# LOG#105. Einstein’s equations.

**Posted:**2013/05/24

**Filed under:**General Relativity, Physmatics |

**Tags:**action, cosmological constant, dark energy, dark matter, Einstein, Einstein's field equations, Einstein-Hilbert action, General Relativity, Physmatics, Relativity, tensor methods, tensors, vacuum energy, variational calculus 2 Comments

In 1905, one of Einstein’s achievements was to establish the theory of Special Relativity from 2 single postulates and correctly deduce their physical consequences (some of them time later). The essence of Special Relativity, as we have seen, is that all the inertial observers must agree on the speed of light “in vacuum”, and that the physical laws (those from Mechanics and Electromagnetism) are the same for all of them. Different observers will measure (and then they see) different wavelengths and frequencies, but the product of wavelength with the frequency is the same. The wavelength and frequency are thus* Lorentz covariant*, meaning that they change for different observers according some fixed mathematical prescription depending on its tensorial character (scalar, vector, tensor,…) respect to Lorentz transformations. The speed of light is **Lorentz invariant**.

By the other hand, **Newton’s law of gravity** describes the motion of planets and terrrestrial bodies. It is all that we need in contemporary rocket ships unless those devices also carry atomic clocks or other tools of exceptional accuracy. Here is Newton’s law in potential form:

In the special relativity framework, this equation has a terrible problem: if there is a change in the mass density , then it must propagate everywhere instantaneously. If you believe in the Special Relativity rules and in the speed of light invariance, it is impossible. Therefore, “Houston, we have a problem”.

Einstein was aware of it and he tried to solve this inconsistency. The final solution took him ten years .

The apparent silly and easy problem is to develop and describe all physics in the the same way irrespectively one is accelerating or not. However, it is not easy or silly at all. It requires deep physical insight and a high-end mathematical language. Indeed, what is the most difficult part are the details of Riemann geometry and tensor calculus on manifolds. Einstein got private aid from a friend called Marcel Grossmann. In fact, Einstein knew that SR was not compatible with Newton’s law of gravity. He (re)discovered the equivalence principle, stated by Galileo himself much before than him, but he interpreted deeper and seeked the proper language to incorporante that principle in such a way it were compatible (at least locally) with special relativity! His “journey” from 1907 to 1915 was a hard job and a continuous struggle with tensorial methods…

Today, we are going to derive the Einstein field equations for gravity, a set of equations for the “metric field” . Hilbert in fact arrived at Einstein’s field equations with the use of the variational method we are going to use here, but Einstein’s methods were more physical and based on physical intuitions. They are in fact “complementary” approaches. I urge you to read “The meaning of Relativity” by A.Einstein in order to read a summary of his discoveries.

We now proceed to derive Einstein’s Field Equations (EFE) for General Relativity (more properly, a relativistic theory of gravity):

**Step 1.** Let us begin with the so-called Einstein-Hilbert action (an ansatz).

Be aware of the square root of the determinant of the metric as part of the volume element. It is important since the volume element has to be invariant in curved spacetime (i.e.,in the presence of a metric). It also plays a critical role in the derivation.

**Step 2.** We perform the variational variation with respect to the metric field :

**Step 3.** Extract out the square root of the metric as a common factor and use the product rule on the term with the Ricci scalar R:

**Step 4.** Use the definition of a Ricci scalar as a contraction of the Ricci tensor to calculate the first term:

A total derivative does not make a contribution to the variation of the action principle, so can be neglected to find the extremal point. Indeed, this is the Stokes theorem in action. To show that the variation in the Ricci tensor is a total derivative, in case you don’t believe this fact, we can proceed as follows:

Check 1. Write the Riemann curvature tensor:

Note the striking resemblance with the non-abelian YM field strength curvature two-form

.

There are many terms with indices in the Riemann tensor calculation, but we can simplify stuff.

Check 2. We have to calculate the variation of the Riemann curvature tensor with respect to the metric tensor:

One cannot calculate the covariant derivative of a connection since it does not transform like a tensor. However, the difference of two connections does transform like a tensor.

Check 3. Calculate the covariant derivative of the variation of the connection:

Check 4. Rewrite the variation of the Riemann curvature tensor as the difference of two covariant derivatives of the variation of the connection written in Check 3, that is, substract the previous two terms in check 3.

Check 5. Contract the result of Check 4.

Check 6. Contract the result of Check 5:

Therefore, we have

Q.E.D.

**Step 5.** The variation of the second term in the action is the next step. Transform the coordinate system to one where the metric is diagonal and use the product rule:

The reason of the last equalities is that , and then its variation is

Thus, multiplication by the inverse metric produces

that is,

By the other hand, using the theorem for the derivation of a determinant we get that:

since

because of the classical identity

Indeed

and moreover

so

Q.E.D.

**Step 6.** Define the stress energy-momentum tensor as the third term in the action (that coming from the matter lagrangian):

or equivalently

**Step 7.** The extremal principle. The variation of the Hilbert action will be an extremum when the integrand is equal to zero:

i.e.,

Usually this is recasted and simplified using the Einstein’s tensor

as

This deduction has been mathematical. But there is a deep physical picture behind it. Moreover, there are a huge number of physics issues one could go into. For instance, these equations bind to particles with integral spin which is good for bosons, but there are matter fermions that also participate in gravity coupling to it. Gravity is universal. To include those fermion fields, one can consider the metric and the connection to be independent of each other. That is the so-called Palatini approach.

Final remark: you can add to the EFE above a “constant” times the metric tensor, since its “covariant derivative” vanishes. This constant is the cosmological constant (a.k.a. dark energy in conteporary physics). The, the most general form of EFE is:

Einstein’s additional term was added in order to make the Universe “static”. After Hubble’s discovery of the expansion of the Universe, Einstein blamed himself about the introduction of such a term, since it avoided to predict the expanding Universe. However, perhaps irocanilly, in 1998 we discovered that the Universe was accelerating instead of being decelerating due to gravity, and the most simple way to understand that phenomenon is with a positive cosmological constant domining the current era in the Universe. Fascinating, and more and more due to the WMAP/Planck data. The cosmological constant/dark energy and the dark matter we seem to “observe” can not be explained with the fields of the Standard Model, and therefore…They hint to new physics. The character of this new physics is challenging, and much work is being done in order to find some particle of model in which dark matter and dark energy fit. However, it is not easy at all!

May the Einstein’s Field Equations be with you!

# LOG#056. Gravitational alpha(s).

**Posted:**2012/11/29

**Filed under:**Cosmology, Physmatics, Quantum Gravity, Relativity |

**Tags:**alpha, alpha strong, asymptotic freedom, atomic physics, confinement, cosmological constant, cosmological constant problem, cosmological gravitational alpha, cosmological parameter fitting, cosmological parameters, Cosmology, coupling constant, de Sitter radius, Einstein's field equations, energy density, energy density ratios, energy ratios, fine structure constant, gravitational alpha, gravitational constant, gravitational fine structure constant, Hubble parameter, Hubble's length, length ratios, naturalness problem, Planck's energy, Planck's length, QCD, QFT, quantum field theory, quantum theory, ratios, Relativity Leave a comment

The topic today is to review a beautiful paper and to discuss its relevance for theoretical physics. The paper is: **Comment on the cosmological constant and a gravitational alpha **by R.J.Adler. You can read it here: http://arxiv.org/abs/1110.3358

One of the most intriguing and mysterious numbers in Physics is the electromagnetic fine structure constant . Its value is given by

or equivalenty

Of course, I am assuming that the coupling constant is measured at ordinary energies, since we know that the coupling constants are not really constant but they vary slowly with energy. However, I am not going to talk about the renormalization (semi)group in this post.

Why is the fine structure constant important? Well, we can undertand it if we insert the values of the constants that made the electromagnetic alpha constant:

with being the electron elemental charge, the Planck’s constant divided by two pi, c is the speed of light and where we are using units with . Here is the Coulomb constant, generally with a value , but we rescale units in order it has a value equal to the unit. We will discuss more about frequently used system of units soon.

As the electromagnetic alpha constant depends on the electric charge, the Coulomb’s electromagnetic constant ( rescaled to one in some “clever” units), the Planck’s constant ( rationalized by since ) and the speed of light, it codes some deep information of the Universe inside of it. The electromagnetic alpha is quantum and relativistic itself, and it also is related to elemental charges. Why alpha has the value it has is a complete mystery. Many people has tried to elucidate why it has the value it has today, but there is no reason of why it should have the value it has. Of course, it happens as well with some other constants but this one is particularly important since it is involved in some important numbers in atomic physics and the most elemental atom, the hydrogen atom.

In atomic physics, there are two common and “natural” scales of length. The first scale of length is given by the Compton’s wavelength of electrons. Usint the de Broglie equation, we get that the Compton’s wavelength is the wavelength of a photon whose energy is the same as the rest mass of the particle, or mathematically speaking:

Usually, physicists employ the “reduced” or “rationalized” Compton’s wavelength. Plugging the electron mass, we get the electron reduced Compton’s wavelength:

The second natural scale of length in atomic physics is the so-called Böhr radius. It is given by the formula:

Therefore, there is a natural mass ratio between those two length scales, and it shows that it is precisely the electromagnetic fine structure constant alpha :

Furthermore, we can show that the electromagnetic alpha also is related to the mass ration between the electron energy in the fundamental orbit of the hydrogen atom and the electron rest energy. These two scales of energy are given by:

1)** Rydberg’s energy** ( electron ground minimal energy in the fundamental orbit/orbital for the hydrogen atom):

2) **Electron rest energy**:

Then, the ratio of those two “natural” energies in atomic physics reads:

or equivalently

R.J.Adler’s paper remarks that there is a cosmological/microscopic analogue of the above two ratios, and they involve the infamous Einstein’s cosmological constant. In Cosmology, we have two natural (ultimate?) length scales:

1st. The (ultra)microscopic and ultrahigh energy (“ultraviolet” UV regulator) relevant **Planck’s length** , or equivalently the squared value . Its value is given by:

This natural length can NOT be related to any “classical” theory of gravity since it involves and uses the Planck’s constant .

2nd. The (ultra)macroscopic and ultra-low-energy (“infrared” IR regulator) relevant **cosmological constant/deSitter radius. **They are usualy represented/denoted by and respectively, and they are related to each other in a simple way. The dimensions of the cosmological constant are given by

The de Sitter radius and the cosmological constant are related through a simple equation:

The de Sitter radius is obtained from cosmological measurements thanks to the so called Hubble’s parameter ( or Hubble’s “constant”, although we do know that Hubble’s “constant” is not such a “constant”, but sometimes it is heard as a language abuse) H. From cosmological data we obtain ( we use the paper’s value without loss of generality):

This measured value allows us to derive the Hubble’s length paremeter

Moreover, the data also imply some density energy associated to the cosmological “constant”, and it is generally called Dark Energy. This density energy from data is written as:

and from this, it can be also proved that

where we have introduced the experimentally deduced value from the cosmological parameter global fits. In fact, the cosmological constant helps us to define the beautiful and elegant formula that we can call the **gravitational alpha/gravitational cosmological fine structure constant **:

or equivalently, defining the cosmological length associated to the cosmological constant as

If we introduce the numbers of the constants, we easily obtaint the gravitational cosmological alpha value and its inverse:

They are really small and large numbers! Following the the atomic analogy, we can also create a ratio between two cosmologically relevant density energies:

1st. **The Planck’s density energy.**

Planck’s energy is defined as

The Planck energy density is defined as the energy density of Planck’s energy inside a Planck’s cube or side , i.e., it is the energy density of Planck’s energy concentrated inside a cube with volume . Mathematically speaking, it is

It is an huge density energy!

**Remark:** Energy density is equivalent to **pressure** in special relativity hydrodynamics. That is,

wiht Pa denoting pascals () and where represents here matter (not energy) density ( with units in ). Of course, turning matter density into energy density requires a multiplication by . This equivalence between vacuum pressure and energy density is one of the reasons because some astrophysicists, cosmologists and theoretical physicists call “vacuum pressure” to the “dark energy/cosmological constant” term in the study of the cosmic components derived from the total energy density .

2nd. **The cosmological constant density energy.**

Using the Einstein’s field equations, it can be shown that the cosmological constant gives a contribution to the stress-energy-momentum tensor. The component is related to the dark energy ( a.k.a. the cosmological constant) and allow us to define the energy density

Using the previous equations for G as a function of Planck’s length, the Planck’s constant and the speed of light, and the definitions of Planck’s energy and de Sitter radius, we can rewrite the above energy density as follows:

Thus, we can evaluate the ration between these two energy densities! It provides

and the inverse ratio will be

So, we have obtained two additional really tiny and huge values for and its inverse, respectively. Note that the power appearing in the ratios of cosmological lengths and cosmological energy densities match the same scaling property that the atomic case with the electromagnetic alpha! In the electromagnetic case, we obtained and . The gravitational/cosmological analogue ratios follow the same rule and but the surprise comes from the values of the gravitational alpha values and ratios. Some comments are straightforward:

1) Understanding atomic physics involved the discovery of Planck’s constant and the quantities associated to it at fundamental quantum level ( Böhr radius, the Rydberg’s constant,…). Understanding the Cosmological Constant value and the mismatch or stunning ratios between the equivalent relevant quantities, likely, require that can be viewed as a new “fundamental constant” or/and it can play a dynamical role somehow ( e.g., varying in some unknown way with energy or local position).

2) Currently, the cosmological parameters and fits suggest that is “constant”, but we can not be totally sure it has not varied slowly with time. And there is a related idea called quintessence, in which the cosmological “constant” is related to some dynamical field and/or to inflation. However, present data say that the cosmological constant IS truly constant. How can it be so? We are not sure, since our physical theories can hardly explain the cosmological constant, its value, and why it is current density energy is radically different from the vacuum energy estimates coming from Quantum Field Theories.

3) The mysterious value

is an equivalent way to express the biggest issue in theoretical physics. A **naturalness problem** called the** cosmological constant problem**.

In the literature, there have been alternative definitions of “gravitational fine structure constants”, unrelated with the above gravitational (cosmological) fine structure constant or gravitational alpha. Let me write some of these alternative gravitational alphas:

1) **Gravitational alpha prime**. It is defined as the ratio between the electron rest mass and the Planck’s mass squared:

Note that . Since , we can also use the proton rest mass instead of the electron mass to get a new gravitational alpha.

2) **Gravitational alpha double prime.** It is defined as the ratio between the proton rest mass and the Planck’s mass squared:

and the inverse value

Finally, we could guess an intermediate gravitational alpha, mixing the electron and proton mass.

3) **Gravitational alpha triple prime**. It is defined as the ration between the product of the electron and proton rest masses with the Planck’s mass squared:

and the inverse value

We can compare the 4 gravitational alphas and their inverse values, and additionally compare them with . We get

These inequations mean that the electromagnetic fine structure constant is (at ordinary energies) 42 orders of magnitude bigger than , 39 orders of magnitude bigger than , 36 orders of magnitude bigger than and, of course, 58 orders of magnitude bigger than . Indeed, we could extend this analysis to include the “fine structure constant” of Quantum Chromodynamics (QCD) as well. It would be given by:

since generally we define . We note that by 3 orders of magnitude. However, as strong nuclear forces are short range interactions, they only matter in the atomic nuclei, where confinement, and color forces dominate on every other fundamental interaction. Interestingly, at high energies, QCD coupling constant has a property called asymptotic freedom. But it is another story not to be discussed here! If we take the alpha strong coupling into account the full hierarchy of alphas is given by:

**Fascinating!** Isn’t it?** Stay tuned!!!**

*ADDENDUM:* After I finished this post, I discovered a striking (and interesting itself) connection between and . The relation or coincidence is the following relationship

Is this relationship fundamental or accidental? The answer is unknown. However, since the electric charge (via electromagnetic alpha) is not related a priori with the gravitational constant or Planck mass ( or the cosmological constant via the above gravitational alpha) in any known way I find particularly stunning such a coincidence up to 5 significant digits! Any way, there are many unexplained numerical coincidences that are completely accidental and meaningless, and then, it is not clear why this numeral result should be relevant for the connection between electromagnetism and gravity/cosmology, but it is interesting at least as a curiosity and “joke” of Nature.

*ADDENDUM (II):
*

Some quotes about the electromagnetic alpha from wikipedia http://en.wikipedia.org/wiki/Fine-structure_constant

“(…)There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! (…)”. R.P.Feynman, *QED: The Strange Theory of Light and Matter*, Princeton University Press, p.129.

“(…) If alpha [the fine-structure constant] were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.(…)” Max Born, in A.I. Miller’s book *Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung. *p. 253. Publisher W.W. Norton & Co.(2009).

“(…)The mystery about *α* is actually a double mystery. The first mystery – the origin of its numerical value *α* ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.(…)” Malcolm H. Mac Gregor, M.H. MacGregor (2007). *The Power of Alpha.*