Sep 13, 2014

Universe in the Size-Mass Diagram


The fundamental goal of science is to describe the material world. Our knowledge of the natural world can be categorized in different ways. The conventional wisdom is division of fields, such as physics, chemistry, biology etc. Fields of subject is not the only logical way to organize our knowledge of the Universe. Chronology, for example, presents the topic in a unified theme, time. The Universe was created from a quantum singularity called the Big Bang. As time went on, fundamental forces and matters were created and formed stars, planets, galaxies and clusters. Since the birth the Earth, the focus can be shifts to the geological history and then biological history of the Earth. At last, the latest chapter is the history of humanity.

Fig. 1, The Chronology of the Universe.

Complementing to time, the structure of matter is another unified theme to understand our Universe. In this description, physics, again, plays a particular role, as it offers insights to most natural entities including the smallest (subatomic particles) and the largest (cosmos itself). Traditionally, the study of physics can be divided into two paradigms: the first one concerns the fundamental laws of the Universe; the second one concerns the evolution and, particularly, the structure of physical systems governed by the law of physics. The fundamental rules of the nature, to our best knowledge today, are summarized to the standard model (SM) of particle physics and Einstein's general theory of relativity (GR). As it turns out, the SM recipe has to include what we call the elementary particles such as electron, quark and photons, whose structure are completely unknown. Then, in the study of matter structures, the fundamental laws of physics (including the elementary particles) are taken as the first principle.

Fig. 2, The Recipe of the Microscopic World - the Standard Model of Particle Physics. LEFT: the prescription, RIGHT: the ingredients.

Size-Mass Diagram

Our universe has been an extremely diverse "ecosystem". Matters come in vastly different scales in terms of either size or mass or any other characteristics. To provide a schematic view of the structures of matters, we need some universal physical characteristics. Mass is the first on the list. Einstein's special theory of relativity unified mass and energy (related by $E = mc^2$). Therefore, our "mass" would naturally includes "energy". For the second property, we chose the characteristic size of the system. Every matter has a finite spatial extension, except for elementary particles, which are the first principle by definition. Moreover, the size of a physical system is usually closely related to the structure. Knowing the mass and the size, the structure of the system is roughly known. But the size-mass diagram (see Fig. 3) can reveal more than the mass and size hierarchies themselves. Matters share the same structural mechanism tend to align in a line.

Fig. 3, Size-Mass Diagram.


The ancients thought the world was made of five elements. Not until the modern era, it was realized that there are hundreds of elements, making of most matters we are familiar with. Matters from elements lie between the cyan line (gas) and the brown line (condensed matter - such as liquids and solids) in the diagram. Hydrogen atom is the smallest and best known atom. Here is also where most of the science delve. Physics focus on systems in the scale of single atom and below. Molecules constitute the realm of chemistry. Carbon based large molecules (such as proteins) lay the foundation of biology (or so-called life science). Biologists concerns a vast domain from virus to human being. Human community is also the subject of research - the so-called social sciences - arguably the most renowned ones, which anthropology, sociology, economy, linguistics and political sciences. The largest coherent community is our world, which includes 70 billion homo sapiens populating the land surface of the Globe. Our engineering, extends more vast scales than ourselves. Transistors in newest commercial silicon chips are only 14 nm in size, representing the state-of-the-art micro-fabrication. One of the largest man-made projects, the USS Nimitz aircraft carrier is of 333 meters, 100 million kilograms. Anything greater than humanity was considered deity. Large amount of matter, mountains and lakes for example, are the subject of geology. Moving up the scale, large objects are usually in the sky. Moon is the satellite of our planet. Mars, Venus and others are peer planets. Sun is a star. Gravity becomes important in the structure.

Greatness and Beyond

Heaven has its own hierarchy. Moon circles around the Earth. Earth and other planets circle around the Sun. This is the area of astronomy and astrophysics. Not until Issac Newton, the hidden force behind stars and planets are known, the gravity. The gravity of Sun pulls Earth, Mars, Venus and other planets around it and makes a planetary system. Stars are also bound by gravity. Gazillions of stars form a galaxy. Galaxies make up groups, clusters and then superclusters. Superclusters connects into filaments and great walls. These structures extends vast distance in the vacuum. Even light, the record-holder of speed, takes millions of year to travel across them. The large scale structure ends at about 100 Mpc or $3\times 10^{24}$ meters. This is known as the end of greatness. Above 100 Mpc, the universe looks pretty smooth and isotropic. Then there rises the study of the Universe itself, cosmology.

But stars and galaxies may not the whole story of the Universe. Study of the galaxy motion and other evidence suggests the Universe has its dark side.

A New Front

The 20th century witness the discovery of a new front in the subatomic scale. In the microscopic realm, objects (microscopic objects are usually called particles) start getting peculiar as we move close to the line of Compton wavelength (green line). The Compton wavelength is define as $\lambda_c = \frac{h}{mc}$, where $m$ is the rest mass, $h$ and $c$ are two constants - Planck constant and the speed of light (Why don't we call it Einstein's constant or Maxwell's constant or Michelson's constant?). Compton wavelength is a universal limit on the measurement of the position of a particle. Due to quantum mechanics, the position of a particle around and below the Compton wavelength scale becomes fuzzy. This phenomenon is called "zitterbewegung", German for "trembling motion". Therefore, if the size of a particle is near or smaller than its Compton wavelength, it has to be described by quantum mechanics. If the (special) relativistic effect is also prominent, quantum field theory applies. Microscopic particles, from the hydrogen atom to the family of hadrons, all reside in this realm. Particularly, the elementary particles, who have been treated as point-like particles, also obey quantum mechanics and Einstein's special theory of relativity. In fact, the entire playbook of these particles, the standard model of particle physics is a manifestation of the quantum field theory.

To study the structure of a microscopic system, the probe has to be smaller than its Compton wavelength. The experimental probes used in these fields are high-energy beams (including light - from laser to X-Ray and $\gamma$-Ray etc al.). Physicists like labeling their probes by their energy $E$ instead of their wavelength $\lambda$. The particle of the beams are highly relativistic such that $\lambda$ and $E$ obey the de Broglie relation: $\lambda = \frac{hc}{E}$, which is similar to the Compton wavelength as the rest energy $E_0 = m c^2$. Therefore, to probe proton ($m_p \simeq 1 \mathrm{GeV}/c^2$), it is necessary to use  > GeV beams. The German probe HERA, for example, used 27.5GeV electron beams. These probes need so much energy that billion-dollar facilities are built to accelerate the probing particle beams - that's why they are called accelerators.

Fig. 4, Evolution of Large Colliders. from: Jordan Nash, "Current and Future Developments in Accelerator Facilities".
The standard model describes physics around 100 GeV or $10^{-17}$ meters. LHC, the largest collider ever built, with maximum energy 8 TeV (or 8000 GeV - to present day - it will be upgraded to 14 TeV), is dedicated to study the standard model and beyond. Physics beyond TeV scale could only be subjected to speculation.

It is believed that the strong and electroweak (electromagnetic and weak force are unified at > 246 GeV scale, known as the electroweak scale) forces are unified at $10^{16}$ GeV, the GUT (Grand Unification Theory ) scale. As the Compton wavelength decreases, the mass of the particle increases. Both factors increase the local gravitational field $f \sim \frac{GM}{r^2}$. At the length scales we know today, the local gravitation field of a microscopic particle is negligible. However, at about ~$10^{19}$ GeV scale (or $10^{-35}$ meters, known as the Planck scale), the local gravity field of the particles would be strong enough to challenge our current understanding of the law of physics. In fact, the space and time below the Planck scale is so twisted that we are not sure if it is still meaningful to talk about size.

The Horizon

The gray line describes black holes. This line is defined by the event horizon (Schwartzschild radius for most objects) $r_s = \frac{2G M}{c^2}$. Inside the event horizon, the gravity is so strong that even light can not get out. Therefore, no information inside the event horizon can be revealed to the outside observers, as its name suggests. It is believed that inside the black hole nothing can stop matter from forever collapsing towards the center. Einstein's general theory of relativity forbids us even to go beyond the line. Along this line, there lives the monsters of the Universe (or rather black hole candidates). Cygnus X-1 is one of the most famous black hole. Here X stands for X-Ray. Like other compact objects, stellar black holes are usually powerful X-Ray sources. This is because objects are heated up (due to the loss of gravitational energy) as they falling into compact stars. The list of black holes extends from the extremely compact stellar objects to the less compact yet super-massive black holes sitting in the center of galaxies. At one end of the line, there lies the observable universe! It suggests that we may be living inside a super-duper black hole. The line cross the Compton wavelength line at the other end. It has been suggested that general relativity has to be modified to accommodate the quantum effect. The hypothetical particle sitting at the intersection is called a Planck particle, which is also a microscopic black hole.

White dwarfs and neutron stars are other examples of the compact objects. In a white dwarf, gravity is balanced by the degenerate press from the electron. If the mass of a star exceeds 1.456 solar mass, known as the Chandrasekhar limit, the star would collapse into a neutron star. Neutron stars are made of nuclear matters. The density of neutron stars are similar to that of nuclei, the microscopic members of the nuclear matter family. In fact, neutron stars can be viewed as gigantic nuclei. A lot of their properties can be estimated (or rather extrapolated) from nuclei. On the other hand, neutron stars do show strong local gravity. This makes them look like black holes.