• Every physics student knows ODEs.
• A force proportional and opposite to displacement \(y\) leads to
• the ODE of simple harmonic motion (primes denote derivatives):

• The 2nd order ODE (1) has a unique solution
• if we prescribe initial data

• (initial displacement and initial velocity)

  \begin{align} y(0) &= y_0 , \nonumber\\ y'(0) &= y_1. \label{initial} \end{align}

• Above system can be solved forward in time from initial data
• or backward in time from final data
• or in any direction from intermediate data.

• Prescription of state ("initial data") at any point of time
• fully determines the solution (state) for all past and future times.
• Stateof the world at one instant determines all past and future (on Newtonian physics).

• The above considerations generalize to a system of \(n\) particles in 3D space
• results in \(3n\) second order ODEs
• or \(6n\) first-order ODE's
• state ("initial data") is \(6n\) numbers: \(3n\) positions, and \(3n\) momenta.

• These \(6n\) numbers prescribe a single point in
• \(6n\) dimensional space called phase space.
• Uniqueness of solutions means
• there is a unique trajectory through each point of phase space.
• All this changes with FDEs.

• Simplest FDE is

• this is also called a retarded FDE or a delay differential equation
• the value of \(y'\) is related to PAST values of \(y\).

• Suppose we try to solve the above equation like we solve an ODE.
• Symbolically

• To actually carry out the integration, we need to know the valus of \(y(t-1)\) between 0 and 1.

• That is, we need to know \(y(t)\) between -1 and 0, i.e., we need past data
• a whole function \(\phi\) over the relevant past interval [-1, 0].

• even if "initial data" same.
• Makes sense intuitively: an FDE relates rate of change NOW to something in past.
• So, changing past will change solution.

• Retarded FDEs can only be solved forward in time
• unlike ODEs they cannot be solved backward in time

• \(y(1) = 0\), no matter what \(k\) was.
• But, since \(b(t)=0\) for \(t \ge 1\),
• the FDE again reduces to the ODE \(y' (t) = 0\), for \(t \ge 1\),
• so that \(y(1) = 0\) implies \(y(t) = 0\) for all \(t \ge 1\).

• Hence, the past of a system modeled by a retarded FDE
• cannot be retrodicted from a knowledge of the entire future.
• Suppose the future data (i.e., values of the function for \textit{all} future times \(t \ge 1\))
• are prescribed using a function \(\phi\)

• if \(\phi\) is different from 0 on \([1,~ \infty ]\),
• then the retarded FDE admits no backward solutions for \(t \le 1\).
• If, on the other hand, $φ ≡ 0 $ on \([1,~ \infty ]\),
• then there are an infinity of distinct backward solutions.

• In either case, knowledge of the entire future furnishes no information about the past
• of a system modeled by retarded FDEs.

• Lack of information measured by average number of yes-no question you must ask to determine truth

• E.g. what is the missing letter?

  p

  ?

  t

• The missing letter must be a vowel,so the word must be one of pat, pet, pit, pot, put
• Suppose "u" is excluded, and others are equally likely (probability \(pᵢ = \frac{1}{4}\)).

• 1. Is it one of O or I?
• Then follow the tree.
• 2 questions are needed
• \(2 = -\log₂ \frac{1}{4} = - \sum_{i=1}^{4} {}~ pₜ \log pᵢ\)

• Note that information-theoretic or Shannon entropy is
• similar to Gibbs entropy (up to a multiplicative constant)

\[S_G(ρ) = -k ∫_{R^{6n}}ρ(x) \ln ρ(x) dx\]

• and Boltzmann entropy \(S_B = k \ln W\) can be seen as a special case of Gibbs entropy.

• Note that in Gibbs case, \(\rho(x)\) is a probability density
• which applies to an ensemble (multiple mental copies of the system) in phase space.
• In Boltzmann case, \(W\) is number of "microstates" consistent with a given "macrostate".
• Both definitions are a bit vague (which ensemble?, what is a microstate?)

• Solvability of retarded FDEs from past data
• + irreversibility (impossibility of solving retarded FDEs backward in time)
• means on retarded FDE model we have MORE information about past than about future.
• Means on retarded FDE model entropy increases towards future.

• Discussions in Pune Univ. canteen with colleagues in 1980's
• Pramod Joag, Naresh Dadhich etc.
• On issue of time in physics.
• Understanding time is tricky. Failed to get across my point of view in YEARS.

• Hence, wrote a series of 10 articles in Physics Education
• UGC journal brought out by Physics Department, Pune U.
• since the aim was to "carry on the discussion" with people around me
• and foreign journals were expensive, not easily available.

• You are talking about big things, changing physics, challenging Einstein
• but publishing in a local Journal.
• Valid science decided NOT by experiment/observation
• but by PUBLICATION by a "prestigious" (=Western) publisher!
• Science about prestige? Not truth?

• So, I added an article on mundane time, and published with the prestigious Kluwer academic series
• on fundamental theories of physics volume 65.

• Physics needs to be reformulated using FDEs (mixed-type)
• which explains most puzzling qualitative features of QM.
• Even just retarded FDEs are history-dependent
• i.e., need past data not just initial data
• Hence, doing physics using just retarded FDEs involves a paradigm shift in physics.

• Meeting (on "Retrocausality day")was stalled after I claimed paradigm shift due to FDE
• H. D. Zeh (Heidelberg): Why are FDEs needed in electrodynamics?
  • Heaviside-Lorentz force law (ODE) + Maxwell's equations (PDEs) enough
  • both need only "initial data"
• Zeh: Hence, no history dependence in physics, no paradigm shift.

• with the particle picture.
• And I said, I SOLVED the FDEs
• using past data.
• Challenged Zeh to solve them differently.
• (He said getting solution was a mathematician's job!)

• Q1. Are FDEs needed or not?
• Q2. Does physics need past data?
• So, let us re-examine Zeh's arguments.

• A charge moving in an electromagnetic field moves on Heaviside-Lorentz force

• Motion of charge determined by applying Newton's law of motion (and solving ODEs).

• The fields \(\vec{E}, \vec{B}\) due to motion of each charge determined using Maxwell's equations:

• We must solve these PDEs for each charge.
• Fields \(\vec{E}\) and \(\vec{B}\) are calculated as the derivatives of a scalar potential \(V\) and a vector potential \(\vec{A}\).

• Middle two of Maxwell's equations are automatically satisfied.
• The choice of potential is not unique, and we can add an extra condition (called a gauge condition) to simplify the remaining two equations.
• In the Lorenz gauge, the first and the last of Maxwell's equations turn into inhomogeneous wave equations
• for the scalar and vector potential respectively.

• The solutions are called Lienard-Wiechert potentials
• The retarded L-W potentials are given by the expressions:

• Here, \(\vec{R} = \vec{r} - \vec{r}_p(t_r)\), where \(\vec{r}_p (t_r)\)
• denotes the position of the charge \(q\) at the retarded time \(t_r\).
• The subscript ``ret'' emphasizes that \(\vec{v}\), the velocity of the charge $q$ is also to be evaluated at retarded time \(t_r\),
• so that \(\vec{v} =\dot{\vec{r}}_p(t_r)\).

• The retarded time \(t_r\) satisfies

• Similarly there are advanced potentials based on advanced (or future) times)

• Examining the above formula immediately shows foolish mistake made by Zeh
• Professor of Physics at Heidelberg and on editorial board of Foundations of Physics
• "Initial data" for Maxwell's equations means fields \(\vec{E}\), \(\vec{B}\) must be prescribed for one instant of time
• across all space.

• we need to know (retarded) positions and (retarded) velocities of both charges
• for ALL past time.
• So, past data needed.

• On electrodynamic 2-body problem
• It involves COUPLING of Heaviside-Lorentz force law with Maxwell's equations.
• "Initial (Cauchy) data" for fields = PAST data for particles.

• Coupling essential for 2-body problem of classical electrodynamics
• because motion of charge 1 depends on fields generated by motion of charge 2
• and vice versa.
• Little studied; not found in stock electrodynamics texts, Jackson etc.

• Explained history dependence and tilt (QM) for a layperson.
• Also explained how Christian church altered beliefs about time to suit its political needs.
• And how these religio-political belief creep into current science through Stephen Hawking etc.
• (Won't discuss it today; needs deeper knowledge of formal math and general relativity.)

• (But if valid science decided by blind belief in social prestige and myth
• that is a superstition which can be exploited.)

• Also, explained why Einstein did not understand that relativity forces use of FDEs
• which Poincaré, the true originator of relativity understood.
• Openly called Einstein a habitual plagiarist
• This shocked many people caused a major distraction for FDEs.

• Einstein is god, can't be critical of him.

This is an act of heresy bigger than Galileo's

• (for science is about worshipping scientists about whom there are big myths,
• not about critically understanding scientific theories).
• Publicized my viewpoint, e.g. 2004 news report.

• titled "On the electrodynamics of moving bodies".
• Michael Atiyah, former President, Royal Society, Abel Laureate,Fields Medalist gave his "Einstein Centenary Lecture"
• in which he repeated my point: FDEs can perhaps explain QM

• but added, "Don't forget I suggested this"
• To drive point home, in a socially savvy way
• he got it named "Atiyah's hypothesis", not "Einstein's mistake"

• did not acknowledge plagiarism but was indicted for it (case 2 of 2007).
• For pointing out "Einstein's mistake"
• I later received an award in Hungary in 2010.
• (But still no one believes Einstein was wrong.)

• on time travel
• on radiative damping
• on retarded gravitation

• again in Physics Education
• Retarded Gravitation Theory 2.0 was developed during Covid but not published.
• Why not? Because I am busy changing math education (impacts more people)
• and physicists may take very long to correct their time beliefs.

• Church politics of time (and related dogmas) have crept deep into Western thought AND science
• because of (1) long-term church hegemony over West (till 19th c.)
• and(2) Western hegemony over us and science after colonialism.
• Hence, conflicts about time in physics are hard to resolve.

• Conflicts about time dogged 19th c. physics
• especially thermodynamics
• resulting in Boltzmann's suicide in 1906.

• Entropy is non-decreasing.
• If \(S\) is entropy, \(\Delta S ≥ 0\)
• Boltzmann tried to prove H-theorem
• Entropy INCREASES until it reaches a maximum in state of thermodynamic equilibrium.

• Gas in a box is a collection of \(n\) molecules
• colliding elastically with each other and with walls of container.
• Elastic collisions means energy is conserved.
• Molecules motions obey Newtonian physics.

• Solution of ODEs is reversible
• so if entropy increase towards future
• it must also increase towards past
• Hence, entropy must stay constant.

• For formal math treatment see Time book, chp. 4, and appendix, and this explanation and this one.
• Here we give only the usual intuitive account in physics texts.
• Through every point of phase space there is a unique trajectory
• so trajectories in phase space never intersect.

• imagine a swarm of points flowing in phase space.
• Since phase space trajectories never intersect this swarm behaves like the flow of an incompressible fluid.
• That is, flow in phase space preserves volume (Liouville's theorem).

• For a gas in a box with finite energy phase space volume is finite
• therefore, the flow must return arbitrarily close to its initial state
• hence infinitely often.(Poincaré recurrence theorem.)
• Therefore, entropy cannot increase.

• E.g., if the gas is initially in one half of the box
• this state will recur repeatedly.

• Retarded FDEs are irreversible, hence reversibility paradox is resolved
• Retarded FDEs exhibit phase collapse, hence volume NOT preserved in phase space
• Hence, recurrence theorem does NOT apply.

• With retarded FDEs entropy increase perfectly possible
• Boltzmann need not have committed suicide.

• Explained entropy increase by assuming retarded potentials
• Derived thermodynamic arrow of time from electrodynamic arrow of time.

• We can restore time asymmetry by considering advanced solutions of wave equation.
• These result in advanced FDEs which will reduce entropy.
• Are these solutions physical?
• Yes, see paper on time travel. Will result in spontaneity (shown by living organisms)

• Most general case is that of mixed-type FDEs (with a "tilt")
• a convex combination of retarded and advanced propagator, where retarded interactions dominate.
• Domination of retardediteractions is an observation, not an assumption.

• A gas consists of molecules which consist of atoms
• which have a positively charged nucleus
• surrounded a by a negatively charged electron cloud
• An atom/molecule is electrically neutral only as an approximation at large-distance from its nucleus.
• At small distances, it may involve induced or permanent dipole, quadrupole, multipole interactions.

• which are both attractive and repulsive (because electron clouds repel each other).
• Keesom force (perm dipole- perm dipole)
• Debye force (perm dipole-induced dipole)
• London force (fluctuating dipole-induced dipole)

\[ V = \frac{A}{r^{12}} - \frac {B}{r^6} \]

• \(r\) is distance between two particles
• (\(A = 4ε \sigma^{1/12}\), \(B = 4ε \sigma^{1/6}\)

• since 19th c. physics was about Newtonian physics
• But the molecules in question are moving
• so molecular interactions should involve retarded action at a distance between dipoles etc.
• Modelling via retarded FDEs will add a small velocity-dependent component but explain entropy increase.