Physical Sciences and Mathematics Commons^™

General Relativity, 5, David Peak

General Relativity

No abstract provided.

General Relativity, 9, David Peak

General Relativity

Cosmic development

As discussed in GR 8, the cosmic scale factor a in the FLWR s-t obeys the Friedmann equation

General Relativity, 7, David Peak

General Relativity

The expanding universe

The fact that the vast majority of galaxies have a spectral redshift can be interpreted as implying that the universe is expanding. This interpretation stems from the Doppler effect in which the relative motion of an emitter and a detector produces a frequency shift of the detected light with respect to the emitted light.

Rudiments Of Quantum Mechanics (Qm), David Peak

Structure of Matter

Rudiments of Quantum Mechanics (QM)

Structure Of Matter, 6, David Peak

Structure of Matter

Quantum Flavor Dynamics (QFD), I

That each generation of the quark and lepton periodic tables (i.e., electron, muon, and tauon–see SM 1, p.1) has two members (a neutrino and a charged particle) that can be flipped into one another by emission or absorption of W bosons is reminiscent of how the angular momentum spin-1/2 component of a charged particle can be flipped between “up” and “down” orientations along some direction ( z ) in space by emission or absorption of photons. This analogy is made more graphic by labeling flavor rows by a new kind of “spin” (completely unrelated to …

Structure Of Matter, 9, David Peak

Structure of Matter

Beyond the Standard Models

As noted in SM 8 there are problems with the standard model of particle physics that beg for resolution. Indeed, there are significant unknowns concerning the standard model of cosmology as well—for example, what really is “inflation” and what preceded it? The latter suggests the need for the unification of gravity and quantum mechanics. In addition, both models involve bizarre empirical parameters. Why are the elementary particle masses so wildly different? Why are the strengths of the interactions so different? What is dark matter? What is dark energy and why is its density so small? Why …

Structure Of Matter, 7, David Peak

Structure of Matter

More about the matter with mass

To repeat, the basic premise of QED is that the physical world doesn’t care about the phase of the electron wavefunction; calculations and observations strongly support that idea. The basic premise of QCD is that the physical world doesn’t care about the color of the quark wavefunction; calculations and observations strongly support that idea. The basic premise of QFD is that the physical world doesn’t care about the flavor (isospin) of the quark or lepton wavefunctions. But that’s not true! For example, a “free” d quark can certainly emit a (virtual) W particle and …

Physics 3710 – Problem Set #2, David Peak

Problems

Physics 3710 – Problem Set #2 Newtonian relativity

Problems 1-4 refer to: Sound travels at about 330 m/s in still air. Observer O is at rest with respect to still air, observer O′ travels with constant velocity +50 m/s in the common x, x′ direction. Event A is the emission of a sound pulse from a stationary source at the origin of O; it occurs at x_A = 0 at t_A = 0. Event B is the reflection of the pulse at x_B = +100 m. Event C is the detection of the reflected pulse at x …

Physics 3710 – Problem Set #3, David Peak

Problems

Physics 3710 – Problem Set #3 Relativistic kinematics, I

Physics 3710 – Problem Set #10, David Peak

Problems

Physics 3710 – Problem Set #10 Relativistic gravity, III

Problems 1-3 refer to: The maximum measured z value for a galaxy is 11.1. As on page 1, GR7, z = λ_d − λ_e / λ_e.

Physics 3710 – Problem Set #6, David Peak

Problems

Physics 3710 – Problem Set #6 Relativistic dynamics, II

Problems 1-5 refer to: The mass of the neutron is 1.008664 u and that of the proton is 1.007276 u, where 1 u = 931.5 MeV.

Physics 3710 – Problem Set #12, David Peak

Problems

Problem Set #12 Quarks and gluons

In the following solid lines represent quarks or antiquarks and dotted lines represent gluons. Time increases upward.

Physics 3710 – Problem Set #7, David Peak

Problems

Physics 3710 – Problem Set #7 Newtonian gravity

Special Relativity, 4, David Peak

Special Relativity

More kinematic consequences of the Lorentz transformations

Light cones: A “light cone” is a set of world lines corresponding to light rays emanating from and/or entering into an event.

Special Relativity, 6, David Peak

Special Relativity

Energy-momentum vector

Special Relativity, 5, David Peak

Special Relativity

No abstract provided.

Physics 3710 – Problem Set #5, David Peak

Problems

Physics 3710 – Problem Set #5 Relativistic dynamics, I

Problems 1-5 refer to: One mass, m₁ = 1 (in some units), collides head-on with a second mass, m₂ = 2 , and sticks to it, forming a composite body of mass M . There are no external forces. Observer O records m₁ as initially moving with dimensionless velocity, u1 = +0.9 in the x - direction, while m₂ is recorded to be at rest. Do not make unwarranted assumptions about M , please; that’s the point of this set of problems.

Structure Of Matter, 1, David Peak

Structure of Matter

In the hot early universe, prior to the epoch of nucleosynthesis, even the most primitive nuclear material—i.e., protons and neutrons—could not have existed. Earlier than 10–5 s or so after t = 0 , the universe would have been a hot soup consisting of the most elementary of particles—photons, electrons, positrons, neutrinos, quarks, and gluons. We now turn to the,” “Standard Model of Particle Physics,” our current understanding of these elementary building blocks and their interactions. The Standard Model of Particle Physics (SMPP), developed in fits and starts over the past 50 years, is a quantitatively predictive theory of subatomic …

Structure Of Matter, 4, David Peak

Structure of Matter

Antiscreening: The triumph of lattice QCD

QED is a phenomenally accurate theory of the interactions of electrically charged particles with photons. The way interactions are described in QED—by adding electromagnetic potential fields to the energy and momentum operators in the charged particle field equations— is essentially exactly correct given that the detailed calculations that can be made in QED agree so well with observation. These calculations are possible because simple processes (involving small numbers of interaction vertices) are significantly more important than complicated processes. That is, QED is a “perturbative” theory. Higher order QED effects, therefore, invariably consist of small …

Physics 3710 – Problem Set #8, David Peak

Problems

Physics 3710 – Problem Set #8 Relativistic gravity, I

Special Relativity, 1, David Peak

Special Relativity

Newton and Maxwell

We have seen that simple magnetic forces are incompatible with Newtonian mechanics. There are far more profound incompatibilities between electromagnetism and Newton. In particular, Maxwell’s equations predict that an accelerating charge produces a time changing magnetic field, which, in turn, produces a time changing electric field, which, in turn, produces a time changing magnetic field, which, in turn, produces … . This self-sustaining production of magnetic and electric fields propagates away from the source charge at a finite speed given by 1 µ0ε 0 (in empty space), the numerical value of which is approximately 3x108 m/s. But, …

Special Relativity, 2, David Peak

Special Relativity

Events not connected by light propagation

Previously, we considered two events (A, with s-t coordinates (x_A, y_A,z_A ,T_A ) = (x^′_A, y_^′A,z^′_A,T ^′_A ) = (0,0,0,0) according to both O and Oʹ [moving relative to O with constant velocity β along the mutual x, x^′ -axes], and B, with coordinates (x_B,0,0,T_B) and (x^′_B,0,0,T^′_B)) connected by a light pulse.

Special Relativity, 7, David Peak

Special Relativity

Relativistic electromagnetism

Let’s return to the problem posed at the end of BK2.

General Relativity, 2, David Peak

General Relativity

Newton’s law of gravitostatics is incompatible with special relativity. To see this, suppose at time t in frame O m₁ is at x₁(t) and m₂ is at x₂(t). Newton’s gravitational force law says F_{1on 2}(t) = Gm₁m₂ / [x₂(t) − x₁(t)]² and relativistic dynamics says dp₂ /dt = F_{1on 2}. Transforming to another frame Oʹ moving relative to O leads to dp′₂/dt′ = F′_{1on 2}. But what is F′_{1on 2} ? If x₁(t) and …

General Relativity, 3, David Peak

General Relativity

Gravity as geometry: part II

Even in a region of space-time that is so small that tidal effects cannot be detected, gravity still seems to produce curvature. The argument for this point of view starts with the recognition that, for mechanical systems, it is impossible to distinguish a frame of reference with a uniform gravitational field from a uniformly accelerating frame that has no gravity. Thus, for example, in a (small) rocket ship with no windows it is not possible to determine whether the weight one reads standing on a scale at the tail of the rocket is due to …

Background, 1, David Peak

Background

This course deals with the structure of matter at its extreme length scales: cosmological on the large end (on the order of 1026 m), sub-nuclear on the small (less than 10-19 m). It also deals with the now firmly established realization that the organization of matter on these two phenomenally different scales is actually intimately connected. This course is about science in its most alive and vibrant state: what we think we know about the big and small of the universe changes virtually daily. Satellite observatories and ground-based particle accelerators make what was formerly “common knowledge” obsolete at a rapid …

History Graph, David Peak

Background

History graph image.

General Relativity, 1, David Peak

General Relativity

In special relativity, events occur in the arena of space-time which may be coordinatized differently by different observers, but which is otherwise immutable. Adding gravity to relativity provides an amazing result: space-time becomes “organic,” taking its form from the matter and energy it contains. This is Einstein’s general theory of relativity and it has the capacity to tell us about the past and future of the universe. Embedded in the history book of the cosmos are several chapters on the origins of matter. As a result, relativity + gravity unites the structures of matter on the largest and smallest scales.

General Relativity, 4, David Peak

General Relativity

Orbital motion of small test masses

The starting point for analyzing free fall trajectories in the (2-space, 1-time) Schwarzschild spacetime is Equation (3) from GR 3:

General Relativity, 6, David Peak

General Relativity

Modern cosmography

The “normal” matter in the universe—i.e., stuff made of protons, neutrons, and electrons— consists, approximately, of lumps floating in a dilute fog. The lumps are galaxies, clusters of 10⁷ to 10¹¹ stars bound together by gravity. In the currently observable universe, it is estimated that there are roughly 10¹¹ galaxies. The dilute fog is primarily neutral atomic hydrogen gas with some helium-4 mixed in (making up a total of 98% or more of the fog); there are also very small fractions of ²H (deuterium), ³He , and ⁷Li . The ratio …