# 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#057. Naturalness problems.

**Posted:**2012/12/02

**Filed under:**Physmatics, Quantum Gravity, The Standard Model: Basics |

**Tags:**CKM matrix, cosmic coincidence, cosmological constant, cosmological constant problem, critical energy density, curvature, dark energy, dark energy density, Dirac large number hypothesis, electroweak scale, energy, energy density, flatness problem, flavour problem, gauge hierarchy problem, Higgs boson, Higgs mechanism, Hubble constant, inflation, inflationary cosmologies, little hierarchy problem, mass, matter density, naturalness, naturalness problem, neutrino mass hierarchy, neutrino masses, neutrino oscillations, NO, NOSEX, parameter, parameter space, Planck era, Planck scale, PMNS matrix, QCD, QFT, quark-lepton complementarity, SM, Standard Cosmological Model, Standard Model, strong CP problem, theta term, types of naturalness, vacuum, vacuum energy, W boson, Z boson 8 Comments

In this short blog post, I am going to list some of the greatest “naturalness” problems in Physics. It has nothing to do with some delicious natural dishes I like, but there is a natural beauty and sweetness related to naturalness problems in Physics. In fact, they include some hierarchy problems and additional problems related to stunning free values of parameters in our theories.

**Naturalness problems** arise when the “naturally expected” property of some free parameters or fundamental “constants” to appear as quantities of order one is violated, and thus, those paramenters or constants appear to be very large or very small quantities. That is, naturalness problems are problems of untuning “scales” of length, energy, field strength, … A value of 0.99 or 1.1, or even 0.7 and 2.3 are “more natural” than, e.g., Equivalently, imagine that the values of every fundamental and measurable physical quantity lies in the real interval . Then, 1 (or very close to this value) are “natural” values of the parameters while the two extrema or are “unnatural”. As we do know, in Physics, zero values are usually explained by some “fundamental symmetry” while extremely large parameters or even can be shown to be “unphysical” or “unnatural”. In fact, renormalization in QFT was invented to avoid quantities that are “infinite” at first sight and regularization provides some prescriptions to assign “natural numbers” to quantities that are formally ill-defined or infinite. However, naturalness goes beyond those last comments, and it arise in very different scenarios and physical theories. It is quite remarkable that naturalness can be explained as numbers/contants/parameters around 3 of the most important “numbers” in Mathematics:

**REMEMBER: Naturalness** of X is, thus, being 1 or close to it, while values approaching 0 or are unnatural. Therefore, if some day you heard a physicist talking/speaking/lecturing about “naturalness” remember the triple and then assign “some magnitude/constant/parameter” some quantity close to one of those numbers. If they approach 1, the parameter itself is natural and unnatural if it approaches any of the other two numbers, zero or infinity!

I have never seen a systematic classification of naturalness problems into types. I am going to do it here today. We could classify naturalness problems into:

1st.** Hierarchy problems**. They are naturalness problems related to the energy mass or energy spectrum/energy scale of interactions and fundamental particles.

2nd. **Nullity/Smallness problems**. These are naturalness problems related to free parameters which are, surprisingly, close to zero/null value, even when we have no knowledge of a deep reason to understand why it happens.

3rd.** Large number problems (or hypotheses).** This class of problems can be equivalently thought as nullity reciprocal problems but they arise naturally theirselves in cosmological contexts or when we consider a large amount of particles, e.g., in “statistical physics”, or when we face two theories in very different “parameter spaces”. Dirac pioneered these class of hypothesis when realized of some large number coincidences relating quantities appearing in particle physics and cosmology. This Dirac large number hypothesis is also an old example of this kind of naturalness problems.

4th. **Coincidence problems**. This 4th type of problems is related to why some different parameters of the same magnitude are similar in order of magnitude.

The following list of concrete naturalness problems is not going to be complete, but it can serve as a guide of what theoretical physicists are trying to understand better:

1. **The little hierarchy problem**. From the phenomenon called neutrino oscillations (NO) and neutrino oscillation experiments (NOSEX), we can know the difference between the squared masses of neutrinos. Furthermore, cosmological measurements allow us to put tight bounds to the total mass (energy) of light neutrinos in the Universe. The most conservative estimations give or even as an upper bound is quite likely to be true. By the other hand, NOSEX seems to say that there are two mass differences, and . However, we don’t know what kind of spectrum neutrinos have yet ( normal, inverted or quasidegenerated). Taking a neutrino mass about 1 meV as a reference, the little hierarchy problem is the question of why neutrino masses are so light when compared with the remaining leptons, quarks and gauge bosons ( excepting, of course, the gluon and photon, massless due to the gauge invariance).

Why is

We don’t know! Let me quote a wonderful sentence of a very famous short story by Asimov to describe this result and problem:

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

2. **The gauge hierarchy problem.** The electroweak (EW) scale can be generally represented by the Z or W boson mass scale. Interestingly, from this summer results, Higgs boson mass seems to be of the same order of magnitue, more or less, than gauge bosons. Then, the electroweak scale is about . Likely, it is also of the Higgs mass order. By the other hand, the Planck scale where we expect (naively or not, it is another question!) quantum effects of gravity to naturally arise is provided by the Planck mass scale:

or more generally, dropping the factor

Why is the EW mass (energy) scale so small compared to Planck mass, i.e., why are the masses so different? The problem is hard, since we do know that EW masses, e.g., for scalar particles like Higgs particles ( not protected by any SM gauge symmetry), should receive quantum contributions of order

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

3. **The cosmological constant (hierarchy) problem.** The cosmological constant , from the so-called Einstein’s field equations of classical relativistic gravity

is estimated to be about from the cosmological fitting procedures. The Standard Cosmological Model, with the CMB and other parallel measurements like large scale structures or supernovae data, agree with such a cosmological constant value. However, in the framework of Quantum Field Theories, it should receive quantum corrections coming from vacuum energies of the fields. Those contributions are unnaturally big, about or in the framework of supersymmetric field theories, after SUSY symmetry breaking. Then, the problem is:

Why is ? Even with TeV or PeV fundamental SUSY (or higher) we have a serious mismatch here! The mismatch is about 60 orders of magnitude even in the best known theory! And it is about 122-123 orders of magnitude if we compare directly the cosmological constant vacuum energy we observe with the cosmological constant we calculate (naively or not) with out current best theories using QFT or supersymmetric QFT! Then, this problem is a hierarchy problem and a large number problem as well. Again, and sadly, we don’t know why there is such a big gap between mass scales of the same thing! This problem is the biggest problem in theoretical physics and it is one of the worst predictions/failures in the story of Physics. However,

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

4.** The strong CP problem/puzzle. **From neutron electric dipople measurements, theoretical physicists can calculate the so-called -angle of QCD (Quantum Chromodynamics). The theta angle gives an extra contribution to the QCD lagrangian:

The theta angle is not provided by the SM framework and it is a free parameter. Experimentally,

while, from the theoretical aside, it could be any number in the interval . Why is close to the zero/null value? That is the strong CP problem! Once again, we don’t know. Perhaps a new symmetry?

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

5. **The flatness problem/puzzle.** In the Stantard Cosmological Model, also known as the model, the curvature of the Universe is related to the critical density and the Hubble “constant”:

There, is the total energy density contained in the whole Universe and is the so called critical density. The flatness problem arise when we deduce from cosmological data that:

At the Planck scale era, we can even calculate that

This result means that the Universe is “flat”. However, why did the Universe own such a small curvature? Why is the current curvature “small” yet? We don’t know. However, cosmologists working on this problem say that “inflation” and “inflationary” cosmological models can (at least in principle) solve this problem. There are even more radical ( and stranger) theories such as varying speed of light theories trying to explain this, but they are less popular than inflationary cosmologies/theories. Indeed, inflationary theories are popular because they include scalar fields, similar in Nature to the scalar particles that arise in the Higgs mechanism and other beyond the Standard Model theories (BSM). We don’t know if inflation theory is right yet, so

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

6. **The flavour problem/puzzle.** The ratios of successive SM fermion mass eigenvalues ( the electron, muon, and tau), as well as the angles appearing in one gadget called the CKM (Cabibbo-Kobayashi-Maskawa) matrix, are roughly of the same order of magnitude. The issue is harder to know ( but it is likely to be as well) for constituent quark masses. However, why do they follow this particular pattern/spectrum and structure? Even more, there is a mysterious lepton-quark complementarity. The analague matrix in the leptonic sector of such a CKM matrix is called the PMNS matrix (Pontecorvo-Maki-Nakagawa-Sakata matrix) and it describes the neutrino oscillation phenomenology. It shows that the angles of PMNS matrix are roughly complementary to those in the CKM matrix ( remember that two angles are said to be complementary when they add up to 90 sexagesimal degrees). What is the origin of this lepton(neutrino)-quark(constituent) complementarity? In fact, the two questions are related since, being rough, the mixing angles are related to the ratios of masses (quarks and neutrinos). Therefore, this problem, if solved, could shed light to the issue of the particle spectrum or at least it could help to understand the relationship between quark masses and neutrino masses. Of course, we don’t know how to solve this puzzle at current time. And once again:

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

7. **Cosmic matter-dark energy coincidence.** At current time, the densities of matter and vacuum energy are roughly of the same order of magnitude, i.e, . Why now? We do not know!

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

And my weblog is only just beginning! See you soon in my next post! 🙂