Einstein's Field Equations allow for the derivation of Newton's law and this, together with the velocity profile of the stars within the galaxies and the galaxies within the galaxy clusters, leads to the introduction of unknown dark matter. We haven't found this dark matter yet, so insisting on questioning whether this introduction is valid and unavoidable is reasonable.

There is MOND (modified Newton dynamics), which works quite well for galaxies (not well for galaxy clusters), but seems not to be a real alternative as it does not deal with curved space-time which is an everyday experience for people working on GPS.

There are some modifications of the Einstein Field Equations which introduce new fields instead of unknown dark matter—but there is an "equivalence between dark matter particles gravitationally coupled to the Standard Model fields and modified gravity theories designed to account for the dark matter phenomenon", which was impressively shown in the paper of Calmet and Kuntz in 2017 in Physics & Astronomy (arXiv: 1702.03832 [gr-qc]).

There is essentially more space near the mass than further away (the prefactor $A$ of the space-part in the metric $$\mathrm{d}s^2 = - B \mathrm{d}t^2 + A \mathrm{d}r^2 + r^2 (\mathrm{d}\theta^2 + \sin^2{\theta}\mathrm{d}\phi^2)$$ is bigger than 1). $A$ is approaching 1 in the infinity, leading to Newton's law.

  1. There is curved space-time and the field equations are valid in the solar system.
  2. Dark matter theories and modifications of Einstein's Field Equations introducing new fields are impossible to distinguish experimentally - and neither of them has been found yet.
  3. There is more space near the center of gravity than further away.
  4. The concept of "space-time is approaching 'flat space-time' in infinity" works well for the solar system.

Rethinking these points leads to the question of whether it could be possible to simply assume other boundary conditions (4.) for galaxies using the Einstein field equations than used for the solar system to explain the dark matter effect.

Using not flat space-time, but "vanishing space-time" (A approaching 0 instead of 1) as the boundary condition for galaxies leads to the introduction of a vacuum-energy which has to be smaller than 0. The space itself would have to be regarded as a field with negative energy. Regarding space-time of galaxies as a potential well. Obviously, this thinking seems to be very far-fetched as it isn't discussed anywhere in the community. Please, help me to understand why.

Is that (vacuum energy < 0) known to be forbidden per se? Is it obviously impossible? Why?

  • $\begingroup$ About what factor $A$ are you talking? Can you show us the metric you have in mind? $\endgroup$ Jan 29, 2022 at 16:18
  • $\begingroup$ The general static, spherically symmetric metric: $$ds^2 = -Bdt^2 + Adr^2 + r^2(d\theta^2 + sin^2\theta d\phi^2) \tag2$$ $\endgroup$ Jan 29, 2022 at 16:41
  • $\begingroup$ Btw, there are other hints to a vacuum energy smaller than zero, for example moving the Ricci-scalar into the RHS (energy-term) of the Einstein field equations leads to a negative energy of the space(time). See physics.stackexchange.com/questions/684085/… $\endgroup$ Jan 30, 2022 at 8:29

1 Answer 1


In physics there is nothing forbidden per se but it has to fulfill some restrictions, please see the informative contribution entitled "Quantum vacuum: Less than zero energy" on https://www.sciencedaily.com/releases/2019/10/191002102750.htm . Your idea to use the negative cosmological constant (a prerequisite for your A approaching to zero) to explain the galaxy rotational curves has been recently discussed in the paper "Negative cosmological constant in the dark sector?", https://arxiv.org/abs/2008.10237 by other physicists, too. Maybe you will find there some support of your argumentation.

  • $\begingroup$ Thanks for the links! Very interesting and promising. "Your idea is to use the negative cosmological constant..." is yet not exactly what I'm looking for - the cosmological constant (as it is used in the standard model) is tighly interwoven with a pressure in the stress-energy-tensor ... Yet, for the idea of A approaching zero "simply" a negative energy density $T_{00} < 0$ is supposedly sufficient. $\endgroup$ Feb 3, 2022 at 21:06

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