Phys 4390: General Relativity

Transcription

1 Phys 4390: General Relativity Dr. David McNutt, 1 (call me Dave) 1 Department of Physics Saint Mary s University January 9, / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

2 My Background I am a mathematician with an interest in differential geometry, dynamical systems and their applications in physics and engineering. My PhD focused on the classification of space-times in General Relativity (GR), in particular my focus was on certain "toy-models" of gravitational radiation: the pp-wave and Kundt wave solutions. At the moment my work in GR has been limited, however I do have considerable experience with the theory. Figure: A purely electromagnetic plane-wave spacetime causing a null cone from event Q to be focused at a second vertex R 2 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

3 Outline 1 Course Goals Course Texts Academic Policies 2 Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity 3 When is Gr Important Some Context Special Relativity From SR to GR 3 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

4 Course Goals Course Goals Course Texts Academic Policies At completion of this course, students will be able to: Employ tensor analysis to approach problems in general relativity. Understand and explain the underlying physical principles of general relativity (GR) Have a quantitative understanding of the application of GR in modern astrophyics. 4 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

5 Course Goals Course Texts Academic Policies Introduction. Review of Special Relativity (SR), and use of tensor notation. Tensor algebra and Calculus: metrics, curvature, covariant differentiation. Fundamental concepts in GR: Principle of Equivalence, Mach s Principle, Principle of Covariance, Principle of Minimal Coupling. Energy momentum tensor and Einstein s equations. Scwarzchild solutions and black holes. Applications of GR in astrophysics: compact objects, gravitational waves, lensing cosmology. This outline, especially the last point, is subject to change depending on student s interest and time constraints 5 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

6 Course Texts Course Goals Course Texts Academic Policies I will primarily use the textbook "Introducing Einstein s Relativity" by Ray D Inverno, as this is a good introduction to GR with some advanced topics covered. That said, I may also take material from: Robert M. Wald s "General Relativity", John Stewart s "Advanced General Relativity", James Hartle s "Gravity: An Introduction to Einstein s General Relativity". 6 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

7 Approach to Teaching Course Goals Course Texts Academic Policies After this class I will not be using slides. Instead I will work on the board and follow class notes. As I am left handed and calligraphy was never a strong point, I will provide typed class notes. These notes may follow after the class is done, but it is my sincere hope to prepare them prior to class. These notes will be posted on my personal website, and blackboard. 7 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

8 Academic Integrity Course Goals Course Texts Academic Policies I encourage students to discuss work with each other in order to mutually understand the topics taught. Similarly, I see no harm with people working together on assignment questions. I object to solutions being copied word-for-word without any attempt to rehash to reflect the student s understanding. 8 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

9 Marking Scheme Course Goals Course Texts Academic Policies The current marking scheme is: Assignments 60% Final Exam 40% There will be a total of 5 assignments, with one every two weeks. 9 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

10 A Big List Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity 1 Precision gravity in the solar system 2 Relativistic stars 3 Black holes 4 Cosmological Models 5 Gravitational lensing 6 Gravitational waves 7 Quantum gravity 10 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

11 Precision Gravity Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity To measure the gravity field of the earth, NASA and the German Aerospace Center launched GRACE:Gravity Recovery and Climate Experiment. This has become an important tool for studying the ocean, geology and climate of the earth. Combining the data from GRACE with that of LAGEOS:LAser GEOdynamics Satellites, it is hoped that the relativistic effect of frame-dragging could be measured. These satellites are intended to provide an orbiting laser ranging benchmark for geodynamical studies of the Earth. Figure: Gravity Anomaly map from GRACE; taken from Wikipedia Figure: LAGEOS-1 satellite; taken from Wikipedia 11 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

12 Relativistic Stars Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity White dwarfs and neutron stars employ non-thermal pressure sources, i.e., electron and neutron degeneracy respectively, to resist contraction. At low masses white dwarves may be analyzed without relativity, however at higher masses these models are inaccurate. Neutron stars are relativistic. Recent computational models on the ignition of supernovae are considering general relativistic effects. Figure: Chandrasekhar Limit Graph; taken from wikiepdia 12 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

13 "Global" Cosmology Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity In order to describe curved spacetimes we must use GR This requires working on scales much larger than galactic clusters; often dust or perfect fluid models are considered for the spacetimes. In the weak field approximation it can be shown that the Newtonian description is sufficient, with great accuracy. The Friedmann equations dictate cosmic expansion and allow for one to study a number of possible scenarios for curvature of the universe. The cosmological constant, which originally frustrated Einstein s vision of a static universe has been shown to be a necessary part of cosmology. Figure: Two dimensional examples of surfaces with negative, vanishing, and positive curvature respectively; taken from Cosmology tutorial 13 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

14 Gravitational Lensing Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity While originally discussed by Chwolson (1924) and Klin (1936), Einstein published a canonical article on the subject in Three classes of gravitational lensing: strong lensing, weak lensing, and microlensing. Figure: Artist conception of gravitational lensing from a galaxy cluster; taken from NASA Figure: Einstein s Cross: An example of strong gravitational lensing, where four images of the same distant quasar appear around a foreground galaxy. 14 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

15 Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity Gravitational Waves GR predicts that ripples in spacetime can propagate at the speed of light, these ripples are called gravitational waves. Mergers of compact objects should produce immense amounts of gravitational radiation All mass produces gravitational waves, in this sense if one could detect gravitational waves, the universe would be very "bright". Unfortunately these are incredibly difficult to detect due to weak coupling F of matter Fgrav Figure: Artist conception of LISA (Laser Interferometer Space Antenna) spacecraft; taken from NASA elec 15 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

16 Planck Scale and Quantum Gravity Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity Using the fundamental constants of nature, it is possible to derive units associated with an era when quantum gravity is relevant on the "Planck" Scale. Taking, G and c, we can produce Planck length, mass and time. l p = G c = m c m p = G = kg t p = lp c = G c 5 = s 16 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

17 Hand-waving argument When is Gr Important Some Context Special Relativity From SR to GR As a simple argument consider the following: In the Newtonian approximation of a test particle in a closed orbit with speed v, radius R around a mass M GM R 2 = v 2 R v 2 = GM R Dividing v 2 by c 2 yields a dimensionless ratio v 2 c 2 = GM Rc 2 17 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

18 Comparison of Gm/Rc 2 values When is Gr Important Some Context Special Relativity From SR to GR Blackholes 1 Neutron stars 10 1 Sun 10 6 Earth 10 9 Figure: Taken from Fig 1.1 of Hartle provides a 18 / 27 Dr. David McNutt, (call me Dave) Phys comparison 4390: General of Relativity masses and distances.

19 When is Gr Important Some Context Special Relativity From SR to GR Successes and Failures of Newtonian Picture Replaced the Aristotelian picture that: Objects move when acted upon by a force, and tend to a stationary state when force is removed. Could not explain the force of gravity, which is a constant force but objects were accelerated. Newton s First Law gave us the first clue about relativity. If the force is such that F = 0 then v = C where C is a constant vector This allows for the concept of inertial frames of reference. Any frame for which v = C is defined to be an inertial frame of reference Newton s Laws cannot impose the constancy of the speed of light, which led to an erroneous believe in absolute simultaneity instead of a relative one. 19 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

20 When is Gr Important Some Context Special Relativity From SR to GR Newtonian Transformations between Inertial Frames of Reference Making the transformation: x n = x vt, y n = y, z n = z, t n = t it is easily shown that the second derivative of x w.r.t. t satisfies d2 x dt 2 = d2 x n dt 2 n and F = F n 20 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

21 The Light Cone in Special Relativity When is Gr Important Some Context Special Relativity From SR to GR Speed of light is the same in all inertial frames Normal matter is restricted to speeds less than c New concept of simultaneity, namely Relative Simultaneity 21 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

22 When is Gr Important Some Context Special Relativity From SR to GR Coordinate Transformations in Special Relativity Consider the Lorentz transformation, x n = x vt, yn = y, zn = z, tn = t vx/c2 1 (v/c) 2 1 (v/c) 2 As before, the frame will be boosted by speed v along x axis relative to the original frame at O 22 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

23 When is Gr Important Some Context Special Relativity From SR to GR Space-time Diagram Under Lorentz Boost 23 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

24 When is Gr Important Some Context Special Relativity From SR to GR Correspondence of electric and (Newtonian) Gravitational Force Newtonian Gravity Electrostatics Forces Between Sources Fg = GMm r 2 ē Mm Fe = qq 4πɛ 0 r 2 ēqq Force Derived From Potential Fg = m Φ g( x m) Fe = q Φ e( x q) Potential Outside a Spherical Source Φ g = GM r Φ e = Q 4πɛ 0 r Field Equation 2 Φ g = 4πGρ m 2 Φ e = ρ e/ɛ 0 Note: if ḡ( x = Φ g( x) then g( x) = 4πρ m( x), which is similar to Ē = ρe( x)/ɛ / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

25 When is Gr Important Some Context Special Relativity From SR to GR Moving Charges: Maxwell s Equations and Lorentz Force The Lorentz force describes how a collection of moving charges are affected by a velocity dependent force from magnetic fields. F e = q(ē + v B) The velocity dependent term is missing in Newtonian Gravity. For any frame chosen, acceleration depends on mass alone in Newtonian gravity, this implies it is not relativistic. It is possible to add B g term, however this becomes very complicated. 25 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

26 Measuring Ē and B Fields When is Gr Important Some Context Special Relativity From SR to GR Using neutral charges one can construct an inertial frame. Any particle at rest can be used to measure Ē, since F e = qē Once in motion one can measure B through the expression F e = q(ē + v B) Sadly this approach cannot be used to measure gravity, since there is no form of matter that is "neutral" to gravity. 26 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity

27 Steppin Stone: SR to GR When is Gr Important Some Context Special Relativity From SR to GR In the presence of gravity, freely falling frames are locally inertial, we call this the Principle of Equivalence. These particles will follow geodesics, which can be seen as the path of least resistance. Returning to the point of no "neutral" charges. Any particle is a source of gravitational field, so as they move through spacetime they also bend it. We can approach GR by formulating SR in this new frame and taking our physical laws and applying the Principle of Covariance (Physical Laws are preserved under changes of coordinates.) Some people believe we need additional principles. 27 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity