Are there any known α'-exact string backgrounds with NO covariantly constant null Killing vector?

The external spacetime field produced by an object of mass M, the Schwarzschild spacetime metric solution, is usually obtained as follows [1]:

1) Assumes a spherically symmetric spacetime metric, and is static and time invariant;

2) Assumes a vacuum conditions outside, with T_µν = 0;

3) Solve the Einstein field equation, R_µν - (1/2)g_µνR=T_µν...... (EQ.1)

4) Utilize the boundary condition: the Newtonian potential ф = -GM/r, which introduces the mass M. Obtain the result:

ds² = -(1-2GM/r)dt² + (1-2GM/r)^-1dr² + r²dΩ²...... (EQ.2)

Overall, the Schwarzschild metric employs a priori derivation steps. The solution is unique according to Birkhoff's theorem.

Einstein does not explain why M leads to ds², our questions are:

a) The spacetime metric is containing the energy-momentum T_spacetime , which can only originate from T_µν and is conserved. Why then must spacetime receive, store, and transmit energy-momentum by curvature^* ?

b) The implication of condition 2) is that the spacetime field energy-momentum is independent of M or can be regarded as such. Comparing this to the electric field of an electron is equivalent to the fact that the energy contained in the electron's electric field is independent of the electron itself. Since T_spacetime is also bound to M, is it not part of M?

c) For complex scenarios, in the T_µν of Einstein's field equation EQ.1, should one include the spacetime energy momentum at the location? With the above Schwarzschild solution, it seems that there is none, otherwise both sides of the equation (EQ.1) become a deadly circle. So, should there be or should there not be? Does the field equation have a provision or treatment that T_µν can only contain non-spacetime energy momentum?

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Notes

General Relativity field equations [1]:

G_µν = G*T_µν...... (EQ.1).

It is a relation between the matter field (energy-momentum field) T_µν and the spacetime field G_µν, where the gravitational constant G is the conversion factor between the dimensions [2].Einstein constructed this relation without explaining why the spacetime field and the matter field are in such a way, but rather assumed that nine times out of ten, they would be in such a way. He also did not explain why the spacetime field G_µν is described by curvature and not by some other parameter. Obviously, we must find the exact physical relationship between them, i.e., why T_µν must correspond to G_µν, in order to ensure that the field equations are ultimately correct.

We know that matter cannot be a point particle, it must have a scale, and matter cannot be a solid particle, it must be some kind of field. The fact that matter has a scale means that it has to occupy space-time; the fact that matter is a field means that it is mixed with space-time, i.e., matter contains space-time. So, when applying Einstein's field equations, how is matter's own spacetime defined? Does it change its own spacetime? If its own energy-momentum and structure have already determined its own spacetime, should the way it determines its own spacetime be the same as the way it determines the external spacetime? If it is the same, does it mean that the spacetime field is actually a concomitant of the matter field?

If one were to consider a gravitational wave, one could think of it as a fluctuating spacetime field that propagates independently of the material source after it has been disconnected from it. They have decoupled from each other and no longer continue to conform to the field equations (EQ.1). Although gravitational waves are the product of a source, the loss of that source prevents us from finding another specific source for it to match it through the equation (EQ.1). Just as after an electron accelerates, the relationship between the radiated electromagnetic wave and the electron is no longer maintained. Does this indicate the independence of spacetime field energies?

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Planck Scales (Planck's 'units of measurement') are different combinations of the three physical constants h, c, G, Planck Scales=f(c,h,G):

Planck Time: tp=√ℏG/c^5=5.31x10^-44s ......(1)

Planck Length: Lp=√ℏG/c^3=1.62x10^-35m ......(2)

Planck Mass: Mp=√ℏc/G=2.18x10^-8 kg ......(3)

“These quantities will retain their natural meaning for as long as the laws of gravity, the propagation of light in vacuum and the two principles of the theory of heat hold, and, even if measured by different intelligences and using different methods, must always remain the same.”[1] And because of the possible relation between Mp and the radius of the Schwarzschild black hole, the possible generalized uncertainty principle [2], makes them a dependent basis for new physics [3]. But what exactly is their natural meaning?

However, the physical constants, the speed of light, c, the Planck constant, h, and the gravitational constant, G, are clear, fundamental, and invariant.

c: bounds the relationship between Space and Time, with c = ΔL/ Δt, and Lorentz invariance [4];

h: bounds the relationship between Energy and Momentum with h=E/ν = Pλ, and energy-momentum conservation [5][6];

G: bounds the relationship between Space-Time and Energy-Momentum, with the Einstein field equation c^4* Gμν = (8πG) * Tμν, and general covariance [7].

The physical constants c, h, G already determine all fundamental physical phenomena‡. So, can the Planck Scales obtained by combining them be even more fundamental than they are? Could it be that the essence of physics is (c, h, G) = f(tp, Lp, Mp)? rather than equations (1), (2), (3). From what physical fact, or what physical imagination, are we supposed to get this notion? Never seeing such an argument, we just take it and use it, and still recognize c,h,G fundamentality. Obviously, Planck Scales are not fundamental physical constants, they can only be regarded as a kind of 'units of measurement'.

So are they a kind of parameter? According to Eqs. (1)(2)(3), c,h,G can be directly replaced by c,h,G and the substitution expression loses its meaning.

So are they a principle? Then who are they expressing? What kind of behavioral pattern is expressed? The theory of quantum gravity takes this as a " baseline ", only in the order sense, not in the exact numerical value.

Thus, Planck time, length, mass, determined entirely by h, c, G, do they really have unquestionable physical significance?

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‡ Please ignore for the moment the phenomena within the nucleus of the atom, eventually we will understand that they are still determined by these three constants.

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[1] Robotti, N. and M. Badino (2001). "Max Planck and the 'Constants of Nature'." Annals of Science 58(2): 137-162.

Do we need to find a motivation for symmetry: {?} → {invariance} → {conservation} → {symmetry} →

Should there be an ultimate symmetry that is identical to the conservation, structure invariance, and interaction invariance of the energy-momentum primitives and that determines all other symmetries?

Symmetry, invariance, and conservation are, in a sense, the same concept [1][2][3] and will generally be described in this order, as if symmetry were dominant.

As commonly understood, energy-momentum conservation was the first physics concept to be developed. It exists as a matter of course in mechanics, thermodynamics, and electricity. However, after physics entered the twentieth century, from quantum mechanics to general relativity, the conservation of energy-momentum has been repeatedly encountered with doubts [5][6][7][8][9][10], and so far it still can't be determined as a universal law by physics. Some of the new physics is insisting on "something out of nothing"[11][12][13][14] or spontaneous vacuum fluctuations[15], which equals to the rejection of energy-momentum conservation. The important reasons for this may be: First, Energy-momentum conservation cannot be proved† . Second, energy-momentum in physics has never been able to correspond to a specific thing, expressed by a unified mathematical formula‡, and it can only be the "equivalence" of various physical forms that are converted and transferred to each other [16]. Third, we have a biased understanding of the status of energy-momentum conservation, such as "These symmetries implied conservation laws. Although these conservation laws, especially those of momentum and energy, were regarded to be the most important of all. Although these conservation laws, especially those of momentum and energy, were regarded to be of fundamental importance, these were regarded as consequences of the dynamical laws of nature rather than as consequences of the symmetries that underlay these laws."[17]. Conservation of energy-momentum was relegated to a subordinate position. Fourth, it is believed that the Uncertainty Principle can be manifested as a " dynamics ", which can cause various field quantum fluctuations in the microscopic domain, and does not have to strictly obey the energy-momentum conservation.

"Symmetry" refers to the "invariance under a specified group of transformations" of the analyzed object [4]. Symmetry is always accompanied by some kind of conservation, but conservation does not only refer to the conservation of energy-momentum, but also, under different conditions, to the conservation of other physical quantities, such as charge, spin, or the conservation of other quantum numbers. Thus, "conservation" is usually the constant invariance of something at some level, and Wigner divided symmetries into classical geometrical symmetries and dynamical symmetries, which are associated with specific types of interactions, every interaction has a dynamical invariance group. "It may be useful to discuss first the relation of phenomena, laws of nature, and invariance principles to each other. This relation is not quite the same for the classical invariance principles, which will be called geometrical, and the new ones, which will be called dynamical."[1]. According to Wigner, we can define the "geometric invariance" of everything as the manifestation of interactions filtered through the absoluteness of the spatio-temporal background. This interaction exhibits itself whenever you assume an observer*. displacement invariance, Lorentz invariance are typical. We can define all "dynamical invariance" as manifestation when the background absolutes of the potential field are filtered out. gauge invariance, the diffeomorphism invariance are typical manifestations." from a passive role in which symmetry is the property of interactions, to an active role in which symmetry serves to determine the interactions themselves --a role that I have called symmetry dictates interaction." "Einstein's general relativity was the first example where symmetry was used" actively to determine gravitational interaction" [2]. This expresses the same idea, that the role of symmetry is elevated to the status of "force". Gross says that the secret of nature is symmetry. The most advanced form of symmetries we have understood are local symmetries-general coordinate invariance and gauge symmetry. The most advanced form of symmetries we have understood are local symmetries-general coordinate invariance and gauge symmetry. unified theory that contains both as a consequence of a greater and deeper symmetry of which these are the low energy remnants [18]. He regards the unification of general relativity and quantum field theory as a unification of symmetries. He regards the unification of general relativity and quantum field theory as a unification of symmetries. If we define generalized invariance as the completeness of the structure, properties, and laws of interaction of the analyzed objects when they interact, i.e., the undecomposability of the whole as a whole, the conservation of the properties (charge, spin, other quantum numbers, etc.), and the consistency of the interaction relations (laws), it is clear that the invariance in this case is special invariance, which means only the invariance of the laws of interaction.

While symmetry, conservation, and invariance are almost equivalent expressions at the same level, there are subtle but important differences. If unbounded, it is the order in which the three are expressed, who actually determines whom, and who ultimately determines the laws of physics. In any case, when we currently speak of symmetry, it must correspond to specific invariance and conservation, not to broad invariance and conservation. This in fact greatly limits the claim that "symmetry dictates interaction", since interaction is much more general. There is no such thing as a failure of interaction, but there is often a failure of symmetry, unless we decide that there will be an ultimate symmetry that determines all other symmetries.

"A symmetry can be exact, approximate, or broken. Exact means unconditionally valid; approximate means valid under certain conditions; broken can mean different things, depending on the object considered and its context. different things, depending on the object considered and its context."[19] "It is not clear how rigorous conservation laws could follow from approximate symmetries"[1]. This reflects the uncertainty of the relationship between conservation currents ( charges) and symmetries, and if we know that conservation currents can still be maintained even with approximate symmetries, it should be understood that this must be a function of the fact that conservation currents have a more universal character. From a reductionist point of view, the conservation charge at all levels will gradually decompose with the decomposition of matter, until finally it becomes something that cannot be decomposed. Such a thing can only be the most universal energy-momentum and at the same time be the ultimate expression that maintains its conservation as well as the invariance of interactions. Otherwise, we will pursue the questions:

1) If energy-momentum conservation is not first, where does the power to move from one symmetry to another, symmetry breaking [11] [12], come from? How can symmetry violations [13] in physics be explained?

2) If symmetry fully expresses interactions, how do we evaluate "symmetry implies asymmetry", "imperfect symmetry", " approximate symmetry", " hidden symmetry"? hidden symmetry"?

3) One of the implications of energy-momentum conservation is that they have no origin, are a natural existence, and do not change with scale and energy level or temperature; symmetry has an origin, and is related to scale, temperature and energy level. How are they equivalent to each other?

4) Must there be an ultimate symmetry which will determine everything and be consistent with conservation and invariance?

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Notes

† We will analyze this separately, which is its most important physical feature [20].

‡ Can different forms of energy be unified?[16]

* We can define the actual observer to be the object of action and the abstract observer to be the object of action for analysis. For example, when we analyze the Doppler effect, we are analyzing it in the abstract; if you don't actually detect it, no Doppler effect occurs in the object of analysis.

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References

[1] Wigner, E. P. (1964). "Symmetry and conservation laws." Proceedings of the National Academy of Sciences 51(5): 956-965.

[2] Yang, C. N. (1996). "Symmetry and physics." Proceedings of the American Philosophical Society 140(3): 267-288.

[3] Yang, C. N. (1980). "Einstein's impact on theoretical physics." Physics Today 33(6): 42-49.

[4] Brading, K., E. Castellani and N. Teh (2003). "Symmetry and symmetry breaking."

[5] Bohr, N., H. A. Kramers and J. C. Slater (1924). "LXXVI. The quantum theory of radiation." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47(281): 785-802.

[6] Dirac, P. A. M. (1927). "The quantum theory of dispersion." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 114(769): 710-728.

[7] Carroll, S. M. and J. Lodman (2021). "Energy non-conservation in quantum mechanics." Foundations of Physics 51(4): 83.

[8] Bondi, H. (1990). "Conservation and non-conservation in general relativity." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 427(1873): 249-258.

[9] Maudlin, T., E. Okon and D. Sudarsky (2020). "On the status of conservation laws in physics: Implications for semiclassical gravity." Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 69: 67-81.

[10] Pitts, J. B. (2022). "General Relativity, Mental Causation, and Energy Conservation." Erkenntnis 87.

[11] Hoyle, F. (1948). "A new model for the expanding universe." Monthly Notices of the Royal Astronomical Society, Vol. 108, p. 372 108: 372.

[12] Vilenkin, A. (1982). "Creation of universes from nothing." Physics Letters B 117(1): 25-28.

[13] Josset, T., A. Perez and D. Sudarsky (2017). "Dark energy from violation of energy conservation." Physical review letters 118(2): 021102.

[14] Singh Kohli, I. (2014). "Comments On: A Universe From Nothing." arXiv e-prints: arXiv: 1405.6091.

[15] Tryon, E. P. (1973). "Is the Universe a Vacuum Fluctuation?" Nature 246(5433): 396-397.