Skip to main content

Is the cosmological constant an eigenvalue?

Abstract

We propose to reinterpret Einstein’s field equations as a nonlinear eigenvalue problem, where the cosmological constant \(\Lambda \) plays the role of the (smallest) eigenvalue. This interpretation is fully worked out for a simple model of scalar gravity. The essential ingredient for the feasibility of this approach is that the classical field equations be nonlinear, i.e., that the gravitational field is itself a source of gravity. The cosmological consequences and implications of this approach are developed and discussed.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    An equation almost identical to Eq. (5) (apart from a factor of 2) can be derived directly from Einstein’s field equations by considering a metric where only the time-time component differs from its Minkowski value and requiring it to approach the Minkowski metric for large spatial distances [21, 22].

  2. 2.

    Equation (9) implies a modification of Newton’s acceleration in the gravitational field generated by a pointlike mass m, which becomes: \(g(r)=-Gm/r^2 + c^2 \Lambda \,r\). Such modification becomes sizeable only on cosmological scales. See [27] for further details.

References

  1. 1.

    Rich, J.: Fundamentals of Cosmology. Springer, Berlin (2010)

    Book  Google Scholar 

  2. 2.

    Milgrom, M.: A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J. 270, 365 (1983)

    ADS  Article  Google Scholar 

  3. 3.

    Famaey, B., McGaugh, S.S.: Modified Newtonian dynamics MOND: observational phenomenology and relativistic extensions. Living Rev. Relativ. 15, 10 (2012)

    ADS  Article  Google Scholar 

  4. 4.

    Carroll, S.M.: The cosmological constant. Living Rev. Relativ. 4, 1 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Caldwell, R.R., Dave, R., Steinhardt, P.J.: Cosmological imprint of an energy component with general equation of state. Phys. Rev. Lett. 80, 1582 (1998)

    ADS  Article  Google Scholar 

  6. 6.

    Casado, J.: Linear expansion models vs. standard cosmologies: a critical and historical overview. Astrophys. Space Sci. 365, 16 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Milne, E.A.: World-structure and the expansion of the universe. Zeitschrift für Astrophysik 6, 1 (1933)

    ADS  MATH  Google Scholar 

  8. 8.

    Melia, F., Shevchuk, A.S.H.: The \(R_h=ct\) universe. Mon. Not. R. Astron. Soc. 419, 2579 (2012)

    ADS  Article  Google Scholar 

  9. 9.

    Villata, M.: On the nature of dark energy: the lattice Universe. Astrophys. Space Sci. 345, 1 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Benoit-Lévy, A., Chardin, G.: Introducing the Dirac-Milne universe. Astron. Astrophys. 537, A78 (2012)

    ADS  Article  Google Scholar 

  11. 11.

    Chardin, G., Manfredi, G.: Antimatter and the Dirac-Milne universe. Hyperfine Interact. 239, 45 (2018)

    ADS  Article  Google Scholar 

  12. 12.

    Manfredi, G., Rouet, J.-L., Miller, B., Chardin, G.: Cosmological structure formation with negative mass. Phys. Rev. D 98, 023514 (2018)

    ADS  Article  Google Scholar 

  13. 13.

    Manfredi, G., Rouet, J.-L., Miller, B.N., Chardin, G.: Structure formation in a Dirac-Milne universe: comparison with the standard cosmological model. Phys. Rev. D 102, 103518 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Einstein, A.: Zur Theorie des statischen Gravitationsfeldes. Ann. Phys. 343, 443 (1912)

    Article  Google Scholar 

  15. 15.

    Riess, A.G., Filippenko, A.V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P.M., Gilliland, R.L., Hogan, C.J., Jha, S., Kirshner, R.P., Leibundgut, B., Phillips, M.M., Reiss, D., Schmidt, B.P., Schommer, R.A., Smith, R.C., Spyromilio, J., Stubbs, C., Suntzeff, N.B., Tonry, J.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009 (1998)

    ADS  Article  Google Scholar 

  16. 16.

    Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R.A., Nugent, P., Castro, P.G., Deustua, S., Fabbro, S., Goobar, A., Groom, D.E., Hook, I.M., Kim, A.G., Kim, M.Y., Lee, J.C., Nunes, N.J., Pain, R., Pennypacker, C.R., Quimby, R., Lidman, C., Ellis, R.S., Irwin, M., McMahon, R.G., Ruiz-Lapuente, P., Walton, N., Schaefer, B., Boyle, B.J., Filippenko, A.V., Matheson, T., Fruchter, A.S., Panagia, N., Newberg, H.J.M., W. J. Couch and The Supernova Cosmology Project: Measurements of \(\Omega \) and \(\Lambda \) from 42 high-redshift supernovae. Astrophys. J. 517, 565 (1999)

    ADS  Article  Google Scholar 

  17. 17.

    Nielsen, J.T., Guffanti, A., Sarkar, S.: Marginal evidence for cosmic acceleration from Type Ia supernovae. Sci. Rep. 6, 35596 (2016)

    ADS  Article  Google Scholar 

  18. 18.

    Lovelock, D.: The four-dimensionality of space and the Einstein tensor. J. Math. Phys. 13, 874 (1972)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Bianchi, E., Rovelli, C.: Why all these prejudices against a constant? (2010). arXiv:1002.3966

  20. 20.

    Adamek, J., Durrer, R., Kunz, M.: N-body methods for relativistic cosmology. Class. Quantum Gravity 31, 234006 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    Giulini, D.: Consistently implementing the field self-energy in Newtonian gravity. Phys. Lett. A 232, 165 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Giulini, D.: Einstein’s Prague field equation of 1912: another perspective. In: Relativity and Gravitation, pp. 69–82. Springer, Berlin (2014)

  23. 23.

    Franklin, J.: Self-consistent, self-coupled scalar gravity. Am. J. Phys. 83, 332 (2015)

    ADS  Article  Google Scholar 

  24. 24.

    Harvey, A., Schucking, E.: Einstein’s mistake and the cosmological constant. Am. J. Phys. 68, 723 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Nowarowski, M.: The consistent Newtonian limit of Einstein’s gravity with a cosmological constant. Int. J. Mod. Phys. D 10, 649 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  26. 26.

    Chiappinelli, R.: What do you mean by nonlinear eigenvalue problems? Axioms 7, 39 (2018)

    Article  Google Scholar 

  27. 27.

    Gurzadyan, V.G.: The cosmological constant in the McCrea–Milne cosmological scheme. Observatory 105, 42 (1985)

    ADS  Google Scholar 

  28. 28.

    Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Prob. 25, 123004 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Melia, F.: The cosmic horizon. Mon. Not. R. Astron. Soc. 382, 1917 (2007)

    ADS  Article  Google Scholar 

  30. 30.

    Melia, F.: On recent claims concerning the \({R}_h=ct\) universe. Mon. Not. R. Astron. Soc. 446, 1191 (2014)

    ADS  Article  Google Scholar 

  31. 31.

    Indelicato, P., Chardin, G., Grandemange, P., Lunney, D., Manea, V., Badertscher, A., Crivelli, P., Curioni, A., Marchionni, A., Rossi, B., Rubbia, A., Nesvizhevsky, V., Brook-Roberge, D., Comini, P., Debu, P., Dupré, P., Liszkay, L., Mansoulié, B., Pérez, P., Rey, J.-M., Reymond, B., Ruiz, N., Sacquin, Y., Vallage, B., Biraben, F., Cladé, P., Douillet, A., Dufour, G., Guellati, S., Hilico, L., Lambrecht, A., Guérout, R., Karr, J.-P., Nez, F., Reynaud, S., Szabo, C.I., Tran, V.-Q., Trapateau, J., Mohri, A., Yamazaki, Y., Charlton, M., Eriksson, S., Madsen, N., van der Werf, D., Kuroda, N., Torii, H., Nagashima, Y., Schmidt-Kaler, F., Walz, J., Wolf, S., Hervieux, P.-A., Manfredi, G., Voronin, A., Froelich, P., Wronka, S., Staszczak, M.: The Gbar project, or how does antimatter fall? Hyperfine Interact. 228, 141 (2014)

    ADS  Article  Google Scholar 

  32. 32.

    Bertsche, W.A.: Prospects for comparison of matter and antimatter gravitation with ALPHA-g. Phil. Trans. R. Soc. A 376, 20170265 (2018)

    ADS  Article  Google Scholar 

  33. 33.

    Kellerbauer, A., Amoretti, M., Belov, A., Bonomi, G., Boscolo, I., Brusa, R., Büchner, M., Byakov, V., Cabaret, L., Canali, C., Carraro, C., Castelli, F., Cialdi, S., de Combarieu, M., Comparat, D., Consolati, G., Djourelov, N., Doser, M., Drobychev, G., Dupasquier, A., Ferrari, G., Forget, P., Formaro, L., Gervasini, A., Giammarchi, M., Gninenko, S., Gribakin, G., Hogan, S., Jacquey, M., Lagomarsino, V., Manuzio, G., Mariazzi, S., Matveev, V., Meier, J., Merkt, F., Nedelec, P., Oberthaler, M., Pari, P., Prevedelli, M., Quasso, F., Rotondi, A., Sillou, D., Stepanov, S., Stroke, H., Testera, G., Tino, G., Trénec, G., Vairo, A., Vigué, J., Walters, H., Warring, U., Zavatarelli, S., Zvezhinskij, D.: Proposed antimatter gravity measurement with an antihydrogen beam. Nucl. Instrum. Methods Phys. Res. Sect. B 266, 351 (2008)

    ADS  Article  Google Scholar 

  34. 34.

    Rubin, V.C., Ford Jr., W.K., Thonnard, N.: Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R = 4 kpc) to UGC 2885 (R = 122 kpc). Astrophys. J. 238, 471 (1980)

    ADS  Article  Google Scholar 

  35. 35.

    Capozziello, S., Garattini, R.: The cosmological constant as an eigenvalue of f(R)-gravity Hamiltonian constraint. Class. Quantum Gravity 24, 1627 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  36. 36.

    Garattini, R.: The cosmological constant and the Wheeler–DeWitt equation (2009). arXiv:0910.1735

  37. 37.

    Garattini, R.: Cosmological constant as an eigenvalue of the Hamiltonian constraint in Hořava–Lifshitz theory. Phys. Rev. D 86, 123507 (2012)

    ADS  Article  Google Scholar 

  38. 38.

    Zecca, A.: The Wheeler–DeWitt equation as an eigenvalue problem for the cosmological constant. Eur. Phys. J. Plus 129, 59 (2014)

    Article  Google Scholar 

  39. 39.

    Garattini, R., De Laurentis, M.: The cosmological constant as an eigenvalue of the Hamiltonian constraint in a varying speed of light theory. Fortschr. Phys. 65, 1600108 (2017)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Bergshoeff, E.A., Rosseel, J., Townsend, P.K.: Gravity and the spin-2 planar Schrödinger equation. Phys. Rev. Lett. 120, 141601 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  41. 41.

    Fitzpatrick, P., et al.: Klaus Deimling, nonlinear functional analysis. Bull. (New Ser.) Am. Math. Soc. 20, 277 (1989)

    Article  Google Scholar 

  42. 42.

    Ambrosetti, A., Malchiodi, A., et al.: Nonlinear analysis and semilinear elliptic problems. In: Nonlinear Analysis and Semilinear Elliptic Problems, vol. 104. Cambridge University Press, Cambridge (2007)

  43. 43.

    Lindqvist, P.: A nonlinear eigenvalue problem. Top. Math. Anal. 3, 175 (2008)

    Article  Google Scholar 

  44. 44.

    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)

    Book  Google Scholar 

Download references

Acknowledgements

I wish to thank Gabriel Chardin for his thorough reading of the manuscript and many insightful comments. I also thank Omar Morandi and Raffaele Chiappinelli for helping with some mathematical issues. Needless to say, I am solely responsible for the errors or inaccuracies that may still remain in this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Giovanni Manfredi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: mathematical digressions

Appendix A: mathematical digressions

Appendix A.1: nonlinear eigenvalue problems

Nonlinear eigenvalue problems occur frequently in the mathematical literature. A very readable review was published recently [26]. More extensive discussions can be found in the two monographs [41, 42]. A nonlinear eigenvalue problem can be written generally as: \({{\mathcal {F}}}(\lambda ,u)=0\), where \(\lambda \) is the eigenvalue, \({{\mathcal {F}}}\) is a nonlinear function, and u usually belongs to a Banach space. In our case, the problem takes the special form: \({{\mathcal {F}}}(u)=\lambda u\). A typical example is the p-Laplacian eigenvalue problem with Dirichlet boundary conditions:

$$\begin{aligned}&\text {div} (|\nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2} u = 0, \quad \mathrm{in}\,\, \Omega , \end{aligned}$$
(A1)
$$\begin{aligned}&u = 0, \quad \mathrm{on }\,\, \partial \Omega \end{aligned}$$
(A2)

where \(\Omega \subset {\mathbb {R}}^3\), and \(p>1\) is an integer [43]. Note that the above equation is homogeneous in u, just as Eq. (8) is homogeneous in \(\Phi \). This means that if u is a solution, then Cu is also a solution, for any real or complex number C. This is a property shared with linear equations.

Appendix A.2: relevant functional spaces

Let us consider the following linear eigenvalue problem, which corresponds to Eq. (10) in 1D, with \(2\pi G/c^2=1\) and \(\lambda =\Lambda /2\):

$$\begin{aligned}&-u_{xx} + \rho u = \lambda u, \quad x \in I \equiv [0,2\pi ], \end{aligned}$$
(A3)
$$\begin{aligned}&u(0)=u(2\pi )=1, \end{aligned}$$
(A4)
$$\begin{aligned}&u_x(0)=u_x(2\pi )=0, \end{aligned}$$
(A5)

with \(\rho \ge 0\) and \(\int _I \rho (x)dx < \infty \), and the subscript stands for differentiation. Elliptic PDEs are usually defined in Sobolev spaces [44], because such spaces guarantee the existence of the derivatives, at least in a weak sense. Here, the appropriate space seems to be \(H^1\), which is also a Hilbert space equipped with the inner product \((u,v)=\int _I uv dx + \int _I u_x v_x dx\) and the related norm \(\Vert u\Vert =\sqrt{(u,u)}\).

We first consider the simple case where \(\rho =0\) (no matter density). Then, the eigenfunctions of Eq. (A3) are cosines: \(\varphi _n = \cos (nx)\), with eigenvalues \(\lambda _n = n^2\). Then, if a solution u(x) exists for the full Eq. (A3) with non-vanishing \(\rho \), it can be represented as:

$$\begin{aligned} u(x) = \frac{\sum _n a_n \varphi _n(x)}{\sum _n a_n}, \end{aligned}$$
(A6)

which satisfies the required boundary conditions (\(a_n\) are real numbers). Of course, we have not proven that such a solution actually exists, which is a nontrivial mathematical problem.

In an infinite space (\(I = {\mathbb {R}}^3\)), the norm generally diverges, because of the boundary condition on the first derivative (A5). However, this point should not be considered crucial: for instance, non-integrable wave functions are routinely used as solutions of the standard Schrödinger equation to describe propagating plane waves. In addition, as we have seen in the main text, these boundary conditions are to be applied on the Hubble sphere, i.e. in a finite volume, so this problem should not actually arise.

Appendix A.3: elliptic equations with Cauchy boundary conditions

An elliptic PDE, such as Eq. (A1), with Dirichlet boundary conditions, constitutes a well-posed problem. In contrast, if one takes Cauchy boundary conditions, the related problem is ill-posed [28]. Cauchy boundary conditions correspond to specifying both the function and its normal derivative on the boundary, e.g. \(u = \partial u/\partial n=0\) on \(\partial \Omega \). In that case, the problem does not always have a solution. Our Eqs. (10), (12), and (13) fall in this category.

However, as an eigenvalue problem, the problem makes perfect sense. The eigenvalue \(\lambda \) is determined precisely by the requirement that the problem does possess a solution for Cauchy boundary conditions. We can illustrate this on a simple problem in 1D. Let us consider the following first-order nonlinear differential equation:

$$\begin{aligned}&u^2 u_x + u = \lambda u, \quad x \in [0,1], \end{aligned}$$
(A7)
$$\begin{aligned}&\quad u(0) = 0, \quad u_x(1) =\sqrt{2}/2 . \end{aligned}$$
(A8)

The problem is obviously overdetermined, as a 1D differential equation admits only one boundary condition. Hence, for fixed \(\lambda \), Eq. (A7) does not generally admit a solution respecting both boundary conditions (A8). This is easily verified by checking that a solution of Eq. (A7) is \(u(x) = u_0 \sqrt{x}\), with \(u_0=\pm \sqrt{2(\lambda -1)}\), which satisfies the boundary condition in \(x=0\) but not (for arbitrary \(\lambda \)) in \(x=1\). But if one treats Eq. (A7) as an eigenvalue problem, the second boundary condition becomes a constraint that fixes the eigenvalue, in this case \(\lambda =2\). This is precisely what happens for our problem: adding the extra boundary condition on the gradient of \(\Psi \) on the boundary of the domain determines the value of \(\Lambda \).

As a further ‘physical’ example, let us consider the 1D heat equation with Cauchy boundary conditions:

$$\begin{aligned}&T_t = T_{xx} - S(x) T + \lambda T, \quad x \in [0,L], \end{aligned}$$
(A9)
$$\begin{aligned}&T(x,0) = T_0, \end{aligned}$$
(A10)
$$\begin{aligned}&T(0,t) = T(L,t) = T_0, \end{aligned}$$
(A11)
$$\begin{aligned}&T_x(0,t) = T_x(L,t) = 0 , \end{aligned}$$
(A12)

where T(xt) is the local temperature at an instant t, \(-S(x) T\) is a heat sink, and \(\lambda T\) is a heat source. The above problem corresponds to a system initially at temperature \(T_0\) everywhere, which evolves under the action of the sinks and sources of heat. The boundary conditions prescribe that the temperature must remain equal to \(T_0\) at \(x=0\) and \( x=L\) [Eq. (A11)] and that the heat flux must vanish at the boundaries [Eq. (A12)]. This is not physically realizable, of course: if no heat can escape the system, then the temperature at the boundaries cannot be fixed arbitrarily, but will be determined by the interplay of the sinks and sources. However, if \(\lambda \) is not fixed but rather considered as an eigenvalue, then the Cauchy boundary conditions can indeed be satisfied. Physically, this means that the source term \(\lambda T\) is tuned precisely so as to keep the temperature equal to \(T_0\) at the two boundaries. This determines the value of \(\lambda \).

We also note that a steady-state solution of Eq. (A9) corresponds to a solution of our model Eq. (10), in a 1D planar geometry. Indeed, the numerical results shown in this work were obtained by propagating the field \(\Psi (r,t)\) according to a heat-type equation like Eq. (A9), so that the solution relaxes naturally to the lowest-order eigenfunction.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Manfredi, G. Is the cosmological constant an eigenvalue?. Gen Relativ Gravit 53, 31 (2021). https://doi.org/10.1007/s10714-021-02800-8

Download citation

Keywords

  • Cosmological constant
  • Scalar gravity
  • Cosmology
  • Dark matter
  • Dark energy