Concerning the relationship between science and philosophy

This post consists of the  elements of an answer I wrote at quora.com ; the question there was: “Is philosophy the top of all kinds of sciences?”

I think it would be convenient to distinguish between the general term “science”, referring to the state or fact of knowing, or to knowledge acquired by study and learning, and the modern meaning of “science”, mostly referring to mathematics and to the exact sciences using the rules of the scientific method (astronomy, physics…).

Philosophy and science were not separate in Antiquity.

In the original sense, philosophy meant the love, study, or pursuit of wisdom, or the knowledge of things and their causes, theoretical as well as practical.

Pythagoras was a mathematician, and at the same time it is said that he was the first one to call himself a philosopher, or “lover of wisdom”.

Plato was a philosopher who recommended the knowledge and the study of geometry. In The Republic, Plato thought that the best ruler was the king-philosopher.

Aristotle studied nature and wrote works about physics, biology, logic, etc, from a philosophical point of view.

According to the OED:

   “In the Middle Ages, ‘the seven (liberal) sciences’ was often used synonymously with ‘the seven liberal arts’, for the   group of studies comprised by the Trivium (Grammar, Logic, Rhetoric) and the Quadrivium (Arithmetic, Music, Geometry, Astronomy).”

The expression Natural philosophy was frequently used for centuries :

   “Natural philosophy or philosophy of nature (from Latin philosophia naturalis) was the philosophical study of nature and the physical universe that was dominant before the development of modern science. It is considered to be the precursor of natural science.

    From the ancient world, starting with Aristotle, to the 19th century, the term “natural philosophy” was the common term used to describe the practice of studying nature. It was in the 19th century that the concept of “science” received its modern shape with new titles emerging such as “biology” and “biologist”, “physics” and “physicist” among other technical fields and titles; institutions and communities were founded, and unprecedented applications to and interactions with other aspects of society and culture occurred. Isaac Newton‘s book Philosophiae Naturalis Principia Mathematica (1687), whose title translates to “Mathematical Principles of Natural Philosophy”, reflects the then-current use of the words “natural philosophy”, akin to “systematic study of nature”. Even in the 19th century, a treatise by Lord Kelvin and Peter Guthrie Tait, which helped define much of modern physics, was titled Treatise on Natural Philosophy (1867).

In the last few centuries, alchemy separated from chemistry, astrology separated from astronomy, and there was also a certain separation between philosophy on one side, and mathematics and the exact sciences on the other side.

Mathematics became progressively the most prominent and the essential scientific discipline, it is acknowledged as the language of science and of the physical world.

Philosophy is nowadays often regarded as a reflection, view or study of the general principles of a particular branch of knowledge, or activity. There is a philosophy of science, philosophy of mathematics, philosophy of education ,etc.

Some theories or views related to epistemology (which is concerned with the general theory and the study of knowledge) and philosophy, such as rationalism, empiricism, and positivism, share a number of principles with the scientific approach to events and phenomena.

photo of Kant

(Source of the image above: Wikimedia Commons)

A scientist or a physicist can also be a philosopher. Important thinkers can be philosophers and create philosophical systems, but modern philosophers must take into account the advances, discoveries and theories in modern science. A historical example would be Immanuel Kant elaborating his philosophical system and philosophical ideas at the end of the eighteenth century in light of and in relation to the exact sciences known at that time, especially Euclidean geometry and Newtonian classical physics and mechanics.

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A poem I wrote years ago

I was fifteen- soon to be sixteen- years old ; I had been reading (important) books about science, physics, philosophy , and other similar topics,  and all those ideas in my head intermingled and inspired me to write a poem involving particle physics and particle collisions and combining elements of science and philosophy .

I wrote the poem in French , using the French alexandrine poetic meter of twelve syllables, but I didn’t follow the poetic rules very closely.

I will provide the final version of the poem here , with a line by line English translation. Different people have different tastes and opinions , I hope it will be liked .

The hydrogen-1 atom mentioned in the title of the poem is also called “protium” , but this last word is not much used in French. Protium is the most common hydrogen isotope, having one proton ( and one electron) and no neutrons.

A proton is supposed to be talking or telling the story in the poem . I think I was a little inspired by the poem ” Le Bateau ivre ” by Arthur Rimbaud .

Here it is :

Bombardement d’atomes par un proton d’hydrogène 1H
Bombardment of atoms by a proton of protium 1H

Synchrotrons , canons à électrons, cyclotrons
Synchrotrons, electron guns, cyclotrons

Soyez prêts, particules, deutons, neutrons, hélions
Be prepared, particles, deutons/deuterons, neutrons, helions

En attendant que les hommes préparent les canons
Until men prepare the guns

Le moment est arrivé, l’appareil frappe
The moment has come, the apparatus strikes

Dans son coeur vidé moi, le proton j’attrape
In its emptied heart I , the proton take

Le coup et je vais croiser les atomes en grappe
The blow and I go meet the atoms in clusters

Je fuis dans l’espace et le temps calculables
I flee in computable space and time

Ma vitesse est vertigineuse, incroyable
My speed is vertiginous, incredible

Non pas celle de la lumière, infranchissable
Not that of light, insurmountable

C’est le lieu de la relativité impie
It is the place of impious Relativity

Masses, longueurs, lois de la physique varient
Masses, lengths, physical laws vary

Ma trajectoire déterminée sera suivie
My particular/determined path will be followed

Par d’autres microcosmes malheureux
By other unfortunate microcosms

Le trajet est terminé, le choc a eu lieu
The journey is over, the shock/collision occurred

Je donne la vie à de nouveaux corps heureux
I give life to new happy/fortunate bodies

Quanta de matière utilisés pour la paix
Quanta of matter used for peace

Dans le monde de la science un pas est fait
In the world of science a step/discovery has been made

L’humanité en marche en connaît les bienfaits
Humanity in motion/advancing knows the benefits (of this discovery)

Books about physics, astrophysics and astronomy regarded as important classics

This post is mostly inspired (with some additions and modifications) from and answer I wrote at quora.com .

I will try to give a list of famous , influential books or classic books having a significant historical importance in the fields of physics , astrophysics and astronomy . It’s a somewhat extensive list but it’s not exhaustive.

Starting with Antiquity :

Then advancing to more recent times:

Al Sufi stars

Copernicus book

Below is a page from the Astronomia Nova (in 1609) showing the three models of planetary motion known in the seventeenth century (free image from Wikipedia) :

Astronomia Nova

Newton's Principia

Hydrodynamica

Carnot reflexions

  • Recherches sur la théorie des quanta (Researches on the quantum theory) , and The Current Interpretation of Wave Mechanics: A Critical Study , by Louis de Broglie .
  • Collected papers , The interpretation of Quantum Mechanics , and Statistical Thermodynamics , by Erwin Schrödinger .
  • The Physical Principles of the Quantum Theory , by Werner Heisenberg .
  • Books and papers by Paul Dirac , such as The Principles of Quantum Mechanics and Lectures on Quantum Field Theory .
  • Space, Time and Gravitation: An Outline of the General Relativity TheoryThe Internal Constitution of Stars , and The Nature of the Physical World , by Arthur Eddington .
  • Problems of Cosmology and Stellar Dynamics , An Introduction to the Kinetic Theory of Gases , and  The Growth of Physical Science , by James Hopwood Jeans .
  • The Theory of Sound , by John William Strutt, 3rd Baron Rayleigh .
  • Problems of Atomic Dynamics , Atomic Physics , Principles of Optics , Experiment and Theory in Physics , and A General Kinetic Theory of Liquids , by Max Born .
  • Books and papers by David Bohm , such as Quantum Theory , Causality and Chance in Modern Physics , The Undivided Universe.

Some more recent well known , insightful and/or widely used books would include :

  • The Large Scale Structure of Space-Time , by Stephen Hawking and George F. R. Ellis .
  • Speakable and Unspeakable in Quantum Mechanics , by John Stewart Bell .
  • Classical-Mechanics , by Herbert Goldstein .
  • Classical Electrodynamics , by J.D. Jackson .
  • Galactic Dynamics , by Binney and Tremaine .
  • The Quantum Theory of Motion: an account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics , by Peter Holland .
  • Photons and Atoms: Introduction to Quantum Electrodynamics , by Claude Cohen-Tannoudji , Gilbert Grynberg and Jacques Dupont-Roc .
  • Introduction to Elementary Particles , by D.J. Griffiths .
  • Condensed Matter Field Theory , by Alexander Altland .
  • The Standard Model and Beyond , by Paul Langacker .
  • The Road to Reality , by Roger Penrose .
  • Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law , by Peter Woit .
  • The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next , by Lee smolin .
  • Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth , by Jim Baggott .

Additional relevant links :

https://en.wikipedia.org/wiki/Hi…

https://en.wikipedia.org/wiki/Hi…

Astronomy in the medieval Islamic world

Indian astronomy

Chinese astronomy

Regarding gravitational waves

Gravitational waves have become a popular topic recently , and this post presents material I have written for an answer at quora.com (with a few modifications).

Gravitational waves are changes in curvature similar to ripples in space-time . They are an indirect result of the special theory theory of relativity , and were explicitly proposed by Einstein in 1916 in the framework of his theory of general relativity . He showed that the acceleration of mass generates gravitational fields which are time-dependent and are capable of transporting energy (as gravitational radiation ) from their source at the speed of light .
Gravitational waves are quadrupolar in nature , whereas electromagnetic waves are dipolar . Oscillating multipole moments of the mass distribution of a system produce gravitational radiation .
Many attempts have been made to detect gravitational waves , but no direct evidence of their existence has been observed until their recent detection in February 2016 .

The Einstein field equations describe the interactions between space-time curvature and mass , as Maxwell’s equations describe and specify the relationship between electric charge and electromagnetic fields .
The field equations have a solution represented by a weak oscillating perturbation to the curvature of space-time , and this solution is a gravitational wave .
These waves can be regarded as an oscillating perturbation to a flat Minkowski space-time metric , or also as a tidal force oscillating between free test masses , or as a strain oscillating in space-time .
More explicitly , one can show that a wave equation represents the solutions in free space for the metric perturbations of a nearly flat space-time , with waves propagating at the speed of light ( this is a weak gravitational field approximation) .
One can take a coordinate system where the metric has components :
g_ {\mu\nu} = \eta_ {\mu\nu} + h_ {\mu\nu}
where
\eta_ {\mu\nu}  is the Minkowski metric in special relativity , and
h mu nu

After some calculations the  solution to Einstein’s equations in free space can be written as :

wave eq

where

hbar mu nu

So the metric perturbations propagate in free space as waves at the speed of light .
A primary example of a source of gravitational waves is a pair of neutron stars , or two black holes , or one of each type of these astrophysical objects .
Observing supernova explosions or the orbital motion of binary pulsars may possibly give and indirect proof of the existence of gravitational waves .
The image below represents gravitational waves generated by two neutron stars orbiting each other (image source : File:Wavy.gif ) :
wavy anim

 

Ways of detecting gravitational waves include resonant mass detectors , free mass detectors , detectors in space , cosmic background measurements , and monitoring pulsar signals .
External disturbances and the effects of thermal noise in the detecting system should be avoided , the possible interaction between detectors and gravitational waves being very weak .
In 1974 Russell Hulse and Joseph Taylor discovered and observed the orbital period of a binary pulsar . They confirmed that the orbit was accelerating at the rate predicted by the emission of gravitational waves according to the theory of general relativity .
The LIGO (Laser Interferometer Gravitational-Wave Observatory) detectors
are used to attempt to observe directly cosmic gravitational waves . They can detect extremely small strains (of the order of  one part in 10²¹ ).
In the quantum theory of gravity , a quantum field whose excitations are gravitons represents the gravitational field .
Gravitons may be regarded as the normal modes of oscillation of a (gravitational) gauge field , produced by a mass current of accelerating masses .
Some (online) links and resources :
Gravitational wave
McGraw-Hill Encyclopedia of Science and Technology , 10th Edition .

To make this answer complete  , it should be noted that the expression gravity waves is also used to refer to waves studied in oceanography , meteorology and fluid dynamics .
Used in this sense , a gravity wave is a liquid surface layer wave controlled by gravity and not by surface tension .
The surface tension of water becomes unimportant at wavelengths greater than a few centimeters . On the ocean surface or interfaces , all significant waves are gravity waves .
In meteorology , gravity waves are transverse atmospheric waves where the restoring force is caused by the effect of gravity on density and pressure fluctuations .
See for example the Wikipedia article Gravity wave .
The expressions gravity waves and gravitational waves are sometimes used interchangeably for both meanings (i.e. for waves related to general relativity and waves related to fluid dynamics) , so this might cause some confusion.

As an update to the information above , something new took place in the history of the detection of gravitational waves on 11 February 2016 .
For the first time, scientists have observed ripples in the fabric of spacetime called gravitational waves, arriving at Earth from a cataclysmic event in the distant universe. This confirms a major prediction of Albert Einstein’s 1915 general theory of relativity and opens an unprecedented new window to the cosmos.[…]
The gravitational waves were detected on Sept. 14, 2015 at 5:51 a.m. EDT (09:51 UTC) by both of the twin Laser Interferometer Gravitational-wave Observatory (LIGO) detectors, located in Livingston, Louisiana, and Hanford, Washington.[…]
Based on the observed signals, LIGO scientists estimate that the black holes for this event were about 29 and 36 times the mass of the sun, and the event took place 1.3 billion years ago. About three times the mass of the sun was converted into gravitational waves in a fraction of a second—with a peak power output about 50 times that of the whole visible universe. […]
The discovery was made possible by the enhanced capabilities of Advanced LIGO, a major upgrade that increases the sensitivity of the instruments compared to the first generation LIGO detectors, enabling a large increase in the volume of the universe probed—and the discovery of gravitational waves during its first observation run.
As and additional note , it is generally preferable to have other precise experiments confirming the detection and presence of gravitational waves.

The Lagrangian of the Standard Model of particle physics

I will present some notes and explanations related to the Standard Model of particle physics and its Lagrangian . The text in this post is inspired from two answers I gave at quora.com .

The Standard model and its Lagrangian form a vast topic . I will attempt to give relevant and accurate information about it.

The story of the Standard Model started in the 1960s with the elaboration of the theory of quarks and leptons  , and continued for about five decades until the discovery of the Higgs boson in 2012.
For a timeline of the history of the Standard Model see the Modern Particle Theory timeline .
The formulation of the Lagrangian of the Standard Model with its different terms and parts mirrored the theoretical and experimental advances associated with particle physics and with the Standard Model.

The Lagrangian function or Lagrangian formalism is an important tool used to depict many physical systems and used in Quantum Field Theory . It has the action principle at its basis .

In simple cases the Lagrangian essentially expresses the difference between the kinetic energy and the potential energy of a system .

The Standard Model of particle physics describes and explains the interactions between the essential components and the fundamental particles of matter , under the effect of the four fundamental forces: the electromagnetic force , the gravitational force , the strong nuclear force , and the weak (nuclear) force.
However , the Standard model is mainly a theory about three fundamental interactions , it does not fully include or explain gravitation .
The Standard model (or SM) is  a gauge theory representing fundamental interactions as changes in a Lagrangian function of quantum fields.  It depicts spinless , spin-(1/2) and spin-1 fields interacting with one another in a way governed by the Lagrangian which is unchanged by Lorentz transformations.

The Lagrangian density or simply Lagrangian of the Standard Model contains kinetic terms , coupling and interaction terms (electroweak and quantum chromodynamics sectors) related to the gauge symmetries of the force carriers (i.e. of the elementary and fundamental particles which carry the four fundamental interactions) , mass terms , and the Higgs mechanism term .

Explicitly , the parts forming the entire Lagrangian generally consist of :
Free fields : massive vector bosons , photons , and leptons.
Fermion fields describing matter.
The Lepton-boson interaction.
Third-order and fourth order interactions of vector bosons.
The Higgs section.

Leptons are the elementary particles not taking part in strong interactions.
All leptons are fermions. They include the electron , muon , and tauon , and the electron neutrino , muon neutrino , and tauon neutrino.
All leptons are color singlets , and all quarks are color triplets.

In the Standard model , the Higgs mechanism provides an explanation for the generation of the masses of the gauge bosons via electroweak symmetry breaking.

Different reference works , books , e-books or textbooks use different or slightly different notations and symbols to describe or designate the entities and terms within the Lagrangian of the SM .

Below is a detailed image of the Lagrangian of the Standard Model  (Source: http://einstein-schrodinger.com/Standard_Model.pdf ).
However I have rearranged it and modified it with the help of Photoshop to make it look more presentable and more readable.

lagrangian of the standard model

The Lagrangian function in the Standard Model , as in other gauge theories , is a function of the field variables and of their derivatives.

G_ {\mu \nu} is the gauge field strength of the strong SU(3) gauge field.
Gluons are the eight spin-one particles associated with SU(3).
A particle which couples to the gluons and transforms under SU(3) is called ‘colored’ or ‘carrying color’.
Gluons and quarks are confined in hadrons.

W_ {\mu \nu} is the gauge field strength of the weak isospin SU(2) gauge field .

The field strength tensor W_ {\mu \nu} is given by :

field strength tensor

where g_2 is the electroweak coupling constant , a dimensionless parameter.

The charged W^+ and W^- bosons and the neutral Z boson represent the quanta of the weak interaction fields between fermions , they were discovered in 1983 .

B_ {\mu \nu} is the gauge field strength of the weak hypercharge U(1) gauge field.
The field strength tensor B_ {\mu \nu}  is given by :

B_ {\mu \nu} = \frac {\partial B_ {\nu}} {\partial\mu} - \frac {\partial B_ {\mu}} {\partial\nu}

In the Standard model , electrons and the other fermions are depicted by spinor fields .
The group U(1) is the set of one-dimensional unitary complex matrices .
U(1) represents the symmetry of a circle unchanged by rotations in a plane.

SU(2) is called ‘the special unitary group of rank two’. It is a non commutative group related to SO(3) , the sphere symmetry in 3 dimensions.
SU(2) is the set of two-dimensional complex unitary matrices with unit determinant.

SU(3) , the special unitary group of rank three , is used in quantum chromodynamics (QCD) .
SU(3) is the set of three-dimensional complex unitary matrices with determinant equal to 1 .
The natural representation of SU(3) is that of 3×3 matrices acting on complex 3D vectors.
The generators of the group SU(3) are eight 3×3 , linearly independent , Hermitian , traceless matrices called the Gell-Mann matrices . These generators can be created from Pauli spin matrices (which are used with the group SU(2) ) .

The SM Lagrangian displays invariance under SU(3) gauge transformations for strong interactions , and under SU(2)xU(1) gauge transformations for electroweak interactions.

The electromagnetic group is not directly the U(1) weak hypercharge group component of the standard model gauge group. The electric charge is not one of the basic charges carried by particles under the unitary product group SU(3)xSU(2)xU(1) , it is a derived quantity.

All the masses vanish in the absence of the Lagrangian term related to the Higgs , due to the invariance of SU(3)xSU(2)xU(1) .

In some texts the gauged symmetry group of the SM is written with subscripts such as:
\text {SU} _c (3)\times\text {SU} _L (2)\times U_Y (1)
In the notation above , the subscript ‘c’ denotes color.
The subscript ‘L’ denotes left-handed fermions.
The subscript ‘Y ‘  distinguishes the group related to the quantum number  of weak hypercharge , expressed by the letter Y , from the group associated with ordinary electric charge, expressed by Q .
U_ {\text {em}} (1) denotes the electromagnetic group.

The Higgs field in the Standard model is a complex scalar doublet. It is generally represented by :

 Higgs field doublet

In the image of the SM Lagrangian above , the Higgs field has the form

Higgs field form

The field h(x) is real .

In the SM Lagrangian image above , \phi _ 0  is equal to v .

As an additional note , the  equation of the Lagrangian  is usually made of a definite number of terms and Lagrangians.
In order to make such an equation look less like a big behemoth and make it more compact ,  it would be simpler to view it or write it first as the sum of Lagrangians :

\mathcal{L}=\mathcal{L}_1+\mathcal{L}_2+\text{...}+\mathcal{L}_n

 or equivalently :

\mathcal {L} = \sum _ {i = 1}^n\mathcal {L} _i

Then each Lagrangian in the equation could be expanded and explained.

Some helpful resources about the Standard Model and its Lagrangian :
Standard Model

The Standard Model of Particle Physics

Standard Model (mathematical formulation)

http://arxiv.org/pdf/hep-ph/0304186v1.pdf

Gauge Theory of Weak Interactions: Walter Greiner, Berndt Müller: 9783540878421: Amazon.com: Books

The Structure and Interpretation of the Standard Model, Volume 2 (Philosophy and Foundations of Physics): Gordon McCabe: 9780444531124: Amazon.com: Books

The Standard Model: A Primer: Cliff Burgess, Guy Moore: 9781107404267: Amazon.com: Books

An Introduction to the Standard Model of Particle Physics: W. N. Cottingham, D. A. Greenwood: 9780521852494: Amazon.com: Books

The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics: Robert Oerter: 9780452287860: Amazon.com: Books

Here is also a link to one of the  important papers in the history of the Standard Model written in 1967 by Weinberg and entitled ‘A Model of Leptons’ :
http://physics.princeton.edu/~mcdonald/examples/EP/weinberg_prl_19_1264_67.pdf

The equations of electrodynamics , tensors , and gravitation-Second Part

I’ll start this continuation of the topic of a previous post by  an exposition and explanation of the concept of tensors in relativity physics  given by Albert Einstein , from his 1916 paper The Foundation of the General Theory of Relativity :

” The laws of physics must be of  such a nature  that  they apply  to systems of reference in any kind of motion.

The general laws of nature are to be expressed by equations which hold good  for all systems of coordinates, that is, are co-variant  with respect to any substitutions whatever (generally covariant).

Let certain things (” tensors “)  be defined with respect to any  system of  coordinates  by  a  number of functions of  the  coordinates , called  the  ” components ”  of the  tensor. There  are  then   certain  rules  by  which   these components can  be calculated   for  a new system of  coordinates , if they are known  for the original  system  of coordinates, and if  the  transformation  connecting  the two systems is known. The things  hereafter  called    tensors are further characterized by the fact  that the equations of transformation for their  components are linear  and  homogeneous.  Accordingly,  all  the components in  the  new system  vanish,  if  they all  vanish in the  original system. If,  therefore, a law  of nature is expressed  by equating all the components of a tensor to zero, it  is  generally covariant. By  examining the  laws of  the formation of  tensors, we acquire  the  means  of  formulating generally covariant laws. “

Source : The Principle of Relativity (Dover Books on Physics)
Einstein’s paper can also be found here:

The Foundation of the Generalised Theory of Relativity

I will deal in this post with the derivation of the Einstein field equations of General Relativity and with the related equations of motion.

As in the previous post , I will not show all the long and detailed calculations but I will give an overview of the equations , calculations and derivations .

We start with the derivation of the field equations free from the sources generating the field.The equations are deduced from the principle of least action.

A small remark : in the equations below , the symbol g written in two ways is the same letter in two different fonts , it represents either the metric tensor (with subscript or superscript) or the determinant of its matrix representation.

The action of the gravitational field is given by:

action gravitation

where χ is the gravitational constant appearing in Einstein’s field equations.
Ω is the product of the differentials of the coordinates.
In four-dimensional space we have :

d \Omega =\text{dx}^0 \text{dx}^1 \text{dx}^2 \text{dx}^3 

In the action integral above , The scalar curvature (or the Ricci scalar) R is related to the Ricci tensor by the equation R = gij Rij  .
And the Ricci tensor is related to the Riemann curvature tensor by the equation:

Ricci tensor

Hence we have:

gravitation action integral

The variation of the action yields:

variation of action

After calculation the first integral in the variation of the action above gives:

first integral in action

The third integral in the variation of the action is equal to zero:

third integral in action

And the variation of the action can be written as:

em-grav-001d

The variations δgij are arbitrary and the Einstein field equations independent of the sources generating the field
(  Rij – (1/2)gij R = 0 ) can be deduced from the integral above.

The action for matter and the electromagnetic field is given by:

action for matter

This action and its variation involve the Lagrangian density:

action variation

After integrating and calculating we get:

action calculation

Defining the energy momentum tensor (or stress–energy–momentum tensor ) by the relation:

energy momentum tensor

We get for the variation of the action:

action with stress energy tensot

The preceding results lead to the condition δ(Sg+Sme)=0 being written as:

action variation result

The equality above must remain valid for arbitrary variations of δgij  ,and we deduce :

field equations

which are the Einstein field equations of General Relativity.

The contravariant form of the energy-momentum tensor is given by the transformation :

contravariant energy momentum tensor

Below is a schematic description of the contravariant components of the energy-momentum tensor (image from Wikipedia) :

components of contravariant tensor

Note that of we take the more general form of the action :

action with cosmological constant

The field equations for the most general case would become:

field equations with cosmological constant

where λ is the cosmological constant .

The Einstein field equations are in agreement with the conservation of the energy-momentum tensor , which means its divergence is null:

null divergence

The energy-momentum tensor (mixed tensor) can be written as the sum of two terms:

The first one is of electromagnetic origin:

electromagnetic mixed tensor

The second one involves the presence of matter:

mixed tensor matter

The condition of null divergence can be written as:

null divergence two terms

After some calculations we get for the first term :

first term of divergence

And for the second term in the null divergence equation:

second term null divergence

By using the last two equations above , equation (1) can be written as:

em-grav-12

which is the equation of motion of a continuous distribution of charge inside a gravitational field and an electromagnetic field.

The references and sources in my Science books page and the problem solvers page can be viewed for books which I have read , worked out or consulted and which are related to the topics in this post.
I found the book by Jean-Claude Boudenot ( Electromagnetisme et gravitation relativistes ) helpful while I was studying these topics in the past.

The equations of electrodynamics , tensors , and gravitation-First Part

The featured image above was made with Mathematica . It represents  the propagation of an electromagnetic wave.

Elements of this post were written for an answer I gave at quora.com , the question was about Maxwell’s equations
in tensor form. I have modified the answer and added material and equations related to General Relativity and gravitation. 

Maxwell’s equations are the fundamental equations of classical electromagnetism and electrodynamics. They  can be stated in integral form , in differential form (a set of partial differential equations) , and in tensor form.

The conventional differential  formulation of Maxwell’s equations in the International System of Units is given by:

Maxwell's equations differential form

In the above equations , E is the electric field (vector field) , B is the magnetic field ( pseudovector field) ,  ρ is the charge density , and  J is the current density.
ε0 is the Vacuum permittivity ,
and μ0 is the Vacuum permeability .

The second and third equations above form the first group of Maxwell’s equations (the generalized Faraday’s law of induction and Gauss’s law for magnetism) , and the the first and last equations above form the second group  (Gauss’s law and Ampere’s circuit law extended by Maxwell) .

The equations of electromagnetism (Maxwell’s equations and the Lorentz force) in covariant form (invariant under Lorentz transformations) can be deduced from the Principle of least action .

The electromagnetic wave equation can be derived from Maxwell’s equations , its solutions are electromagnetic (sinusoidal) waves.

All the steps and equations related to the topic of this post will not be shown here because they would  make the post much too long , but an overview of the equations , calculations and derivations  will be presented.

The development of the components of the Lorentz force and (after some calculations) its formulation in tensor form allows the introduction of the electromagnetic field tensor Fij :

Lorentz force

where uj is the four-velocity four-vector .

The variation of the action δS with respect to the coordinates of a particle gives the equations of motion. 

The action for a charged particle in an electromagnetic field can be expressed as:

action variation
q is the electric charge.

The variation of the action gives:

variation of action

After some calculations and an integration by parts the variation of the action becomes:

variation of action and integral

In the equation above , ui is the four-velocity four-vector.

Since the trajectories of the particle are supposed to have the same initial and final coordinates , the first term in the right-hand side of the equality above is equal to zero.
The potentials are a function of the coordinates , and  the following equalities can be used:

potential variation

Which gives for the variation of the action:

variation action & potential

Thus the electromagnetic field tensor emerges  from the variation above ; in fact the electromagnetic tensor is given by the definition:

electromagnetic tensor

 

Ai is the electromagnetic four-potential comprising the electric scalar potential V and the magnetic vector potential A .

The electromagnetic tensor is antisymmetric:

antisymmetric tensor

And the diagonal components in it are equal to zero:

diagonal components

 

The components of this tensor can be found using:

potential and field definition

For example:

tensor components

The others components can also be calculated and one gets for the covariant electromagnetic tensor:

electromagnetic tensor components

The components of the contravariant tensor can be found using:

contravariant tensor

where g^{\text{ij}} is the Metric tensor.

The transformation laws for the electromagnetic tensor are given by the general relation:

tensor transformation law

The following equation :

Maxwell equations first group

gives the first group of Maxwell’s equations .

 

If the Levi-Civita tensor ( e^{\text{ikmn}} ) is used ( a completely antisymmetric 4th rank tensor) , one gets for the first group of equations:

Levi-Civita tensor

Using the expression of the action with the Lagrangian density and taking the variation of the action as stationary:

Lagrangian

The last relation above represents the Lagrange equations. And the electromagnetic tensor  Fik is assumed to have been found and defined after developing the Lorentz force.

The relations above give:

Lagrangian equations
Thus Lagrange’s equations give the result:

Lagrange's equations Maxwell  result

And the second group of Maxwell’s equations is given by :

Maxwell equations second group

The second group of equations also gives the equation of charge conservation :

charge conservation
Another notation (the comma notation) and an additional way to write Maxwell’s equations in tensor form are the following :

Maxwell equations tensors comma notation

 

In a gravitational field the electromagnetic tensor is given by:

electromagnetic tensor gravitation

\nabla _j is the covariant derivative and \Gamma _{\text{ij}}^k  are Christoffel’s symbols .
So the relation between the electromagnetic tensor and the potentials is unchanged in the presence of a gravitational field.

The first and second groups of Maxwell’s equations become :

first and second group Maxwell equations

 

After calculation the first group can be shown to be equal to the original first group of equations without gravitation.

The second group can be expressed as:

second group with gravitation

where g is the determinant of the metric tensor.

The equations of motion of a particle of charge q in the presence of an electromagnetic field and a gravitational field are given by  :

equations of motion

Or equivalently:

motion of partilce

In General Relativity , the energy-momentum tensor Tij of an electromagnetic field in free space is expressed in the form of the electromagnetic stress- energy tensor:

energy momentum tensor
And the Einstein field equations

Einstein field equations.

are called the Einstein-Maxwell equations.

To be continued…

Question: what is the square root of 36 ? – Part Two

I will continue with answers and results equal to the square root of 36 ( originally answered at quora.com ). This time the results are mostly related to physics.

With physics one has to take into account the units and the corresponding dimensions of the equations and of the constants.

6 and the square root of 36 are dimentionless numbers , so the result must be dimentionless .
If the result is a simple fraction with numerator and denominator , then the units usually cancel out.
In other cases when one deals with logarithms one should multiply with the inverse dimensions to get a dimentionless result.
In one or two results where I didn’t look for the inverse units  I multiplied the equation with a quantity I called (U) ,  which represents the inverse of the units by which one should multiply the result to get a dimentionless number.

Here are the results :

square root of 36 physics

 

One possible way to explain what I have done here is the following:
If some people , living on an isolated fictitious island or on an another hypothetical planet , attached a great importance to and had a fixation on  the square root of 36 (or the number 6) for one reason or another , and got accustomed to the use of 6 as a fundamental constant ,  unit or number , then they would have likely  tried to construct a system of measurement  based on the number 6 , and  to express physics and math formulas ,equations , constants and rules in relation to 6.

After all , 6 or \sqrt{36} is equal to :

  • The floor of  2 \pi :
    2 \pi \approx 6.2831853071795864769 ;6=\lfloor 2 \pi \rfloor
  • \frac{1}{60} of the circumference of a circle in degrees.
  • It is also  one tenth of 60 seconds which make up a minute , one tenth of 60 minutes which make up an hour , one fourth of 24 hours which equal a day on Earth , one half of  12 months which make up a year , etc.
  • A peculiar ‘hexacentric’ system , so to speak.

Or this can be seen as a (creative) exploration of or exercise in advanced math and physics in order to express many equations , formulas and constants in relation to the number 6 (or \sqrt{36} ) .
Or whatever.

Apologies to Isaac Newton , Leonhard Euler , Bernhard Riemann , Einstein , Stokes , Coulomb , Avogadro , Lagrange , and others (wherever they may be) , for playing around with their equations , formulas , constants , and/or functions.

And one more addiction to this answer :

Does the future of humanity depend on answering what is the square root of  36 , or not?
Have philosophers from Antiquity to the present overlooked this fundamental question , which goes beyond the Kantian categories of space and time set out in his Critique of Pure Reason , and beyond Nietzsche’s Beyond Good and Evil , ushering the transmutation of all values and a defining moment for a new era  in the history of Humankind?
It’s just a square root , for common sense’s sake (or is it?).

Anyway , enough philosophizing.

Here are ( 3=\frac{\sqrt{36}}{2} ) more answers to \sqrt{36} , this time with images :

\sqrt{36}  is equal to :


The number subjected to a geometric rotation in the following image (done with Mathematica and some Photoshop) :

number 6 rotated

The number expressing the power and the coefficients in the equation of the curve in the polar plot below :

number 6 polar plot

The number expressing the degree of the root  and the power of the variables in the 3D plot below :

sinc number 6

The rotated number  and the polar plotted curve in the first two images  above seem to exhibit symmetry.
Symmetry is an very important property in science , math , physics , equations , nature , and wherever it is found.

Online sources and reference works related to what I have written in this answer can be found in my pages about Science books problem solvers and philosophy books in this site/blog.

Some other online sources:
http://mathworld.wolfram.com/

http://en.wikipedia.org/wiki/Category:Mathematics

http://en.wikipedia.org/wiki/Category:Physics

Question: what is the square root of 36 ? – Part One

I went over to www.quora.com a few weeks ago to answer a question about calendars , and then I got busy there. Since I am able to answer different types of questions , I started answering one question after the other , and I got stuck . I mean it’s a good way of getting stuck , answering questions about culture and science is useful and educational , but it can  become time consuming and it requires attention and dedication.

Anyway . somebody came up with a question about the square root of 36. This question was obviously a stale unoriginal question , probably meant as a joke , but I decided to spice it up a little and make it more interesting.
So I answered the question my own way ,  and I got a  good amount of likes and ‘upvotes’ .

I will rewrite the answer  I gave in here (with some modifications ).

Here are some results equal to \sqrt{36}  :

 

square root 36 results one

 

An here is another group of results equal to \sqrt{36} .
If one tries to work out or verify  these equalities , it would be a good exercise in intermediate and advanced math ( and physics).

 

square root of 36 results two

To be continued in another post.

About me and comics, comic books, and movies

This is a website and blog about culture , science , and related subjects , and it is supposed to be a serious website . But then again , as one famous jester and villain said :”Why so serious?”

Comics and comic books ( and movies ) are nowadays part of general and popular culture , and they have a place in this blog , given also the fact that I’ve been reading comic books ( and watching movies about them ) for many years.

I started reading comic books when I was a little kid.I read them in Arabic , in French and in English.
I read comics by DC Comics (Superman , Batman , the Flash ,and others) and Marvel Comics (Spiderman ,Fantastic Four ,Thor , and others) .Maybe I read a little more DC than Marvel Comics , but I read both , and I also read other types of comics.One series of comic books I bought and read years ago was about The Death and Return of Superman.
I also read a number of Franco-Belgian comic books , especially the Adventures of Tintin , and Asterix and Obelix.

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From a sociological point of view , comics and comic books reflect in general the beliefs , customs , conventions and accepted ideas of the society which produced them , as well as the sociopolitical milieu and period in which they were created.
Some comics about historical events, historical figures or adventures taking place in the past may be enjoyable but may also be fanciful and inaccurate (which can also be true of a number of historical novels).

Comic books such as those by DC and Marvel Comics display a good amount of facts related to or borrowed from science and physics.Sometimes these comics present stories and fiction with scientific insights and interesting information about physics , astrophysics ,chemistry and the exact sciences , but these stories also contain what could be called ‘technobabble’ , things and statements revolving around science but imprecise , unrealistic or unexplainable according to the known laws and rules of physics.At times the reader has to allow for exceptions to existing scientific and physical laws in order to continue reading the fictional tales presented to him.

One relationship or pairing I liked in DC Comics is the following:

Superman and Wonder woman

Some may prefer Superman staying with Lois Lane and marrying her , others like this new relationship and even imagine Superman and Wonder Woman getting married or having kids  . I think one possible  way to resolve the issue would be  to connect two storylines .In the movie ‘Superman Returns‘ , Superman had a son with Lois Lane , so if this fact could be used as an ending to the relationship between Lois and Clark , and the beginning of the relation between Diana and Clark considered as a continuation of that storyline , then this could be satisfactory for all fans and readers , and  the affair between the Son of Krypton and The Amazon Princess can go on to its completion.

Concerning one particular aspect of fictional science in comic books and movies ,and based on my own readings and analysis of scientific rules and facts , I think that although time travel is an interesting  idea or concept , it is  not possible and will not happen in reality. Perhaps I will write more about this subject in the future.

The Adventures of Tintin are all entertaining ,with two comic books in the series (Destination Moon and Explorers on the Moon) anticipating the first human trip to the Moon before it actually happened.It was a good piece of science fiction , although it contained a few errors.One book in the series , Vol 714 Pour Sydney or Flight 714 , which dealt with the existence of UFOs and Extraterrestrials , was in my opinion scientifically unverifiable and somewhat unrealistic.

Other Franco-Belgian comics I have read include :
Gaston Lagaffe , Spirou and Fantasio , Barbe Rouge , Iznogood , Ric hochet , Lucky Luke , Les Tuniques Bleus , Chick Bill , Les Schtroumpfs (the Smurfs) , Benoît Brisefer , Johan et Pirlouit (Johan and Peewit) , Alix , Les Petits Hommes …

Additional reference work related to this post:
The Physics of Superheroes : Spectacular Second Edition , by James Kakalios.