter a matter dominated universe and would have no reason to introduce dark energy to their model. In fact, three different lines of reasoning (third one being Baryon Acoustic Oscillations) and experimentation within our current framework lead us to the same result for the relative densities of matter and dark energy. You can see in Fig 36 that the three observations have three different regions (probability ellipses) of possible densities, and very conveniently for us, all of them intersect and point to the same result - about 70% of the Universe’s density is dark energy, and about 30% is matter. This gives us strong reason to believe that we have done something correctly w.r.t. our observing and theorizing about the Universe. Figure 36: Intersecting region from BAO, CMB and SN1a results 40 Fig 37 traces out the evolution of the universe within the ΛCDM model. Remember that this model is one of the simplest ways to account for all our current observations within the Big Bang model and hence widely accepted. Newer observations could lead us anywhere! Figure 37: Evolution of the Universe (as we have theorized so far) 41 8 Week 12 So far we have studied the interactions in the universe using Newton’s theory of gravity. But as observations get more refined the theory falls short while trying to describe observed phenomena. These shortcomings were finally addressed when Einstein came up with general relativity and the associated field equations where gravity was not a force but a curvature of spacetime. In this section of the course we will discuss some of the phenomena we observe and understand (to some extent) as a consequence of looking at the world through Einstein’s theory of gravity. Newton’s equations perfectly describe the world at the level most of us will analyse in our lives and Einstein’s field equations reduce to them when the curvature is minimal, there is a profound difference between them. They are philosophically and mathematically very different ways of understanding one of the major forces in the physical world. This is especially evident in astronomy where the implications of general relativity sometimes verge on the edge of science fiction. 8.1 Inertial and non-inertial frames of reference We used special relativity and the Lorentz transformations to translate between measurements made in different frames of references. However, this only holds for inertial frames of reference (observer not accelerating) and not for frames where the observer is accelerating. We try to understand the qualitative difference in the dynamics of moving objects within different types of frames of reference. 8.1.1 Inertial frame Imagine a room in deep space, far away from everything – there are no forces acting on you. You’d be floating somewhere in the room. Figure 38: Inertial frame You get yourself to an elevated step in one corner of the room using a small pocket jet and stick yourself there. What do you think happens to the room if you crash into a wall while using the jet? 42  Figure 39: Motion in an inertial frame If you threw a ball from there, it would move in a straight line in the same direction you threw it. If you ’dropped’ the ball from your hand, it would float around just like you did before you used the pocket jets! Such a frame is not accelerating and has no forces acting on it. 8.1.2