LOG#231. Statistical tools.

Subject today: errors. And we will review formulae to handle them with experimental data.

Errors can be generally speaking:

1st. Random. Due to imperfections of measurements or intrinsically random sources.

2st. Systematic. Due to the procedures used to measure or uncalibrated apparatus.

There is also a distinction of accuracy and precision:

1st. Accuracy is closeness to the true value of a parameter or magnitude. It is, as you keep this definition, a measure of systematic bias or error. However, sometime accuracy is defined (ISO definition) as the combination between systematic and random errors, i.e., accuracy would be the combination of the two observational errors above. High accuracy would require, in this case, higher trueness and high precision.

2nd. Precision. It is a measure of random errors. They can be reduced with further measurements and they measure statistical variability. Precision also requires repeatability and reproducibility.

1. Statistical estimators.

Arithmetic mean:

(1)   \begin{equation*}\boxed{\overline{X}=\dfrac{\displaystyle{\sum_{i=1}^n x_i}}{n}=\dfrac{\left(\mbox{Sum of measurements}\right)}{\left(\mbox{Number of measurements}\right)}}\end{equation*}

Absolute error:

(2)   \begin{equation*}\boxed{ \varepsilon_{a}=\vert x_i-\overline{x}\vert}\end{equation*}

Relative error:

(3)   \begin{equation*}\boxed{\varepsilon_r=\dfrac{\varepsilon_a}{\overline{x}}\cdot 100}\end{equation*}

Average deviation or error:

(4)   \begin{equation*}\boxed{\delta_m=\dfrac{\sum_i\vert x_i-\overline{x}\vert}{n}}\end{equation*}

Variance or average quadratic error or mean squared error:

(5)   \begin{equation*}\boxed{\sigma_x^2=s^2=\dfrac{\displaystyle{\sum_{i=1}^n}\left(x_i-\overline{x}\right)^2}{n-1}}\end{equation*}

This is the unbiased variance, when the total population is the sample, a shift must be done from n-1 to n (Bessel correction). The unbiased formula is correct as far as it is a sample from a larger population.

Standard deviation (mean squared error, mean quadratic error):

(6)   \begin{equation*}\boxed{\sigma\equiv\sqrt{\sigma_x^2}=s=\sqrt{\dfrac{\displaystyle{\sum_{i=1}^n}\left(x_i-\overline{x}\right)^2}{n-1}}}\end{equation*}

This is the unbiased estimator of the mean quadratic error, or the standard deviation of the sample. The Bessel correction is assumed whenever our sample is lesser in size that than of the total population. For total population, the standard deviation reads after shifting n-1\rightarrow n:

(7)   \begin{equation*}\boxed{\sigma_n\equiv\sqrt{\sigma_{x,n}^2}=\sqrt{\dfrac{\displaystyle{\sum_{i=1}^n}\left(x_i-\overline{x}\right)^2}{n}}=s_n}\end{equation*}

Mean error or standard error of the mean:

(8)   \begin{equation*}\boxed{\varepsilon_{\overline{x}}=\dfrac{\sigma_x}{\sqrt{n}}=\sqrt{\dfrac{\displaystyle{\sum_{i=1}^n}\left(x_i-\overline{x}\right)^2}{n\left(n-1\right)}}}\end{equation*}

If, instead of the unbiased quadratic mean error we use the total population error, the corrected standar error reads

(9)   \begin{equation*}\boxed{\varepsilon_{\overline{x},n}=\dfrac{\sigma_x}{\sqrt{n}}=\sqrt{\dfrac{\displaystyle{\sum_{i=1}^n}\left(x_i-\overline{x}\right)^2}{n^2}}=\dfrac{\sqrt{\displaystyle{\sum_{i=1}^n}\left(x_i-\overline{x}\right)^2}}{n}}\end{equation*}

Variance of the mean quadratic error (variance of the variance):

(10)   \begin{equation*}\boxed{\sigma^2\left(s^2\right)=\sigma^2_{\sigma^2}=\sigma^2\left(\sigma^2\right)=\dfrac{2\sigma^4}{n-1}}\end{equation*}

Standard error of the mean quadratic error (error of the variance):

(11)   \begin{equation*}\boxed{\sigma\left(s^2\right)=\sqrt{\sigma^2_{\sigma^2}}=\sigma\left(\sigma^2\right)=\sigma_{\sigma^2}=\sigma^2\sqrt{\dfrac{2}{n-1}}}\end{equation*}

2. Gaussian/normal distribution intervals for a given confidence level (interval width a number of entire sigmas)

Here we provide the probability of a random variable distribution X following a normal distribution to have a value inside an interval of width n\sigma.

1 sigma amplitude (1\sigma).

(12)   \begin{equation*}x\in\left[\overline{x}-\sigma,\overline{x}+\sigma\right]\longrightarrow P\approx 68.3\%\sim\dfrac{1}{3}\end{equation*}

2 sigma amplitude (1\sigma).

(13)   \begin{equation*}x\in\left[\overline{x}-2\sigma,\overline{x}+2\sigma\right]\longrightarrow P\approx 95.4\%\sim\dfrac{1}{22}\end{equation*}

3 sigma amplitude (1\sigma).

(14)   \begin{equation*}x\in\left[\overline{x}-3\sigma,\overline{x}+3\sigma\right]\longrightarrow P\approx 99.7\%\sim\dfrac{1}{370}\end{equation*}

4 sigma amplitude (1\sigma).

(15)   \begin{equation*}x\in\left[\overline{x}-4\sigma,\overline{x}+4\sigma\right]\longrightarrow P\approx 99.994\%\sim\dfrac{1}{15787}\end{equation*}

5 sigma amplitude (1\sigma).

(16)   \begin{equation*}x\in\left[\overline{x}-5\sigma,\overline{x}+5\sigma\right]\longrightarrow P\approx 99.99994\%\sim\dfrac{1}{1744278}\end{equation*}

6 sigma amplitude (1\sigma).

(17)   \begin{equation*}x\in\left[\overline{x}-6\sigma,\overline{x}+6\sigma\right]\longrightarrow P\approx 99.9999998\%\sim\dfrac{1}{506797346}\end{equation*}

3. Error propagation.

Usually, the error propagates in non direct measurements.

3A. Sum and substraction.

Let us define x\pm \delta x and y\pm \delta y. Furthermore, define the variable q=x\pm y. The error in q would be:

(18)   \begin{equation*}\boxed{\varepsilon (q)=\delta x+\delta y}\end{equation*}

Example. M_1=540\pm 10 g, M_2=940\pm 20 g. M_1=m_1+liquid, with m_1=72\pm 1g  and M_2=m_2+liquid, with m_2=97\pm 1g. Then, we have:

M=M_1-m_1+M_2-m_2=1311g as liquid mass.

\delta M=\delta M_1+\delta m_1+\delta M_2+\delta m_2=32g, as total liquid error.

M_0=1311\pm 32 g is the liquid mass and its error, together, with 3 significant digits or figures.

3B. Products and quotients (errors).

If

    \[x\pm \delta x=x\left(1\pm \dfrac{\delta x}{x}\right)\]

    \[y\pm \delta y=y\left(1\pm \dfrac{\delta x}{x}\right)\]

then, with q=xy you get

(19)   \begin{equation*}\boxed{\dfrac{\delta q}{\vert q\vert}=\dfrac{\delta x}{\vert x\vert}+\dfrac{\delta y}{\vert y\vert}=\vert y\vert\delta x+\vert x\vert\delta y}\end{equation*}

If q=x/y, you obtain essentially the same result:

(20)   \begin{equation*}\boxed{\dfrac{\delta q}{\vert q\vert}=\dfrac{\delta x}{\vert x\vert}+\dfrac{\delta y}{\vert y\vert}=\vert y\vert\delta x+\vert x\vert\delta y}\end{equation*}

3C. Error in powers.

With x\pm \delta x, q=x^n, then you derive

(21)   \begin{equation*}\dfrac{\delta q}{\vert q\vert}=\vert n\vert \dfrac{\delta x}{\vert x\vert}=\vert n\vert \vert x^{n-1}\vert \delta x\end{equation*}

and if g=f(x), with the error of x being \delta x, you get

(22)   \begin{equation*}\boxed{\delta f=\vert\dfrac{df}{dx}\vert\delta x}\end{equation*}

In the case of a several variables function, you apply a generalized Pythagorean theorem to get

(23)   \begin{equation*}\boxed{\delta q=\delta f(x_i)=\sqrt{\displaystyle{\sum_{i=1}^n}\left(\dfrac{\partial f}{\partial x_i}\delta x_i\right)^2}=\sqrt{\left(\dfrac{\partial f}{\partial x_1}\delta x_1\right)^2+\cdots+\left(\dfrac{\partial f}{\partial x_n}\delta x_n\right)^2}}\end{equation*}

or, equivalently, the errors are combined in quadrature (via standard deviations):

(24)   \begin{equation*}\boxed{\delta q=\delta f (x_1,\ldots,x_n)=\sqrt{\left(\dfrac{\partial f}{\partial x_1}\right)^2\delta^2 x_1+\cdots+\left(\dfrac{\partial f}{\partial x_n}\right)^2\delta^2 x_n}}\end{equation*}

since

(25)   \begin{equation*}\sigma (X)=\sigma (x_i)=\sqrt{\displaystyle{\sum_{i=1}^n}\sigma_i^2}=\sqrt{\sigma_1^2+\cdots+\sigma_n^2}\end{equation*}

for independent random errors (no correlations). Some simple examples are provided:

1st. q=kx, with x\pm \delta x, implies \boxed{\delta q=k\delta x}.

2nd. q=\pm x\pm y\pm \cdots, with x_i\pm \delta x_i, implies \boxed{\delta q=\delta x+\delta y+\cdots}.

3rd. q=kx_1^{\alpha_1}\cdots x_n^{\alpha_n} would imply

    \[\boxed{\dfrac{\delta q}{\vert q\vert}=\vert\alpha_1\vert\dfrac{\delta x_1}{\vert x_1\vert}+\cdots +\vert\alpha_n\vert\dfrac{\delta x_n\vert}{\vert x_n\vert}}\]

When different experiments with measurements \overline{x}_i\pm\sigma_i are provided, the best estimator for the combined mean is a weighted mean with the variance, i.e.,

(26)   \begin{equation*}\overline{X}_{best}=\dfrac{\displaystyle{\sum_{i=n}^n}\dfrac{x_i}{\sigma^2_i}}{\displaystyle{\sum_{i=1}^n}\frac{1}{\sigma^2_i}}\end{equation*}

This is also the maximal likelihood estimator of the mean assuming they are independent AND normally distributed. There, the standard error of the weighted mean would be

(27)   \begin{equation*}\sigma_{\overline{X}_{best}}=\sqrt{\dfrac{1}{\displaystyle{\sum_{i=1}^n}\dfrac{1}{\sigma^2_i}}}\end{equation*}

Remark: for non homogenous samples, the best estimation of the average is not the arithmetic mean, but the median.

See you in other blog post!

LOG#230. Spacetime as Matrix.

Surprise! Double post today! Happy? Let me introduce you to some abstract uncommon representations for spacetime. You know we usually represent spacetime as “points” in certain manifold, and we usually associate points to vectors, or directed segments, as X=X^\mu e_\mu, in D=d+1 dimensional spaces IN GENERAL (I am not discussing multitemporal stuff today for simplicity).

Well, the fact is that when you go to 4d spacetime, and certain “dimensions”, you can represent spacetime as matrices or square tables with numbers. I will focus on three simple examples:

  • Case 1. 4d spacetime. Let me define \mathbb{R}^4\simeq \mathbb{R}^{3,1}=\mathbb{R}^{1,3}\simeq \mathcal{M}_{2x2}(\mathbb{C}) as isomorphic spaces, then you can represent spacetime X^\mu e_\mu=X as follows

(1)   \begin{equation*} \boxed{X=\begin{pmatrix} x^0+x^3& x^1+ix^2\\ x^1-ix^2& x^0-x^3\end{pmatrix}=\begin{pmatrix} x^0+x^3& z\\ \overline{z}& x^0-x^3\end{pmatrix}}\end{equation*}

and where z\in\mathbb{C}=x^1+ix^2=x^1+x^2e_2=\displaystyle{\sum_{j=1}^2}x^je_j is a complex number (e_1=1).

  • Case 2. 6d spacetime. Let me define \mathbb{R}^6\simeq \mathbb{R}^{5,1}=\mathbb{R}^{1,5}\simeq \mathcal{M}_{2x2}(\mathbb{H}) as isomorphic spaces, then you can represent spacetime X^\mu e_\mu=X as follows

(2)   \begin{equation*} \boxed{X=\begin{pmatrix} x^0+x^5& x^1+ix^2+jx^3+kx^4\\ x^1-ix^2-jx^3-kx^4& x^0-x^5\end{pmatrix}=\begin{pmatrix} x^0+x^5& q\\ \overline{q}& x^0-x^5\end{pmatrix}}\end{equation*}

and where q\in\mathbb{H} is a quaternion number q=x^1+ix^2+jx^3+kx^4=x^1+x^2e_2+x^3e_3+x^4e_4=\displaystyle{\sum_{j=1}^4}x^je_j, with e_1=1.

  • Case 3. 10d spacetime. Let me define \mathbb{R}^{10}\simeq \mathbb{R}^{9,1}=\mathbb{R}^{1,9}\simeq \mathcal{M}_{2x2}(\mathbb{O}) as isomorphic spaces, then you can represent spacetime X^\mu e_\mu=X as follows

(3)   \begin{equation*} \boxed{X=\begin{pmatrix} x^0+x^9& x^1+\sum_jx^je_j\\ x^1-\sum_jx^je_j& x^0-x^9\end{pmatrix}=\begin{pmatrix} x^0+x^9& h\\ \overline{h}& x^0-x^9\end{pmatrix}}\end{equation*}

and where h\in\mathbb{O} is

h=\displaystyle{\sum_{j=1}^8}x^je_j=x^1+x^2e_2+x^3e_3+x^4e_4+x^5e_5+x^6e_6+x^7e_7+x^8e_8 is an octonion number with e_1=1.

Challenge final questions for you:

  1. Is this construction available for different signatures?
  2. Can you generalize this matrix set-up for ANY spacetime dimension? If you do that, you will understand the algebraic nature of spacetime!

Hint: Geometric algebras or Clifford algebras are useful for this problem and the above challenge questions.

Remark: These matrices are useful in

  • Superstring theory.
  • Algebra, spacetime algebra, Clifford algebra, geometric algebra.
  • Supersymmetry.
  • Supergravity.
  • Twistor/supertwistor spacetime models of spacetime.
  • Super Yang-Mills theories.
  • Brane theories.
  • Dualities.
  • Understanding the Hurwitz theorem.
  • Black hole physics.

LOG#229. Mars organics.

Hi, there. Short post today ignites a new category post. Life and Chemistry. The search of life outside Earth, and beyond, is a goal for the current and forthcoming centuries (provided we are not extincted). First targets for life searches in the Solar System include: solar system planets, some of their moons and maybe comets.

Between all the solar system planets (neglecting Moon by naive assumptions, perhaps we should reconsider that if there is ice and water enough on our satellite), Mars is an ideal place to search for life. Other targets like Titan, Ganymede, Ceres, Pluto and others will not be covered today. What do we know about Mars? Mars has a thin atmosphere made of carbon dioxide. And, after decades, we also know some chemical compounds on Mars (with more or less uncertainty):

  • Sulfur-like compounds. A list includes
  1. Thiophen C_4H_4S.
  2. Methyl thiophenes C_5H_6S.
  3. Methanethiol CH_4S.
  4. Dimethyl sulfide C_2H_6.
  5. Benzothiophene C_8H6S.
  • Non sulfur-like compounds.
  1. Benzene C_6H_6.
  2. Toluene (or tropylium ion C_7H_7^+).
  3. Alkylbenzenes (C_8H_9 or benzoate ion C_7H_5O^-).
  4. Chlorobenzene C6H_5Cl.
  5. Nophtahlene C_{10}H_8.

Smaller molecules seen on Mars (of course, beyond CO_2) are:

  1. Carbonyl sulfide COS.
  2. Oxygen O_2.
  3. Carbon disulfide CS_2.
  4. Carbon monoxide CO.
  5. Hydrogen sulfide H_2S.
  6. Sulfur dioxide SO_2.
  7. Methane CH_4. The origin of martian methane is yet a mystery. Proof of life, interior geology or exotics or complex mechanims? We do not know.

There are lots of carbon-chain molecules, with about 1 up to 5 carbon atoms likely on Mars soil. Would we find out azobenzene C_{12}H_{10}N_2 molecules on Mars? Likely not. Of course, we will not found superconductors or MoS_2 in principle, but now we do know there is water below the Mars soils. Ice. And likely salty subterran water cycles. Are there bacteria and other life beings, even microscopic or bigger, on Mars right no? Impossible to say yet! New rovers will try to uncover the biggest mysteries on the red planet and finally to decide if there is some kind of microorganisms or even life beings hidden on Mars! We will know in this century for sure! If not on Mars, Titan and other places of the Solar System like Enceladus have prospective high odds to sustain some king of life. Europe as well. But do not try to land there ;).

See you in another blog post!

LOG#228. The scientific method.

What is the scientific method? There are many definitions out there, but I am providing mine, the one I explain to my students, in this short post.

SCIENTIFIC METHOD (Definition, not unique)

A (cyclic) method/procedure to gather/organize, check (verify or refute) and test, conserve/preserve and transmit/communicate knowledge (both in form of data or organized abstract data/axioms/propositions) or more generally information, based on:

  • Experience. By experience we understand observation of natural phenomena, original thoughts, common sense perceptions and observed data from instruments or data. You can also gather data with emulation or simulation of known data, in a virtual environtment.
  • Intuition and imagination. Sometimes scientific ideas come from experience, sometimes from intuitions and abstractions from real world and/or structures. You can also use imagination to test something via gedanken or thought experiments tied to the previous experiences or new experiences, or use computer/AI/machines to creatively check or do inferences.
  • Logic and mathematical language. Logic, both inductive and deductive, is necessary for mathematical or scientific proofs. Since Galileo, we already know that Mathematics is the language in which Nature is better described with. We can also say that this includes reasoning or reason as a consequence.
  • Curiosity. The will to know is basic for scientists. No curiosity, no new experiments, observations, theories or ideas.

The scientific method has some powerful tools:

  • Computers and numerical simulations. This is new from the 20th century. Now, we can be aided by computer calculations and simulations to check scientific hypothesis or theories. Machine learning is also included here as subtool.
  • Statistics and data analysis. Today, in the era of Big Data and the Rise of AI, this branch and tool from the scientific method gains new importance.
  • Experimental devices to measure quantities predicted or expected from observations and or hypotheses, theories or models.
  • Rigor. Very important for scientists, and mathematicians even more, is the rigor of the method and analysis.
  • Scientific communication, both specialized and plain for everyone. Scientists must communicate their results and findings for testing. Furthermore, they must try to make accessible the uses of their findings or why they are going to be useful or not in the future.

Scientific method can begin from data, or from theories and models. Key ideas are:

  • (Scientific) Hypothesis. Idea, proposition, argument or observation that can be tested in any experiment. By experiment, here, we understand also computer simulations, numerical analysis, observation with telescope or data analysis instruments, machine/robotic testing, automatic check and/or formal proof by mathematical induction or deduction.
  • An axiom is a statement that is assumed to be true without any proof, based on logical arguments or experience.
  • A theory is a set of tested hypotheses subject to be proven before it is considered to be true or false. A theory is also a set of statements that is developed through a process of continued abstractions and experiments. A theory is aimed at a generalized statement or also aimed at explaining a phenomenon.
  • A model is a purposeful representation of reality.
  • A conjecture is proposition based on inconclusive grounds, and sometimes can not be fully tested.
  • A paradigm (Kuhn) is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitutes legitimate contributions to a field.

What properties allow us to say something is scientific and something is not? Philosophy of science is old and some people thought about this question. Some partial answers are known:

  • Falsifiability. Any scientific idea or hypothesis or proposition can be refuted and tested. Otherwise is not science. It is a belief. Scientific stuff can be refutable and argued against with. Experiments or proof can be done to check them. Kuhn defended the addition of additional ad hoc hypotheses to sustain a paradigm, Popper gave up this approach.
  • Verification of data or hypotheses/theories/arguments. Even when you can refute and prove a  theory is wrong, verification of current theories or hypotheses is an important part of scientific instruments.
  • Algorithmic truths and/or logical procedures. Science proceeds with algorithms and/or logic to test things.  Unordered checking looses credibility. Trial and error is other basic procedure of Science.
  • Heuristics arguments based on logic and/or observations. Intuition and imagination can provide access to scientific truths before testing.
  • Reproducibility. Any experiment or observation, in order to be scientific, should be reproducible.
  • Testable predictions. Usually, theories or hypotheses provide new predictions, not observed before.

The scientific method is an iterative, cyclical process through which information is continually revised. Thus, it can be thought as a set of 4 ingredients as well:

  • Characterizations (observations, definitions, and measurements of the subject of inquiry).
  • Hypotheses (theoretical, hypothetical explanations of observations and measurements of the subject).
  • Predictions (inductive and deductive reasoning from the hypothesis or theory).
  • Experiments (tests of all of the above).

Pierce distinguished between three types of procedures:

  • Abduction. It is a mere “guess”, intuitive and not too formal.
  • Deduction. It includes premises, explanations and demonstrations.
  • Induction. A set of classification, probations and sentient reasoning.

From a pure mathematical and theorist way, there are only knowing and understanding facts, analysis, synthesis and reviews or extensions of information/knowledge. From the physical or experimentalist viewpoint, however, we have more:

  • Characterization of experiences and observations.
  • Proposals of hypotheses.
  • Deductions and predictions from hypotheses.
  • Realization of tests and experiments (gathering data).

Note that, from a simple viewpoint, the scientific method and/or main task of Science is to study:

  • Regularities, patterns and relationships between objects and magnitudes.
  • Anomalies or oddities, generally hinting something new beyond standard theories.
  • Reality as something we measure and the link between observers and that reality. What is reality after all? Hard question from the quantum realm side…

By the other hand, a purely bayesianist approach to Science is also possible. In a Bayesian setting, Science is only a set up to test the degree of belief of any proposition/idea/set of hypotheses/model/theory. Theories provide measurable observables and quantities, and scientific predictions are only valid up to certain confidence level with respect some probabilistic distributions. This probabilistic approach to Science does not exclude the existence of purely true or false hypotheses, a frequentist approach to data and error analysis (it complements that tool), and it only focuses on a framework to estimate the probability of propositions, data vectors and experimental parameters fitting certain probability distributions “a prior”.

How to elucidate the degree of (scientific) belief of something? W. K. Clifford discussed this topic with Jaynes in order to give a list. In the Ethics of Belief was argued that: rules or standards that properly govern responsible belief-formation and the pursuit of intellectual excellence are what philosophers call epistemic (or “doxastic”) norms. Widely accepted epistemic norms include:

  • Don’t believe on insufficient evidence.
  • Proportion your beliefs to the strength of the evidence.
  • Don’t ignore or dismiss relevant evidence.
  • Be willing to revise your beliefs in light of new evidence.
  • Avoid wishful thinking.
  • Be open-minded and fair-minded.
  • Be wary of beliefs that align with your self-interest.
  • Admit how little you know.
  • Be alert to egocentrism, prejudice, and other mental biases.
  • Be careful to draw logical conclusions.
  • Base your beliefs on credible, well-substantiated evidence.
  • Be consistent.
  • Be curious and passionate in the pursuit of knowledge.
  • Think clearly and precisely.
  • Carefully investigate claims that concern you.
  • Actively seek out views that differ from your own.
  • Be grateful for constructive criticisms.
  • Question your assumptions.
  • Think about the implications of your beliefs.
  • Persevere through boring or difficult intellectual tasks.
  • Be thorough in your intellectual work.
  • Stick up for your beliefs, even in the face of peer pressure, ridicule, or intolerance.

Unanswered questions by Science are yet to be provided:

  1. Why mathematics is so accurate and precise to describe Nature?
  2. Why is the Universe comprehensible and non-chaotic but regular and structured in general? It could have been very different!
  3. Why numbers and structures are so efficient?
  4. Is Science affected by the Gödel theorems or does it go beyond its applicability?
  5. Can Science explain everything?
  6. Are chaos and other mathematical universes possible and physically realizable or ideally are only unfeasible?

Usually, the scientific method contained theory and experiment only. Now, it also include: computation, big data, machine learning and AI tools!

See you in another blog post!

LOG#227. Cosmic energy.

Short post number two! Surprise!

Have you ever wondered what is the cosmic energy of the Universe? Well, giving up certain General Relativity issues related to the notion of energy in local sense, there is indeed a global notion of energy for the Universe as a whole. I am not considering the Multiverse as an option today. Let me begin with the High School notion of mass and density, particularized for the Universe:

    \[M_U=\rho_UV_U\]

We are considering a closed spherical Universe with 3-d geometry, and then its volume reads

    \[V_U=\dfrac{4}{3}\pi R_U^3\]

What is the radius of the Universe? Well, we could take it as the Hubble radius of the observable Universe, i.e.,

    \[R_U=\dfrac{c}{H}\]

where H\approx 70km/s/Mpc. The density of the Universe can be written as the cosmological value of the vacuum/Hubble scale

    \[\rho_U=\dfrac{\Lambda c^4}{8\pi G}=\dfrac{3c^2H^2}{8\pi G}\]

so \Lambda c^4=3c^2H^2. Therefore, the formula for the mass of the Universe in terms of fundamental constants is

    \[\boxed{M_U=\dfrac{c^3}{2GH}=\dfrac{c^2}{2G}\sqrt{\dfrac{3}{\Lambda}}}\]

and the expression for the cosmic energy follows up from special relativity greatest formula E_U=M_Uc^2 as

    \[\boxed{E_U=\dfrac{c^5}{2GH}=\dfrac{c^4}{2G}\sqrt{\dfrac{3}{\Lambda}}}\]

Also, defining L_\Lambda=\sqrt{3/\Lambda} and L_P ^2=G\hbar/c^3, you get

    \[\boxed{E_U=\dfrac{\hbar}{2}\left(\dfrac{cL_\Lambda}{L_P^2}\right)}\]

Remark: the cosmological constant fixes not only the biggest mass as the Universal mass of the Universe (I am sorry for pedantic expression), but also fixes the smallest possible mass (the so-called Garidi mass in de Sitter group or de Sitter relativity):

    \[M_\Lambda=\dfrac{\hbar}{c}\sqrt{\dfrac{\Lambda}{3}}\]

And you can thus prove that

    \[\dfrac{M_U}{M_\Lambda}=\dfrac{1}{2}\dfrac{c^3}{G_N\hbar}\dfrac{3}{\Lambda}\sim \dfrac{L_\Lambda^2}{L_P^2}\sim 10^{122}\]

Note, that

    \[M_P=\dfrac{\hbar}{c}\dfrac{1}{L_P}\]

    \[M_\Lambda=\dfrac{\hbar}{c}\dfrac{1}{L_\Lambda}\]

    \[M_U=\dfrac{\hbar}{c}\dfrac{L_\Lambda}{L_P^2}\]

    \[V_P\sim L_P^3\]

    \[V_\Lambda=V_U\sim L_\Lambda^3 \]

    \[\dfrac{M_P}{M_\Lambda}\sim\dfrac{L_\Lambda}{L_P}\]

    \[\dfrac{M_U}{M_P}\sim\dfrac{ L_\Lambda}{L_P}\]

    \[\dfrac{M_U}{M_\Lambda}\sim\dfrac{ L_\Lambda^2}{L_P^2}\]

and thus

    \[\dfrac{\rho_P}{\rho_\Lambda}=\dfrac{L_\Lambda^4}{L_P^4}\]

    \[\dfrac{\rho_P}{\rho_U}=\dfrac{L_\Lambda^2}{L_P^2}\]

    \[\dfrac{\rho_U}{\rho_\Lambda}=\dfrac{L_\Lambda^2}{L_P^2}\]

See you in other blog post!!!!!

LOG#226. Higgs vs. lambda.

Hi, there. Short post today!

The dark energy mystery, a.k.a., the cosmological constant problem or why the observed vacuum energy is to tiny is a big problem with no consensus solution yet.

The Standard Model Higgs potential can be written as follows

    \[V_H=-m_H^2\phi^2+\lambda\phi^4+\phi_0\]

Here, m_H is the Higgs mass m_H=125GeV, \lambda=0.13 is the Higgs self-coupling from the given mass and \phi_0 is the vacuum expectation value of the Higgs field, namely

    \[\phi_0=\langle 0\vert \phi\vert 0\rangle=246GeV\]

From High School mathematics, you can easily find out the minimum of the above potential

    \[\dfrac{d V_H}{d\phi}=-2m_H^2\phi+4\lambda\phi^3=2\phi\left(-m_H^2+2\lambda\phi^2\right)=0\]

so the non null solution is given by

    \[\phi_0(min)^2=\dfrac{m_H^2}{2\lambda}\]

and the Higgs potential value at that value becomes

    \[V(\phi(min))=-\dfrac{m_H^2\phi^2}{2\lambda}+\phi_0\approx -\dfrac{m_H^2\phi^2}{4\lambda}\]

The Higgs potential provides a natural source for vacuum energy density. Using the values mentioned before we get

    \[\vert V_H(min)\vert\approx 5\cdot 10^8GeV^4\]

The value we get from Cosmology, using the LCDM model is however very tiny

    \[V(\Lambda)=\dfrac{\Lambda c^4}{8\pi G}\approx 3.1\cdot 10^{-47}GeV^4\]

thus, we have a mismatch between theory and experiment about

    \[\dfrac{\rho(Higgs)}{\rho(\Lambda)}\sim 2\cdot 10^{55}\]

Note that this is, despite the mismatch, better than the crude mismatch due to the even bigger discrepancy between Planck energy density and observed vacuum energy density, since Planck energy density reads:

    \[\rho_P=\dfrac{c^7}{G^2_N\hbar}\approx 5\cdot 10^{113}J/m^3\approx  2.2\cdot  10^{76}GeV^4\]

and thus

    \[\dfrac{\rho_P}{\rho_\Lambda}\sim 10^{122}\]

Or,as well,

    \[\dfrac{\rho_P}{\rho_\Lambda}=\dfrac{\dfrac{c^7}{G^2_N\hbar}}{\dfrac{\Lambda c^4}{8\pi G_N}}=\dfrac{8\pi}{3}\dfrac{c^3}{G_N\hbar}\dfrac{3}{\Lambda}=\dfrac{8\pi}{3}\dfrac{L_\Lambda^2}{L_P^2}\sim \dfrac{L_\Lambda^2}{L_P^2}\]

Solutions? Well, many:

  • The Higgs value is wrong, because the Higgs potential from the SM is not right, but only an approximation.
  • The Cosmological Constant is not the Higgs potential source.
  • Both values are OK, we have bad measurements only.
  • Vacuum energy is scale dependent, you can not compare them without some tricky trick.
  • Some non-perturbative effect in under the floor.
  • A new physics reason not quoted above.

I know ther are some other crude estimates from UV-cutoff given a 122-123 orders of magnitude separation. However, this is much closer. Some SUSY (supersymmetry) theories have some arguments to essentially fit some of the solutions I mentioned. But the issue is not at all clear. A very big cosmological constant would be a disaster for life. The very big vacuum energy of Quantum Field Theory grosser estimates clearly are nonsense, but we do not know yet when the calculation is wrong. Maybe a dS (de Sitter) QFT could provide a better solution using \Lambda as fundamental constant?

Note, do not confuse the Higgs self-coupling \lambda with the cosmological constant \Lambda.

What do you think? A simple minimization of the Higgs potential according the SM gives you a 55 order of magnitude split with observed vacuum energy density? Are they the same or are we lacking something more fundamental?

Off-topic news: it seems the quark-gluon plasma behaves as an ideal relativistic fluid, as the holographic hypothesis suggest, giving experimental increasing support to the bound of shear viscosity to entropy ratio predicted by those holographic models

    \[\dfrac{\eta}{s}\geq\dfrac{\hbar}{4\pi k_B}\]

or in natural units

    \[\dfrac{\eta}{s}\geq \dfrac{1}{4\pi }\]

References:

[1] Jonah E. Bernhard, J. Scott Moreland, Steffen A. Bass, «Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma,» Nature Physics (12 Aug 2019), doi: 10.1038/s41567-019-0611-8

[2]  Kari J. Eskola, «Nearly perfect quark–gluon fuid,» Nature Physics (12 Aug 2019), doi: 10.1038/s41567-019-0643-0.

See you in another blog post!!!!

LOG#225. QCD: the matrices.

Quantum Chromodynamics (QCD). The theory I learned that explains the nuclei as a teenager, i.e, the theory what explained why nuclei were not ripped off away due to nuclear forces. Strong nuclear forces (weak nuclear forces are the cause of radioactivity). Quarks and gluons. Matter and particle force carriers or messengers. I remember yet how some teens asked me and smiled with bad intentions what asked me what I was doing during the weekends and the week out of class. Sad news is that yet I have to keep calm seeing the same smile and similar comments about what I like to read, what I like to do or talk at my current age. Dark fate, I think…

Quarks are the matter fields feeling the strong nuclear force. Leptons do not feel color force. Color charges are just a name, not related to physical color though. It is just a way to work out a triad of gauge charges unlike electric charges (a diad) or gravitational charge (aka mass, a monad). How many colors are there? Three: red (R), blue (B) and green (G). Plus anticolor charges, antired \overline{R}, antiblue \overline{B} and antigreen \overline{G}. When two quarks interact, can swap charges or color via gluons. The interaction is something like a chemical reaction

    \[RB\leftrightarrow R\overline{B}\leftrightarrow BR\]

You can indeed imagine anticolor as color charges backwards in time. Expected gluons can be classified as follows (we will see that there are, in fact, only 8 possible color gluons independent to each other):

    \[\uparrow\uparrow=R\overline{R}\]

    \[\uparrow\downarrow=R\overline{B}\]

    \[\uparrow\downarrow=R\overline{G}\]

    \[\uparrow\uparrow=B\overline{R}\]

    \[\uparrow\uparrow=B\overline{B}\]

    \[\uparrow\downarrow=B\overline{G}\]

    \[\uparrow\downarrow=G\overline{R}\]

    \[\uparrow\downarrow=G\overline{B}\]

    \[\uparrow\uparrow=G\overline{G}\]

Note that these letters allow you to get colorless combinations with different quantum color numbers, thus being coherent with Pauli exclusion principle. States like uuu are possible ir you understand u_Ru_Bu_G, so quarks have different quantum numbers. Color is related as well to hypercharge and isospen, different combinations of isospin and hypercharge also label quarks and hadronic states. Experimentally, as color particles are not observable, only colorless states are possible. This fact settles a problem for B\overline{B},R\overline{R},G\overline{G} states, but, as surely you know, there are two main species of hadrons: baryons (quarky threesomes), and mesons (quarky couples). However, today hadron spectroscopy is much richer. Lattice field theories and experiments show us that there are also resonances AND, likely, exotics. Exotics are states not being purely couples or threesomes of quarks. You can get purely gluonic states called glueballs or gluonium by experts, and also you can get in principle quarky foursomes, quarky pentets and more. I read some ago a paper about heptaquarks and octoquarky states. Crazy! I wish I were such a quarky state sometimes.

What about gauge symmetries for quarks? It is called SU(3)_c symmetry. The fact protons get masses is due not to the Higgs mechanism, as I mentioned some days ago, but to a dynamical non-perturbative process of chiral symmetry breaking called sometimes dimensional transmutation. What a name! Gluons can be described by some 3\times 3 special unitary matrices or “grids”. These matrices have to be traceless (the sum over the main diagonal of its entries adds up to zero!). Let me represent color and anticolor states as column vectors and row vectors, respectively:

    \[R=\begin{pmatrix}1\\ 0\\0\end{pmatrix}\]

    \[B=\begin{pmatrix}0\\ 1\\0\end{pmatrix}\]

    \[G=\begin{pmatrix}0\\ 0\\1\end{pmatrix}\]

    \[\overline{R}=\begin{pmatrix}1& 0&0\end{pmatrix}\]

    \[\overline{B}=\begin{pmatrix}0&1&0\end{pmatrix}\]

    \[\overline{G}=\begin{pmatrix}0& 0&1\end{pmatrix}\]

Then, you can get by tensor product the 3\times 3 matrices

    \[R\overline{R}=R\otimes\overline{R}=\begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\end{pmatrix}\]

    \[R\overline{G}=R\otimes\overline{G}=\begin{pmatrix} 0&1&0\\0&0&0\\0&0&0\end{pmatrix}\]

    \[R\overline{B}=R\otimes\overline{B}=\begin{pmatrix} 0&0&1\\0&0&0\\0&0&0\end{pmatrix}\]

    \[G\overline{R}=G\otimes\overline{R}=\begin{pmatrix} 0&0&0\\1&0&0\\0&0&0\end{pmatrix}\]

    \[G\overline{G}=G\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&1&0\\0&0&0\end{pmatrix}\]

    \[G\overline{B}=G\otimes\overline{B}=\begin{pmatrix} 0&0&0\\0&0&1\\0&0&0\end{pmatrix}\]

    \[B\overline{R}=B\otimes\overline{R}=\begin{pmatrix} 0&0&0\\0&0&0\\1&0&0\end{pmatrix}\]

    \[B\overline{G}=B\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&0&0\\0&1&0\end{pmatrix}\]

    \[G\overline{G}=G\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&0&0\\0&0&1\end{pmatrix}\]

There are 9 matrices, but R\overline{R},B\overline{B},G\overline{G} are not traceless. You get 6 matrices giving up these combinations. The SU(3) group has 8 generators (independent!). Where are the other 2? There are different choices, but you can that the following two matrices provide a good choice

    \[R\overline{R}-G\overline{R}=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}\]

    \[R\overline{R}+G\overline{G}-2B\overline{B}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}\]

There are 8 total color grids/matrices for independent color changes:

    \[R\overline{G},R\overline{B},G\overline{R},B\overline{R},B\overline{G},G\overline{R},G\overline{B},R\overline{R}-G\overline{G},R\overline{R}+G\overline{G}-2B\overline{B}\]

Unlikely photons, gluons also interact with theirselves but not with the Higgs boson! Feynman graphs are typically Y shaped of double Y-shaped for gluon self-interactions. Cinfinement is a complex phenomenon. However, as said before here in TSOR, it can be modeled with a relatively simple potential for two quarks (the quarkonium):

    \[V(q\overline{q})=-\dfrac{4}{3}\dfrac{\alpha_s}{r}+kr\]

where k=\sigma is the string tension. This model is good for “heavy” quarks. This model can be compared to that of QED:

  • In QED, field lines extend up to infinity as \sim1/r^2. In QCD, quark field lines due to color are stretched or compressed into a tiny region between quarks and antiquarks. Breaking a flux tube requires the creation of a quark-antiquark pair.
  • In QED, Electromagnetic flux is conserved to infinity, but in QCD, color flux is trapped between quarks. No strong interations outside the color flux tube! Breaking the string implies a big energy, much larger than that of bounded atoms.
  • Hybrid states are allows, such as \langle q\overline{q}g\rangle, \langle ggg\rangle,\ldots. These exotica are the subject of some simulations of lattice QCD with supercomputers AND, as well, are being tried to mimic some of the resonant states already known. It is complicated to find out a good match, though.

In summary:

  1. QCD is all about quarks, gluons and internal symmetries modeled by matrices, matrices represent transitions between color states.
  2. QCD is the theory of color, the quantum number (not a physical color truly) of strong interactions. Have you ever imagined if we had called them strawberry, blueberry and mint? Or orange, vanilla and chocolate? Maybe, it would taste better…
  3. Gluons interact with theirselves, unlike photons, due to non abelian properties of the gauge group (something they share with weak and electroweak interactions).
  4. Gluons are Higgs transparent, they do not interact with the Higgs field. Why? Nobody knows. But it is fortunate for live to be it so.
  5. Mesons and baryons get masses due to the chiral symmetry breaking and a complex mechanism of dimensional transmutation. However, valence quarks can get a little mass from Higgs particles. So elementary quarks do interact with the Higgs field a little bit, but composite hadrons are getting masses through QCD.
  6. Hybrid states and resonances are expected to arise in the QCD spectrum. However, the Yang-Mills mass-gap problem is yet unsolved. Prove that tha Yang-Mills equations have a mass-gap and you will win a million dollar from Clay Institute!

See you in another blog post!!!!!

LOG#224. Vis tenebris.

What is dark energy? Where does it come from? Is it constant? Is it a quantum field? Is it vacuum energy? Nobody knows what is the 70% of the Universe!

In the beginning of Physics as Science, Aristotle himself introduced the idea of energy (from the Greek \varepsilon\nu\varepsilon\rho\gamma\varepsilon\iota\alpha., meaning capacity of update or act on something). Stones fall down to Earth with the goal of finding their natural place. The capacity to be placed in the natural place requires something called energy. Energy was also implied to a virtuous ending. Today, the physicist definition uses the capability to do some “work”, i.e., the ability to displace something with a force changing the state of the object requires energy. Energy is also a quantity invariant under time translations (first Noether theorem!). Energy spent to create motion is the the same the body acquires to move. That is the origin of the name “vis viva”. Energy is something required to move something.

There are 4 great equations for energy, related to the biggest revolutions in physics:

1st. \dfrac{1}{2}mv^2=E_k. Kinetic energy or true vis viva, in the sense it was the first to be introduced (up to some numerical constant).

2nd. E_t=k_BT. Thermal energy as the result of atomic-molecular hypothesis. The great idea of Boltzmann.

3rd. E=hf=\hbar\omega. Quantum of energy, Planck desperatation made him invent the quanta of radiation to solve the black body problem.

4th. E=mc^2. Special relativity, 20th equation, Einstein greatest contribution to the understanding of the energy concept. Unfortunately, also used in nuclear weapons.

There is a gravitational potential energy formula as well,

    \[E_g=-\dfrac{GM^2}{R}\]

and a electric potential energy

    \[E_e=\dfrac{K_CQ^2}{R}\]

but the above 4 are the most conceptual equations for energy. It was Descartes the first one who discovered  that the followin quantity (or a multiple of it) was constant from Galileo works:

    \[E=\dfrac{mv(t)^2}{2}+mgx(t)\]

and that it was necessary in order to make up a clock! A gravitational clock is a device that requires some vis viva mv^2 to work! If v=\omega L, you get an energy E=M\omega^2L^2 for the clock, and good clocks have fast ticks, so \omega>>1. The clock gravitational frequency can be written as

    \[\omega=\sqrt{\dfrac{g}{L}}\]

L can be done as short as we want … Until quantum mechanics is reached. In quantum mechanics, you get p=mv, and quantization of the action pq=N\hbar.There is a quantum length, critical

    \[L_Q=\dfrac{\hbar}{Mc}\]

Quantum mechanics, thus, allows us to build up clocks with certain uncertainty or fuzziness, instead of gravity, that allows us to build up, in principle, clocks without uncertainty. p=\hbar/L_Q, so then

    \[\omega=\dfrac{\hbar}{ML^2}\]

and

    \[E=ML^2\omega^2=\dfrac{\hbar^2}{ML^2}\]

The critical (optimal) clock in QM is the one with

    \[\omega_Q=c/L_Q=\dfrac{Mc^2}{\hbar}\]

Deep question: can we simulate with classical gravity a quantum clock? How can we understand gravity from a quantum entangled critical system? Hint is that g=\hbar^2/(M^2L^3).

Leibniz formalized the ideas of Descartes with his calculus. Einstein renewed the idea of vis viva and energy with the aid of his special relativity. Time is slower when you move faster. General relativity enhances this fact, showing that the time passes slower closer of a massive body. E=Mc^2 has indeed a classical meaning as “useful energy” or “binding energy” in nuclear or particle physics. Total energy is useful plus unuseful energy. Related to unuseful energy is the idea of entropy. Entropy, from the Greek meaning “transformation”, is a magnitude equal to S=E/T or S=k_B\ln N, where N is the number of possible configurations or microstates with the given energy.. The Max-Ent principle says the total entropy of the universe tends to be maximized. In the end, at lower energies, every possible configuration has the same probability. Then, equipartition of energy holds, as thermodynamics and statistical physics taught us.

Now, dark energy as vis tenebris. Inventing his celebrated General theory of Relativity, a locally special relativistic theory of gravitation, Einstein realized that his equations predicted the Universe was expanding (contracting). That was obviously nonsense, the Universe was static (before Hubble discovery!). Then, he used a freedom his equations had, and introduced a term, called the cosmological term (or cosmic repulsion) that balanced gravity and made the Universe static. After Hubble’s discovery of the expanding Universe of galaxies, Einstein thought he had committed the biggest epic fail of his life…I wonder what he would have thought if alive when in 1998 we discovered the expansion is accelerating (positively) by something that mimics his cosmological constant at this time. The name dark energy is due to the fact we are not, in principle, be forced to believe the cosmological constant is indeed constant, since it could be varying very slowly on cosmological times, and it could be some king of field. Anyway, dark energy (vis tenebris) is permeating the space and it is responsible to the increasing speed of the expansion. Every time the space expands, more vis tenebris is added. If it is the cosmological constant, its addition is such that the density energy is constant in all over the space! Vis tenebris energy density is (take a universe as a sphere for simplicity)

    \[\rho_\Lambda=\dfrac{\Lambda c^4}{8\pi G_N}\]

and then the vacuum energy in a sphere of volume 4\pi R^3/3 reads

    \[E_\Lambda=\dfrac{\Lambda c^4R^3}{6G_N}\]

I you know \Lambda and R, you can compute the dark energy in that volume. As the scale factor increases, the density remains constant so the cosmic energy grows up by

    \[\Delta = \hbar\left(\dfrac{1}{a(t)}-\dfrac{1}{a_0(t)}\right)= \hbar\left(\dfrac{1}{d}-\dfrac{1}{D}\right)\]

Indeed, dark energy (vis tenebris) is the darkly main ingredient of the current Universe (about 2/3 or 3/4 as most). We are dominated by a dark energy “force”. What is it? That is the challenge and the puzzle…

See you in another wonderful post!

LOG#223. Pi-logy.

Hi, there.

Today some retarded Pi-day celebration equations (there is a longer version of this, that I wish I could publish next year). Some numbers and estimates for pi-related equations:

1st. Hawking radiation temperature (Schwarzschild’s 4d black hole case).

(1)   \begin{equation*}T_H=\dfrac{\hbar c^3}{8\mathbf{\pi} G_NMk_B}=6.2\cdot 10^{-8}\left(\dfrac{M}{M_\odot}\right)K\end{equation*}

2nd. Schwarzschild black hole surface area (4d).

(2)   \begin{equation*}4\mathbf{\pi}R_S^2=\dfrac{16\mathbf{\pi}G_N^2M^2}{c^4}=1.1\cdot 10^8\left(\dfrac{M}{M_\odot}\right)^2\end{equation*}

3rd. Black hole power/luminosity (4d).

(3)   \begin{equation*}L_{BH}=P_{BH}=\dfrac{\hbar c^6}{15360\mathbf{\pi}G_N^ 2M^2}=9.0\cdot 10^{-29}\left(\dfrac{M_\odot}{M}\right)^2W\end{equation*}

4th. Black hole evaporation time (4d).

(4)   \begin{equation*}t_{e}=\dfrac{5120\mathbf{\pi}G^2_NM_0^3}{\hbar c^4}=8.41\cdot 10^{-17}\left(\dfrac{M}{1kg}\right)^3s=6.6\cdot 10^{74}\left(\dfrac{M}{1kg}\right)^3s=2.1\cdot 10^{67}\left(\dfrac{M}{1kg}\right)^3yrs\end{equation*}

5th. Time to fall off and arrive to the BH singularity with negligible test mass (4d).

(5)   \begin{equation*}t_f(test)=\dfrac{\mathbf{\pi}}{2c}R_S=\dfrac{\mathbf{\pi}G_NM}{c^3}=1.5\cdot 10^{-5}\left(\dfrac{M}{M_\odot}\right)s\end{equation*}

6th. Time to fall off and arrive to the BH singularity with E=m test mass (4d).

(6)   \begin{equation*}t_f(m)=\dfrac{2}{3}\dfrac{R_S}{c}=\dfrac{4\mathbf{\pi}G_NM}{c^3}=6.2\cdot 10^{-5}\left(\dfrac{M}{M_\odot}\right)s\end{equation*}

7th. Black hole entropy (4d) value in SI units.

(7)   \begin{equation*}S=\dfrac{k_B c^3}{G_N\hbar}A_{BH}=\dfrac{k_BA}{4L_p^2}=\dfrac{4\mathbf{\pi} GM^2}{\hbar c}=\dfrac{\mathbf{\pi}k_Bc^3A_{BH}}{2G_Nh}=1.5\cdot 10^{54}\dfrac{M^2}{M_\odot^ 2}J/K\end{equation*}

8th. M2-M5 brane quantization.

(8)   \begin{equation*}T_{M2}T_{M5}=\dfrac{2\mathbf{\pi}N}{2k_{11}^2}=\dfrac{\mathbf{\pi}N}{k_{11}^2}\end{equation*}

9th. Gravitational wave power or GW luminosity.

    \[L_{GW}=-\dfrac{dE}{dt}=\left(\dfrac{32}{5c^5}\right)G^{7/3}\left(M_c\pi f_{GW}\right)^{10/3}\]

where the gravitational wave frequency is

    \[f_{GW}=2f_{orb}=\dfrac{1}{\mathbf{\pi}}\sqrt{\dfrac{GM}{r}}\]

10th. Chirp frequency or frequency rate.

For circular orbits, you have

    \[\dot{f}_{GW}=\left(\dfrac{96}{5c^5}\right)G^{5/3}\pi^{8/3}M_c^{5/3}f_{GW}^{11/3}\]

11th. Coalescence time for GW merger (circular orbits).

    \[t_c=\dfrac{1}{2^8}\left(\dfrac{GM_c}{c^3}\right)^{-5/3}\left[\mathbf{\pi}f_{GW}\right]^{-8/3}\]

12th. ISCO (inner stable circular orbit) frequency for binary mergers.

    \[f_{max,c}=f_{isco}=\dfrac{c^3}{6^{3/2}\pi GM}\approx 4.4\dfrac{M}{M_\odot} kHz\]

13th. S-matrix in D-dimensions.

    \[S=I+i\dfrac{\left(2\pi\right)^D\delta^D\left(\displaystyle{\sum_fp_f}-\displaystyle{\sum_ip_i}\right)}{\displaystyle{\prod_f}\left(2p_{of}\right)^{1/2}\displaystyle{\prod_i}\left(2p_{oi}\right)^{1/2}}\mathcal{A}\]

14th. Gravitational wave fluxes for gravitons and photons (4d).

    \[F_{GW}=\dfrac{c^3h^2\omega^2}{16\pi G_N}=\dfrac{\pi c^3h^2f^2}{4G_N}\]

where h is the GW strain, and for photons, the GW induced electromagnetic  flux reads

    \[F_{em}=\dfrac{c^3\omega^2 h^4}{8\pi G_N}=\dfrac{\pi c^3 f^2 h^4}{2G_N}\]

15th. Kerr-Newmann black hole area and mass spectrum.

Any massive, rotating, charged black hole have an event horizon given by the following formula

    \[\mathcal{A}_H=4\pi\left[\dfrac{2G_N^2M^2}{c^4}-\dfrac{G_NQ^2}{4\pi\varepsilon_0c^4}+\dfrac{2G_NM}{c^2}\sqrt{\dfrac{G^2_NM^2}{c^4}-\dfrac{G_NQ^2}{4\pi\varepsilon_0c^4}-\dfrac{J^2}{M^2c^2}}\right]\]

This relation can be inverted to obtain the mass spectrum as function of area, charge and angular momentum as follows (exercise!):

    \[\mathcal{M}\left(A_H,Q,J\right)=\sqrt{\dfrac{\pi}{\mathcal{A}}}\left[\dfrac{c^4}{G_N}\left(\dfrac{\mathcal{A}}{4\pi}+\dfrac{G_NQ^2}{4\pi\varepsilon_0c^4}\right)^2+\dfrac{4J^2}{c^2}\right]^{1/2}\]

Challenge: modify the above expressions to include a cosmological constant factor.

16th. Universal quantum gravity potential at low energies.

Quantum gravity at low energy provides the following potential energy

    \[V_{QG}=-\dfrac{GM_1M_2}{r}\left[1+\dfrac{3G_N\left(M_1+M_2\right)}{rc^2}+\dfrac{41G_N\hbar}{10\pi r^2}\right]\]

independent of the QG approach you use!

17th. Running alpha strong.

    \[\alpha_s(Q^2)=\dfrac{\alpha_s(\Lambda^2_{QCD})}{1+\beta\alpha_s(\Lambda^2_{QCD})\log\left(\dfrac{Q^2}{\Lambda^2_{QCD}}\right)}\]

For the general QCD the beta factor reads

    \[\beta=\dfrac{11N_c-2n_f}{12\pi}\]

and the SM gives \beta_0>0 (N_c=3, n_f=6) and slope \beta(\alpha_s)<0 due to asymptotic freedom (antiscreening).

18th. Graviton energy density and single graviton energy density.

The graviton energy density reads off from GR as

    \[\rho_E=\dfrac{c^2\omega^2f^2}{32\pi G_N}\]

and for a single graviton, it reads

    \[\rho_E(single)=\dfrac{\hbar \omega^4}{c^3}=\dfrac{8\pi^3 h f^4}{c^3}\]

where h is the Planck constant, not the strain here.

I have many other pi-logy equations, but let me reserve them for a future longer post!

See you all, very soon!

LOG#222. The New SI.

By the time we will find new physics, we have already redefined the SI in terms of base units and fundamental constants.

The definition of the new SI is the next one: the SI is the system in which the following constants are taken to be exact

  • The unperturbed ground state hyperfine splitting (transition) frequency of the caesium-133 atom \Delta f(Cs-133) is exactly

    \[\Delta f=9192631770 Hz\]

Thus, frequency is fundamental, and the time is a base unit from frequency. One second is the time

    \[1s=\dfrac{9192631770}{\Delta f}\]

and 1 Hz is the reciprocal of the above quantity, exactly too.

  • The speed of light in vacuum c is exactly the quantity

    \[c=299792458m\cdot s^{-1}\]

Using the previous and this definition, you can define the meter to be exactly the amount of length

    \[1m=\dfrac{9192631770c}{299792458\Delta f}\]

  • The elementary charge e is exactly the quantity

    \[e=1.602176565\cdot 10^{-19}\]

Thus, the old electric current unit, base unit, the ampère, is the unit in which you can express charge into current, in corresponding units, with the next conversion constants:

    \[1C=\dfrac{e}{1.602176565\cdot 10^{-19}}=6.241509343\cdot 10^8e\]

    \[1A=\dfrac{1C}{1s}=\dfrac{6.241509343\cdot 10^8e\Delta f}{9192631770}=\dfrac{e}{(1.602176565\cdot 10^{-19})(9192631770)}\]

  • The Planck constant is exactly defined to be

    \[h=6.62607015\cdot 10^{-34}J\cdot s\]

and thus the kilogram is defined in terms of fundamental constants as

    \[1kg=\dfrac{(299792458)^2h\Delta f}{(6.62607015\cdot 10^{-34})(9192631770)c^2}\]

  • The Boltzmann constant is exactly

    \[k_B=1.3806488\cdot 10^{-23}J\cdot K^{-1}\]

so (with 1J=1kg\cdot 1m^2\cdot 1s^{-2} and then

    \[1K=\dfrac{1.3806488\cdot 10^{-23}}{k_B}=\dfrac{1.3806488\cdot 10^{-23}h\Delta f}{(6.62607015\cdot 10^{-34})(9192631770)k_B}\]

  • The Avogadro constant is defined exactly to be

    \[N_A=6.02214129\cdot 10^{23}\]

so the mole is

    \[1mole=\dfrac{6.02214129\cdot 10^{23}}{N_A}\]

  • The luminous efficacy K_{cd} of monochromatic radiation of frequency 540 THz is exactly defined to be 683 lumen/(W\cdot sr). Thus, as stereradian is dimensionless, lm=cd\cdot sr and the candela definition holds exactly to be as well

    \[1cd=1\dfrac{lm}{683W\cdot sr}=\dfrac{K_{cd}}{683}kg\cdot m^2\cdot s^{-3}\cdot sr^{-1}=\dfrac{1\cdot \Delta f^2 hK_{cd}}{(9192631770)^2(6.62607015\cdot 10^{-34})683}\]

See you in another blog post!