Essay Abstract

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. The Einstein tensor within the EFE has proven to be the only tensor that meets a number of properties (divergence-free second rank tensor, etc.) as derived from the Riemann curvature tensor and the metric g. We point out however, simply from summation of tensors and the fundamental theorem of calculus, that this uniqueness of properties does not extend to a differential equation with a non-zero constant of integration. It is important to fully understand why these properties are required to maintain our current cosmological model of attractive gravity. 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.

Author Bio

The author is currently a PhD candidate in Dayton OH USA, after receiving his BSEE in 2008. Recently, during dissertation research, he became interested in elasticity tensors, the history of field theory and the paradox of the accelerating expansion.

Download Essay PDF File

  • [deleted]

Jeff,

You commented about my essay about the unification of gravity with the electromagnetic force and mentioned that my essay dovetailed with your essay. Actually, the book behind my essay covers a broader range of physics topics including the cosmological implications of using the starting assumption that the universe is only spacetime. With this assumption the expansion of the proper volume of the universe is explained as resulting from the transformation of the properties of spacetime. This transformation started with the Big Bang and continues today. This alternative model of cosmology entirely changes the perspective on dark energy and the cosmological constant. It is not possible to explain the implications for dark energy in this short post, but details are available here.

    • [deleted]

    John,

    Found and downloaded your book. Look forward to getting into the details of your explanation as your descriptions seems to match some of the conclusions I have come to through another route.

    • [deleted]

    Dear Jeff Baugher,

    Let me refer you to an excellent essay (topic 1372), titled "Misinterpreting Reality: Confusing Mathematics for Physics" by Robert H McEachern in this contest. He says, "Equations contain very little information. This fact is what makes it possible to symbolically represent them, in a computer memory, by a very small number of bits. As a direct result of this fact, we can conclude, contrary to the fervent belief of most physicists, that equations cannot describe anything other than the most trivial physical phenomenon; those nearly devoid of all information. For complex phenomenon, it is the vast information content of the initial conditions (like the content of an observer's memory) rather than the tiny information content of equations, that really matters."

    In general, I agree with your arguments. However, in my opinion, main strength of the mathematicl model of GR, (including EFE) is that it can be adjusted to simulate the Newtonian gravitation with the additional provision that a finite speed of propagation of gravitational influence is built in the model.

    Regards

    G S Sandhu

      • [deleted]

      GS Sandhu,

      I have obtained a copy of your paper that you linked to. I see where we have differing opinions on whether the manifold can be considered a physical 4 dimensional medium. We are both ending up at the same conclusion, but you are further along than I am so I look forward to learning the details of where the differences lie.

      Regards,

      Jeff Baugher

      • [deleted]

      GS Sandhu,

      I am examining your paper published in Physics Essays more in depth. You state: "The GR is based on Riemannian 3D space in which the points of the space continuum are not considered invariant." Do you not mean a 4D space where the 3 spatial bases are not invariant? If not, can you explain more in depth?

      Regards,

      Jeff

      • [deleted]

      GS Sandhu,

      In reference to equation (24) of your paper, just to make sure, the term with mass in it should >>1 and not

      • [deleted]

      Forum testing of Mimetex.

      Does not seem to recognize addition symbol, there should be a plus sign between two 1s and oplus signs below[math]\oplus 1+1 \oplus [/math]

      Seems to recognize rest of characters ~!@#$%^&*()_+`1234567890-=

      ?,./:";'

        • [deleted]

        Wow, that is not how it appeared in the preview...frustrating forum.

        • [deleted]

        To help explain this essay better, and perhaps invite some more conversation, I present here what I consider to be a mathematical proof of our biggest physical misunderstanding, and a proof against General Relativity (and Einstein's actual biggest blunder).

        (note that the addition sign is not working on my preview so I have substitued the oplus symbol)

        The most general equation formed from the metric guv and the Riemann tensor is

        [math]\Lambda g_{\mu\nu}\oplus G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}[/math]

        In keeping with this essay contest, lets change the symbols for the cosmological constant and the Einstein tensor so that we do not confuse what these are thought to physically represent and thus leave only a mathematical meaning. Then we can have

        [math]\Omega g_{\mu\nu}\oplus L_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}[/math]

        Let the condition

        [math]R_{\mu\nu}=0[/math] apply. This then becomes:

        [math]\Omega g_{\mu\nu}\oplus L_{\mu\nu}=0[/math]

        so that

        [math]L_{\mu\nu}=-\Omega g_{\mu\nu}[/math]

        If we now also add the condition that

        [math]\Omega g_{\mu\nu}=0[/math]

        then this requires

        [math]L_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}[/math]

        This is a special case which results in the Einstein tensor since

        [math]\Omega g_{\mu\nu}=L_{\mu\nu}\oplus G_{\mu\nu}[/math]

        and so [math]G_{\mu\nu}=-L_{\mu\nu}[/math]

        which is substituted back in after pulling a -1 from the coefficients of Luv in the third equation back.

        Therefore, while the regular Einstein field equation with the Einstein tensor and no cosmological constant is correct, no multiple of the metric can be summed with the Einstein tensor. If a multiple of the metric is required, then Luv must be substituted for Guv first. This provides a proof against attractive gravity and numerous other physical explanations of the geometric theory of gravity.

        Thoughts?

        • [deleted]

        Dear Jeff Baugher

        Your essay was well written and I enjoyed reading it, I'll have to reread it again to understand some of your reasoning. This is why I like science we have so many different interpretations of what is real and what is an illusion. Some of the great scientist before us resolved some of these issues but by doing so they created more unresolved questions, that's where we come in.

        In my opinion the paradox of the rapid expansion of the universe may be a simply process of universe rejuvenation, matter is broken down to its infinitesimally smallest parts, bosons, through the weak force interactions, black holes, supernova's etc where there diamagnetic-like mutual repulsion of each other is feeding the expansion of the universe. Dark energy and dark matter is said to make up as much as 95 percent of the universe and growing, yet we know very little about what it is made out of. The way I see it matter is radiating away at a much faster pace than we currently predict the universe will suddenly deflate (deflation) back into a singularity. The deflation won't be caused by gravitational influences, it will be caused by a universe wide collapsing condensate, Higgs field collapse.

        As I said before Jeff I enjoyed reading your essay, good luck.

          • [deleted]

          Oops, I forgot to sign the above thread.

          Ron

          • [deleted]

          Ron,

          Thanks for taking the time to read my essay. The goal I am striving for is to be able to take someone who has an understanding of calculus, give them a mathematical argument, and then let them be able to compare that to published literature on General Relativity prior to the discovery of dark energy. Section 17.3 of Gravitation by Misner, Thorne and Wheeler is probably the leading text and a great section to compare my argument to. Someone who is capable of doing this should come to the same conclusion as I, that the argument corrects a well known (but currently accepted since it came to fit our physical understanding) error induced into GR that comes about from putting the cosmological constant in, but also means that the physical models within GR cannot possibly be correct.

          I agree with your sentiments on being able to consider arguments and analogies put forth in the quest to understand our universe. I look forward to understanding the arguments that you present in your essay also.

          Jeff

          • [deleted]

          Dear Jeff,

          I do not follow your ideas absolutely. What do you try to prove, or what you hope that your calculations will prove? I do not see meaning of your quantities Omega and L in equations (3) and (4). What they mean and what do you try to prove with them? Is vectorial r in eq. (5) and (6) unit vector? In opposite case r^3 should be in denominatror.

          Otherwise, it seems to me, that you have good ideas with divergence free tensor G.

          (BTW: Can you give, as expert, simple explanation for 8Pi factor in EFE?)

          Did you read essay of Petkov? I recommend, maybe it is connected with your ideas.

          What is your opinion of quantum gravity equation of Hadley

          = 8 pi

          http://www2.warwick.ac.uk/fac/sci/physics/staff/academic/mhadley/papers/thesis/thesis.pdf

            • [deleted]

            I could not write Hadley equation properly, so I repeat.

            Its equation is 4.7 on the page 61.

            It is < G > = 8 pi < T >

            He ignored 8 pi.

              • [deleted]

              Janko,

              Thank you for the questions. First for your questions: Yes, the r with the vector symbol is a unit vector. See page 186 of General Relativity:An Introduction for Physicists for the normal equation with the cosmological constant (but as the authors note does not fit with our understanding of current physics) also shows that there is no r3 (since the nature of an Omega term does not allow the volume to be modeled as a point).

              I have not found a definitive derivation of the 8pi, although I have looked.

              I had not run across the Hadley thesis and I find it an excellent read. I would have to go more in depth but his thoughts are similar to mine. What I would add in with his equation is that the expected value of the quantum vacuum is a "potential" energy that might be designated as a multiple of the metric but that there is no known method to combine this with the expected value of Guv.

              I will look at the Petkov more in depth also.

              Let me try another approach to explain this as compared to GR. In some ways, the whole problem also seems to stem from a notation problem.

              Let us take the tensors for Omega and L, looking at this in an abstract manner without consideration of the meaning and derivation of Tuv yet, and let us assign some scalar values as a teaching tool. Looking at equation (3), assign the number 10100 to Omega, and let's refer to this as the curvature potential of spacetime. Then to G we will assign the curvature number due to matter viewed as a particle present at a point in spacetime, and to L we assign the curvature magnitude due to matter viewed as a wave at a point in spacetime. (apologies if a addition sign is missing) You can see that these would be of an opposite nature (can use either/or but not both at same time). [math]\Omega g_{\mu\nu}=G_{\mu\nu} L_{\mu\nu}[/math] From (4) we can see as G increases from zero, which is flat spacetime, the particle density must be increasing since the curvature [math]G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}[/math] is increasing. Likewise for the matter as waves, for flat spacetime Luv is of an equal but opposite value of Omega guv. When the wave density increases, Luv decreases in magnitude away from 10100 so that the curvature increases. [math]\Omega g_{\mu\nu} -L_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}[/math] You can view matter as a particle or a wave, but not both at the same time. This is what I mean by a notation mistake. The most general equation from the Riemann and metric is only written in the literature as [math]\Lambda g_{\mu\nu} G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}[/math] but this is bad notational form, since strictly speaking the tensor "Guv" in this equation does not technically have to equate to the Einstein tensor. It only is equivalent if the multiple of the metric in the equation is zero. Riemannian geometry should certainly have no preference for either way.

              This explanation above might give someone who has studied deSitter space a fit. In deSitter space, there is no matter present so it is stated Guv is zero, but as explained in GR there is a multiple of the metric remaining which of course means that there is curvature, which is now linked to the cosmological constant problem and dark energy, which doesn't seem to make sense mathematically and hasn't yet been explained physically. In addition, MTW explains why simply adding the cosmological constant creates mathematical difficulties. In my view this is akin to epicycles, where new empirical evidence causes one to simply attempt to justify adding new features. While that may have worked for Neptune, I think it may be time to instead go back to the original equation and take another look.

              This leaves several reasons for why, if I have a choice, to prefer the Luv with a multiple of the metric as waves.

              1. The most general equation includes a constant of integration but there is no mathematical logic for preferring this to be zero, even with zero curvature. (Vacuum energy is a physical argument introduced by Zeldovich)

              2. The Luv version allows this multiple of the metric to exist, even with a very flat spacetime as measured by WMAP.

              3. If one estimeates what the vector field for the Newtonian spherical mass gradient would appear like through Luv, then it is of course an opposing gradient as the Newtonian one, but with the minus sign the gradients are mathematically equal. However, this should tend to make repulsive gravity seem like attraction at a distance.

              4. The multiple Omega with the metric cannot be ignored in the Newtonian spherical mass gradient, but it would appear that this results in a point where the unit vector of the gravity switches to an opposite sign at a predictable point.

              • [deleted]

              To make it clearer, when Luv is equal to Omega guv, there are no matter waves present and thus spacetime is flat.

              • [deleted]

              Hi Jeff. A few useful remarks on your essay.

              The way to unify gravity and electromagnetism requires balanced and equivalent attraction and repulsion and balanced and equivalent inertia and gravity.

              The understanding of outer space is significantly limited, as it precludes and destroys thought entirely. You cannot get around this. A lesson to

              FQXi.org as well. Telescopic observations make the unnatural/actually impossible experience of outer space even more incomprehensible. Telescopic observations do tell us something about how we see, however. We would never be able to survive long term/permanently in outer space, by the way.