Dark Energy or a Simple Mistake: Do We Understand How Non-Unique the Einstein Tensor Actually Is? by Jeff Baugher

After reading through some of the essays in this contest, it may be helpful for any public readers to again consider the questions within the contest rules:

"What exactly are the basic physical and mathematical postulates in our "fundamental" physical theories or candidate theories?"

-- My answer to this would of course be that gravity is "attractive" and that there is no physical cosmological medium. The mathematical postulate is that flat spacetime cannot occur with a cosmological constant.

"Which of these could be replaced with a weaker or less restrictive postulate or even removed?"

-- I would state that the less restrictive postulate is to use the full modified Einstein field equation with a flat spacetime.

"What would be the potential benefits (or disadvantages) of these changes?"

-- It would account for the accelerating expansion, provide a basic model for understanding of quantum field theory and change cosmology into a coherent physical theory.

"What are the implicit assumptions we tend to forget we have postulated, or that have become so ingrained that they have become unquestioned dogma?"

-- It is ingrained in us to think of gravity as attractive between two massive bodies (action at a distance) while at the same time to now accept that the space in between has many properties. We implicitly assume that we are "particles" with energy density that are separate from the vacuum of nothingness, but have also come to accept at the very same time that this vacuum has an energy density.

"What are the most interesting current "anomalous" experiments, and what assumptions would we be forced to give up if we took them seriously?"

-- The most interesting anomalous experiment is of course the accelerating expansion. If we took this seriously, we might reconsider how the cosmological constant relates to attractive gravity.

"Which assumptions in physics and in cosmology are interdependent or required for self-consistency, and which could be modified?"

-- Energy density of the vacuum and how energy density relates to gravity.

"Where are there "paradoxes", or apparent contradictions stemming from the combination of seemingly reasonable assumptions? How do we reconcile these?"

-- How can vacuum have such a high energy density, be the cosmological constant, but yet have not only such a small effect on gravity, but a repulsive one at that?

"Are there "meta"-assumptions or criteria (e.g. "simplicity", or "beauty", or "utility") that can or should underlie some current "fundamental" assumptions?"

-- Symmetry of Riemannian geometry.

"(Note: Successful and interesting essays will not use this topic as an opportunity to trot out their pet theories simply because those theories reject assumptions of some other or established theory. Rather, the challenge here is to create new and insightful questions or analysis about basic, often tacit, assumptions that can be questioned but often are not.) "

-- If the Einstein field equation is a differential equation with a constant of integration, why have we never examined the consequences of any other combinations of the Einstein tensor and the cosmological constant?

"Foundational: This Contest is limited to works addressing, in one of its many facets, our understanding of the deep or "ultimate" nature of reality."

-- What exactly are we made of?

"Accessible to a diverse, well-educated but non-specialist audience, aiming in the range between the level of Scientific American and a review article in Science or Nature."

-- Hopefully I have kept this basic enough so that others can manipulate the equations and find flaws and strengths in my logic for themselves.

Another analogy with epicycles...

Like the interpretation of perfect circles to the concept of orbits, we assume that the Einstein tensor is the only interpretation of the full field equation. If we hold onto the perfect circles, then every modification must take those into account. If we hold onto Guv describing particulate matter, every modification to GR must adhere to that concept.

Hi Jeff:

I thoroughly enjoyed your concise and well-written essay.

Based on arguments and results presented in my paper - -" From Absurd to Elegant Universe", your following conclusion is fully vindicated:

"The cosmological constant, within the Einstein field equation (EFE), is the leading method to account for the discovered accelerating expansion of the universe, whether or not an underlying cause is understood. ......... If this were in error it would raise profound questions of how the accelerating expansion relates to not only a gravitationally repulsive cosmological constant, but to our entire understanding of physics."

My paper provides a new fundamental understanding of the Cosmological Constant and relativistic universe expansion as an alternative to the widely accepted linear Hubble expansion. The current paradoxes and inconsistencies are shown to be artifacts of the missing (hidden) physics of the well-known phenomenon of spontaneous decay. A new Gravity Nullification Model for Universe Expansion (GNMUE) is proposed that integrates the missing physics of the spontaneous mass-energy conversion into a simplified form of general relativity. The model predicts the observed expansion of the universe and galaxies and other data. The model provides answers to key fundamental questions and resolves paradoxes among general relativity, quantum mechanics, and cosmology. It also bridges the gap between quantum mechanics and relativity theories via revealing relativistic understanding of the inner workings of quantum mechanics. The impact of the new understanding on widely-accepted fundamental assumptions is discussed and a new wholesome perspective on reality is provided.

I would greatly appreciate your comments on my paper.

Best Regards

Avtar Singh

Avtar,

I have left questions on your essay page.

Regards,

Jeff

After reviewing the thesis some more, I would have to retract what I stated above. It does sound like I agree more with the paper they are criticizing. A specific reason why is if Omega guv is linked to the curvature potential/potential energy of all quantum harmonic oscillators for a point in spacetime, then Luv would represent the remaining unoccupied states. Thus 8pi< T> represents those states that are occupied. Since I think that the Omega guv-Luv is going to model gravity better, then I think that in a very real sense

"where the average is over an ensemble of possible outcomes, but it cannot reasonably give the exact value of the curvature due to a single particle as an average of all the things that might happen!"

is what may actually prove to be true.

I didn't specifically address it in my essay, but it could be asked if this isn't a violation of the Weak Energy Condition or just the unimodular approach. Taking a look at equation (1.33) of The Cosmological Constant Problem,an Inspiration for New Physics we have [math]R-8\pi GT\equiv -4\Lambda[/math]

The difference with what I am proposing is to replace the tensor T with a residual energy density. This would seem to be equivalent to just the energy momentum tensor of matter, but should solve some paradoxes. As for the WEC, this specifies that energy density cannot be negative. Since in this way I am defining energy density that we are familiar with as the difference between higher and lower positive tensors, it isn't a negative energy density.

Regards,

Jeff

Jeff:

I have responded to your questions/comments on my essay page.

Thanks,

Avtar

Jeff. Life is fragile, and it is very precisely regulated and balanced -- this includes in relation to the direct experience of physical force/energy by the body. There is no surviving long term in outer space -- for an unlimited time with reproduction. That is a fact.

The direct experience of the force/energy of physics/physical experience by the body is fundamental/essential to the deepest, integrated, coherent, significant, extensive, and most fundamental ideas/theories/understanding(s) in physics.

My essay will be entered shortly.

The physical/full experience of outer space destroys us and precludes our thought and being entirely. Accordingly, our understanding of outer space is significantly limited. Thought cannot describe/approximate to what it is not, ultimately.

"There is no surviving long term in outer space -- for an unlimited time with reproduction. That is a fact."

Perhaps there are areas of research that I was not aware of but that sounds very much like an opinion, rather than an actual fact. Citation?

I agree with the quote in your essay: "nothing short of a revolution in our understanding of fundamental physics will be required to achieve a full understanding of the cosmic acceleration." In addition, I do not believe that such a revolution can originate at any of the so-called top universities.

Now about your essay. I have two issues with it.

1) In §2 on page 2, you mention that when integrating over a specific interval [a, b] two different integrands (functions) can yield the same integral. Of course this is so, e.g. the function f(x) = 2x and the constant function g(x) = 1 yield the same integral over the interval [0,1]. But then you seem to go off on a tangent. You define f1 = C - f2 and then you show that both f1 and C - f2 yield the same integral over the interval [x1, x2]. But how is that an example of two different functions yielding the same integral? The functions "f1" and "C - f2" are, namely, identical by your own definition. And as a side note, the term "C" in this defintion of yours is a constant function, not a constant of integration (which is a number) as you seem to imply in figure 2. So why this example?

2) In General Relativity (GR), the index-free form of the Einstein field equation (EFE) is [math]G+\Lambda G=\kappa T[/math]

I understand that you want to replace this EFE by this equation:

[math]g\Omega-L=\kappa T[/math]

But why is that an improvement? With the EFE, one can calculate the metric tensor (or the metric tensor and the cosmological constant) given the stress-energy tensor T. But what is the physical interpretation of the tensor L in your equation? You might be able to calculate it, but what is it? You refer to figure 3 as an illustration, but what are the two scalar fields in the bottom picture physically? What is their physical source? What is the corresponding theory of gravitation (what is gravitation according to you)? It seems to me that your idea requires some further development.

All in all, the parallell between our essays is that we both question GR. But there are nuances. I believe that GR is correct in its area of application, but emergent instead of fundamental. You, on the other hand, seem to believe that GR isn't even correct in its area of application, as you suggest another field equation - I take it that you propose these as an improvement of the EFE even at macroscopic scale. Am I correct?

Good luck with the contest.

Best regards, Marcoen

@Steve: hail all hail to the revolution (Jon English).

Hi Marcoen and thanks for the excellent questions:

(1) Perhaps I should state this as actually three different functions f1, f2 and C. I do mean that C is a constant (i.e. a number) and not a function of x (i.e. C(x), a constant can still be integrated, so perhaps when I referred to it as a constant function this was misconstrued as meaning it to be a linear constant function of x, which it is not). A very elementary part of calculus, not something which stands out. The point is that from the answer alone, which would just be area, there is no way to tell whether I used an integration of f1 over x1 to x2 or the integration of C-f2. If one finds the antiderivative of a function, there exists an arbitrary constant of integration which could have any value.

I use this specific example, in that if the cosmological constant is a constant of integration (sometimes called the unimodular approach), then it is a bit of a coincidence that one of the biggest paradoxes about it is that it should be extremely large (according to QFT) but that empirically it appears to be rather tiny. In my example, the constant C could be extremely large or it could be zero and there is no way from anti-differentiation to tell which.

(2) Why would my equation be an improvement over Einstein's tensor G? It could also be fairly asking whether I am making the equation more difficult. To answer this question fully, it might be helpful if we consider a model of a cosmological fluid (see the Dark Energy Task Force report on the use of these). With the modified EFE, there is no known way to put in the estimated value of the cosmological constant (10120 higher than observed) and also find a mechanism which makes it very small. However, by using the alternate equation I propose, a full theoretical value of the "vacuum" could be put in. Suppose then there is matter present at a point in this vacuum. If we model this particle as a reduction in the density of this cosmological fluid, the Luv would be equated to the stress energy tensor of the remaining density kTresuv. Since this constant and remaining stress energy tensor together are mathematically equivalent to the Einstein tensor/stress energy tensor of matter, it should approximate Newtonian gravity, although calling that gravity "attractive" would be qualitatively incorrect. The main point of this though, is that it appears that it would not model it exactly, since at larger radii the Omega term would increase linearly to a point where the gravitational vector switches direction from an apparent attraction to a repulsion. Whether this matches what we empirically observe I do not yet know, but there are too many coincidences in the formation of field theory, action at a distance, GR and now a linear accelerating expansion that has been tied to the cosmological constant to not be suspicious. If this model gives us a more accurate picture of macro gravitational effects, I would also become very concerned about how abstract these cosmological fluid models actually are.

Forgot to answer: "You refer to figure 3 as an illustration, but what are the two scalar fields in the bottom picture physically? What is their physical source? What is the corresponding theory of gravitation (what is gravitation according to you)?"

The scalar fields are generic vector examples of Newtonian gravitational fields.

The first scalar field is a vector representation of the Omega (or equivalently the cosmological constant) term. It should actually be a scalar field with linearly increasing values radiating out in a circular fashion, but it complicates the explanation in that by definition gravitational force vector requires at least another mass. It is an isotropic pressure.

The second scalar field represents a vector formed from the gradient of stress energy tensor equated to Luv. The field scalar values are of a much larger magnitude and the gradient is of an opposing direction, but of course the minus sign in front gives us an equivalent force vector to that of the regular Newtonian gradient. The illustration is to point out that the top vector field is what GR reduces down to in the weak limit, whereas the bottom two fields are closer to what is now empirically known.

Regards,

Jeff

Thinking about some of Marcoen's questions and I realize that my explanations aren't coming across clearly. Let's try this:

--In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) is only defined up to an additive constant, the constant of integration.

Suppose we take a measurement of gravitational force, lensing, etc. and we compare that to what our differential equations give us, and they agree for a given input. We are basically assuming that we are using the correct anti-derivative of our function.

Thinking about the hypothesized size of the cosmological constant, if [math]R-G \equiv -4\Lambda[/math], then from the definition of an indefinite integral, how do we KNOW whether this equation should be [math]R-G \equiv -4\Lambda=0[/math] or [math]R\equiv 4\Lambda-G[/math] If the second is not the correct choice, why?

Hello Jeff,

(1) If you write "f1 = C - f2" then this is a relation between functions. That is, the term "f1" on the left hand side is an element of a function space, and the term "C - f2" on the right hand side is an element of a function space. That latter term "C - f2" is the function obtained by adding the function "- f2" to the constant function "C": in a function space, the binary operation addition takes two functions as an argument. If you want C to be a numerical constant (a number), then you can define the function "f1" by defining its function value as in "f1(x) = C - f2(x)": this is a relation between numbers, so here C is a number (a constant of integration if you want). So subtleties in mathematical notation can make a big difference.

(2) I'm still having difficulties with the interpretation of your two scalar fields, illustrated by your figure 3. It seems to me that the first scalar field doesn't occur in the ontology of Newtonian mechanics. Isn't this something you want to add?

As a side note, in the framework of my own theory, the EPT, the observed space is a material substance which is transcended by matter waves (there are, thus, two different kinds of components). A metric may then be defined in terms of a difference between two scalars (close to what you suggest) but at this point it is not possible to conduct tensor algebra in this framework as the necessary mathematics have yet to be developed.

With best regards, Marcoen

Hi Marcoen,

(1) Yes, you are correct in your statements, but what I am pointing out in my essay applies to either a function space or to just numbers. If I obtain an answer that is the derivative of the function spaces, was the correct physical relation from which the answer came, f1' or (C-f2)'? If I obtain a numerical value at x, was the correct physical relation f1(x) or C - f2(x)? This antidifferentiation, at its most basic level, stems from the fact that human senses are not well adapted for constant magnitudes but rather differences in space and time. By this I mean that when we obtain a physical measurement, we antidifferntiate though our understanding of the laws of physics. Perhaps I did not relate this well in my essay, but I wanted to keep it short as possible.

(2) I assume you are referring to the first scalar field at the bottom of Fig.(3). I do have a counter example of the first scalar field as derived in GR textbooks. As it is only the equation, I am not sure how much it would add to the essay to present it also. My version doesn't appear in the standard ontology of Newtonian mechanics, but the concept very much does so in the reduction of General Relativity to the weak field limit of Newtonian gravity. Others state the first scalar field has become the most paradoxical question ever posed to science (dark energy). I guess if I could rewrite the essay I would point out that the scalar field addition was not my idea, but is a representation of the idea Einstein threw out. It is what occurs when a cosmological constant is added to the standard Einstein field equation in order to account for a linearly growing force that opposes attractive gravity.

There are several profound problems with the entire existence of a cosmological constant, for which my substitution appears to be a case never considered previously, but for a general essay like this those would probably not stir much interest.

Unfortunately there is a balancing act between showing the mundane concept of my questions (which stem from (1)) and the inherent complexity of the Einstein field equation. Too difficult and people's eyes glaze over, but I am still searching for the simplest but most effective way to get this across.

If we take (da)²/a)+c²k/a²=8pi/3Gro+the energy of the quantum hole/3 .

Now if we insert the hubble law and my equations, you can correlate with the expansion/contyraction correlated with the maximum volume of universal sphere. Considering that the density can imply a contraction at this critical point.So we can see where the points of equilibrium are.The SR and the GR are ok and if we consider a system closed, it becomes relevant considering the number of quantum entanglement having the serie of uniquness.

See that the parameters must be precise for a correct universal dynamic. The volumes , spherical are essential like the rotations spinal and orbital.The proportions appear with mass. If the mass polarises the light and so if the entropy, physical increases proportional with mass.We can see where is this maximum volume just before the contraction towards the perfect equilibrium between all physical spheres. The friedman lemaître equation can be optimized.

Good work :) with the integrations, substitutions, extrapolations, additions, multiplications, settings,.......

Regards

Hello Spherical Jedi (I like your nickname),

I haven't yet gotten into the Friedmann equations in depth, but I will be certain to re-examine your concepts once I do. Do you have a website where your material is listed?

Regards,

Jeff

"What could be causing this acceleration? Physicists call it dark energy, and it could make up more than 70 percent of the cosmos. But so much remains unknown about dark energy that some scientists are asking whether it exists at all.

What if, instead of a mysterious unseen energy, "there is something wrong with gravity?" asks Sean Carroll, a theoretical physicist at the California Institute of Technology.

Einstein's theory of general relativity represents gravity as the curvature of space and time. Perhaps this idea "is still right, but we're not solving the equations correctly," suggests Carroll. "

Who's Afraid of the Dark? Alternatives to Dark Energy

Jeff

In the paper: Fedosin S.G. Cosmic Red Shift, Microwave Background, and New Particles. Galilean Electrodynamics, Spring 2012, Vol. 23, Special Issues No. 1, P. 3 - 13, I describe the possible reason for dark energy, red shift and microwave background. The explanation is based at the Theory of Infinite Hierarchical Nesting of Matter.

On the other hand, the cosmological constant was explained in the paper: Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, February 2012, Vol. 35, No. 1, P. 35 - 70.

Other ideas about gravitation see my Essay.

I hope it will be useful for you.

Sergey Fedosin Essay