Open Question on Quantization of Gravity

[Mike Strand]

In Metaphysics, topic "Is time continuous or discrete?", one of our members, Dimebag, with others, is speculating about the quantization of time. What if we define a quantum of time as the time it takes light to travel one unit of Planck distance -- we could call this a Planck unit of time -- and also define one unit of Planck distance as a quantum of distance? In this way we have a trial quantization of both time and space.

Is this a clue to the quantization of gravity? In a couple of posts in Dimebag's topic, I think I showed that using the Planck units of distance and time as quanta, we have also quantized velocity and acceleration. Now, gravity is equivalent to acceleration in the general theory of relativity, and what's more, gravity is a feature of space and time. So it would appear that using the Planck units of distance and time as quanta of distance and time, respectively, might lead to quantization of gravity.

What do you think? Has this approach already been tried? Idle speculation leading nowhere?

Another probably idle speculation: If in the outer reaches of galaxies the changes in acceleration of visible matter fall below the quantum threshold, could this make it appear that gravity (acceleration) is falling off as the inverse of distance, rather than the inverse of distance squared, hence making it appear as if there is more matter than there really is in the galaxy, which scientists call "dark matter"?

At the risk of appearing a fool, I emailed Lee Smolin about this speculation -- he's a noted theoretical physicist interested in quantum gravity and other issues in modern physics and wrote the book, "The Trouble with Physics". If he sends me a reply, I'll share it in this topic.

[Mike Strand]

Here is my email to Lee Smolin, which may prove to make me look like an idiot, but I have nothing to lose, so here goes:

Dear Dr. Smolin,

I've looked at the mathematical implications of quantizing both time and distance for defining both velocity and acceleration. This email corrects and expands upon my earlier emails (below). Here are highlights of my results:

1. "Instantaneous" velocities or accelerations are impossible in the classic sense (calculus), because time intervals can never be smaller than one quantum of time, so the limit as time approaches zero (derivative) in the real numbers cannot be calculated in the usual sense. Thus all velocities and accelerations can be viewed as averages over finite time intervals, the smallest of which is a quantum of time.

2. The set of all possible velocities is countably infinite, and any velocity can be expressed as a rational multiple of the speed of light. If the speed of light is the maximum possible velocity, then the rational multiplier is less than or equal to 1 in magnitude.

3. The set of all possible accelerations is countably infinite. Any acceleration can be expressed as a rational multiple of the speed of light divided by the value of one quantum of time.

4. If the speed of light is the maximum possible velocity, then the rational multiplier in the formula for acceleration cannot exceed 1 in magnitude. This implies that acceleration is bounded - it cannot exceed the speed of light divided by one quantum of time!

I would be happy to show you the details of the algebra, if you are interested.

It has been suggested that the quantum of distance be one unit of Planck distance, and the quantum of time be the time it takes light to travel one unit of Planck distance (the quantum distance). These two have numerical values, and their ratio, divided by the time quantum value, will be the upper bound on the magnitude of acceleration. However, other values for these quanta may be defined based on, for example, the limitations in our instruments for measuring distance and time, and my results would still hold.

I couldn't help but speculate what the upper bound on acceleration might have to do with dark matter. Here's my thought: The accelerations of stars near the center of the galaxy can be huge according to Newton's law, but if they are bounded above, as I believe I've shown, then they are less different from the accelerations of stars in the rim of the galaxy than they would be under strict Newtonian dynamics. In other words, the accelerations appear to fall off more slowly going from center to rim than under inverse squared distance. Hence the apparent extra matter that can't be seen.

This also puts an upper limit on the acceleration of matter into a supposed black hole. Maybe there really is no singularity!

I hope you might have time to look at this, and as I said above, I would be happy to send you details of my math analysis that led to these results. For all I know as an outsider, this may all have been looked at before by others, and may or may not be a dead end. I would surely like to find out, though!

Sincerely,
Mike Strand

That's it, folks! If I get a reply from Dr. Smolin, I'll share it with you. In the meantime, I would appreciate any review, corrections, and other comments from members of this forum!

[Notvacka]

In Metaphysics, topic "Is time continuous or discrete?", one of our members, Dimebag, with others, is speculating about the quantization of time. What if we define a quantum of time as the time it takes light to travel one unit of Planck distance -- we could call this a Planck unit of time -- and also define one unit of Planck distance as a quantum of distance? In this way we have a trial quantization of both time and space.

Is this a clue to the quantization of gravity? In a couple of posts in Dimebag's topic, I think I showed that using the Planck units of distance and time as quanta, we have also quantized velocity and acceleration. Now, gravity is equivalent to acceleration in the general theory of relativity, and what's more, gravity is a feature of space and time. So it would appear that using the Planck units of distance and time as quanta of distance and time, respectively, might lead to quantization of gravity.

What do you think? Has this approach already been tried? Idle speculation leading nowhere?

I have speculated in that direction too. And it seems perfectly reasonable to me. As for leading nowhere, I don't know if it actually changes anything perceivable on a larger scale. It would make the space-time continuum granular rather than continous, but Einstein's general theory of relativity would still apply. Also, since our consciousness is tied to the temporal dimension in a way that's not yet understood, it's perhaps impossible to verify by any kind of observation.

You might think of time quanta as the individual frames of a movie. Human perception can't distinquish between individual frames at the comparatively snail pace of 24 frames per second at the cinema. And a quantum of time would be incredibly short indeed. We can stop a movie and examine a single frame, but we can't "stop" time, because, to carry the analogy to conclusion, we exist within the movie.

[Dimebag]

Interesting concept, and I hope you get a reply from smolin. I can't really comment on the acceleration concept as I haven't really grasped it.

However another thought occured to me regarding the possible granularity of space. Suppose a single light source emits light in all directions, and evenly, and suppose that space is granular, in a similar way to the way hexagons fit together into a grid. Imagine if space is like this, but due to the size of the Planck length, there are many more directions that a photon can take. The fact that space is granular would imply that the path of an emitted photon must follow a certain radial from it's point of emission. Now imagine this light source emits in all directions through these radials. As there is a finite number of radials, that means there will be fixed distances between the photons that travel along these radials, which would lead to a large number of "blind spots" where no photons travel. At short distances these blind spots would be impossible to detect, but at cosmic distances they should be more noticable.

If such an effect could be observed it would imply such a quantum foam.

Now I am merely a layman so I have most likely missed something, or misunderstood something, so if I did, just let me down gently.

[Mike Strand]

Thanks for your comments, Dimebag! Thanks to you, I've been having fun with this lately, and even if it leads nowhere, I'm glad to have run across your thoughts in this forum.

As for your quantum foam, it's an interesting speculation, and if your suggested experiment can be carried out, it may blow quantization of space out of the water! I think the only way out is if the photons (or electromagnetic "wavefronts") can cross grain or grid boundaries somehow. But maybe that would contradict the assumed granularity (??) Hey, I'm a layman too!

By the way, I corrected some of my algebra and sent the results to Dr. Smolin. Acceleration is still bounded, but I hope my corrections refine the bound. Here's my note to Dr. Smolin:

Hi again, Dr. Smolin.

Correction to my algebra: The bound on acceleration is twice the speed of light divided by a quantum of time -- given that light has the greatest possible velocity and given the quantization of time and distance. I had overlooked the fact that |a|<1 and |b|<1 imply |a-b|<2. This upper bound on acceleration assumes that acceleration was measured as close to "instantaneously" as possible; that is, over a single quantum of time.

If acceleration were measured over, say, m quanta of time, where m is an integer >1, then the bound on acceleration is reduced by the factor 1/m. This makes sense, in that an average acceleration measured over a longer time interval together with an upper speed limit imply that the changes in velocity can't get too extreme -- velocity can only change between plus and minus the speed of light (taking into account direction), no matter how long you track it.

Best wishes,
Mike

The factor 1/m is intriguing. It puts the act or process of measurement at center stage. The longer the time interval over which acceleration is measured (difference between velocity at beginning of the time interval and velocity at the end, divided by the time interval length), the smaller the upper bound on the measured acceleration. As m gets larger, the observed average acceleration approaches zero, or constant velocity. Not sure what this purely mathematical outcome means, other than that it is consistent with taking average accelerations of a race car over various-sized portions of a race -- the longer the portion, the smaller the average acceleration. This is related to the facts that the race car has a maximum velocity, and so average velocities become less variable as the times over which they are measured during the race get larger. In the limit, there is only one value for the average velocity over the entire race.

In terms of how astronomers measure accelerations of stars in galaxies -- I have little or no idea. So I'm not sure how our quantization of time and distance and my analysis may relate to their measurement procedures.

[Mike Strand]

My previous upper bound on acceleration was based on movement in one dimension. I went ahead and applied vector algebra to try it out in three dimensions of space and came up again with acceleration having bounded magnitude. Here is my e-mail to Dr. Lee Smolin summarizing my results. It may turn out I'll be proved a bone-head, by him or one of you who participate in this forum, but that's OK -- nothing ventured, nothing gained. As a former math major, I'm used to being corrected.

Dear Dr. Smolin,

I looked at velocity and acceleration in three-space and time (background dependent approach), using vector algebra, with quantization of time and distance and using the velocity of light as an upper bound on velocity. Here is a summary of my findings and their implications, without the mathematical details, along with a suggested experiment to test these implications.

Under the assumptions that (1) the velocity of light in a vacuum is the upper bound on velocity, and (2) time and distance are both quantized, it can be shown using algebraic vector analysis that the magnitude of acceleration is also bounded above.

An upper bound, though not the least upper bound, can be calculated, provided that numerical values are available for one quantum of time and one of distance. The definitions of these quanta may come from theory, or from the practical limits of current measurement devices and techniques, whichever is deemed appropriate.

Suggested experiment: To test the upper bound on the magnitude of acceleration, one would take observations of stars moving near a black hole or center of a galaxy in order to calculate velocities and accelerations. If it should turn out that any of these experimental accelerations exceeds the upper bound, then one or more of assumptions (1) and (2) are violated. That is, either the velocity of light can exceed c (approximately 300 million m/s), or at least one of time and distance is continuous – that is, is not quantized.

If acceleration is bounded in magnitude, so is gravity, which has implications for the theory of black holes – they apparently could not, for one thing, be singular (e.g., infinitely massive).

Quantization of time and space has practical consequences for measuring velocity and acceleration. The classic calculus definitions of instantaneous velocity and acceleration do not apply – only averages over one or more quanta of time.

The change in distance for calculating velocity must be at least one quantum of distance, along with at least one quantum of time for the distance to be traversed. Further, the number of quanta of time required to measure an acceleration has to be at least the sum of the number of quanta over which the two velocities needed for calculating acceleration are measured.

Finally, the upper bound on the magnitude of acceleration is inversely proportional to the number of time quanta needed to measure the acceleration.


I'm curious as to whether this has already been worked out, and how. If you care to see my vector algebra, I'll be happy to send it your way!

Sincerely,
Michael Strand
Williamsburg, Virginia

Now it's time to go back out with my boomerang, for an enjoyable intuitive experience of good, old Newtonian dynamics.

[converge]

I might be wrong, but the way I understood it is that velocity and acceleration don't have to be quantized, as they're not actual "things" with physical representations the way that space and time are. They're just functions that describe things acting in space and time. A function can be smooth, there's no need for it to be quantized. The equations happen "underneath" the reality of space-time and just describe what will happen in space-time. For example, imagine the graph of distance vs time for the velocity 1 plack distance per planck time. On a piece of graph paper, you would label the vertical lines "1, 2, 3" for time and the horizontal lines "1, 2, 3" for distance. Now, whether you draw the velocity as a diagonal line or a series of "steps" doesn't really matter. As long as you have the right values at the intersections of the graph paper lines, it doesn't matter what the line does in between them. Interpolating and just making a simple diagonal line is the easiest way to show the function, and it lets you easily scale it up and down and zoom in and out without changing its shape. Because of that, most people would say that this is the most "correct" version of the function. But technically you could draw steps or just random squiggles in the white space between graph intersections, and it would still be correct at the quantum scale. The white space with the squiggles where you can draw whatever you want is the quantum foam, where position and time cease to have meaning.

As far as acceleration, remember that time and space (and the Planck distance and Planck time) are all relative. So you can only talk about acceleration of something relative to you. Remember also that the acceleration of something relative to you always drops to zero as it approaches the speed of light. So something can't have a constant acceleration of 0.5c per second squared for 4 seconds, because that would result in it going twice the speed of light after four seconds. From your perspective, it will always look like its acceleration drops off and gets slower as it approaches the speed of light, until eventually it appears to be zero (or infinitely close to zero) as it gets infinitely close to the speed of light.

So I'd agree that from your perspective, you might say that something could only appear to have a max acceleration of 1c per planck time per planck time, because the smallest thing you could see would be it jumping from 0 to lightspeed in 1 unit of time. But as before on the graph, you don't necessarily have to state it that way. You could draw it as going from 0 to lightspeed in half a unit of time and then its acceleration drops to 0. Either one would be a correct answer to the function; again you could squiggle however you want when plotting the acceleration as long as you line up the right answers on the intersections of the graph points. But we'd probably say the "correct" version of the function is whatever is the simplest thing that connects the dots. And if we want it to match up with what the people in the accelerating frame experience, then acceleration can approach infinity...

The people in the accelerating ship can keep increasing their velocity from their perspective no matter how fast they go, because the planck distance and time keep changing for them. If they jump from 0 to near light speed relative to you in one placnk unit of time, they would calculate their acceleration as 1c per time unit squared. But now distance is different for them, and light is still moving away from them at speed c, and they can increase their speed to near light speed relative to what they're currently moving at, so they'd calculate their acceleration as 2c per time unit squared. Then time and space have changed yet again, but light is still moving away at c, so they can increase their velocity yet again, etc. They can keep increasing their velocity forever in an attempt to catch up to light, but light keeps moving away from them at the same speed. Time and space continue to contract around them, but the actual planck distances and times stay the same. An inch for them is still measured as an inch on a ruler, but the space that fits inside that inch has compressed.

[converge]

Though after typing all that it occurs to me that I don't quite understand how the planck distance works as you approach light speed. In theory you could continue to accelerate until the whole galaxy is squished infinitely thin, so that a whole building would fit inside one planck length. But that would seem to violate the Pauli exclusion principle... it would make it so that all those particles are all in the same space. Hmm. I have to think about this.

[Mike Strand]

Converge, many thanks for your thoughtful and thorough reply!

I, too, was puzzled about how to deal with the effects of special relativity, and I still am, probably more than you are. I appreciated your graph paper depiction of distance vs. time, having studied calculus and acquired an appreciation for the approximations used to explain derivatives and integrals, to help us students see what happens when we then take limits as the graph paper grid size approaches zero.

My thought at first was to simply ask, "What if there is a limit to how accurately we can measure distance and time?", and what effect that would have on actual measurements of velocity and acceleration. An example would be a ruler with marks separated by 1/8 of an inch, so we can only read off lengths, say, to the nearest 1/16 inch. Then I realized this meant we could only get average velocities and accelerations. So whether the quanta of time and distance represent physical facts of space and time, or whether they just represent practical limitations in measuring space and time, the math is the same. The interpretation, however, differs! I'm afraid I've been tempted to think time and space are truly quantized, without a basis for thinking this, in order to make the upper bound on acceleration appear more significant and important to physics. I couldn't help but get excited at the possibility of an upper bound on actual acceleration, and its implications for gravity.

It may be intuitively obvious that if velocity has an upper bound and we can only calculate average velocities and accelerations, that calculated acceleration will also have an upper bound. But the actual acceleration -- who knows? This depends on whether the space and time quanta are fundamental facts, or whether they only reflect the accuracy of our measurements. If the former, acceleration appears truly bounded, but if the latter, we can only say our calculated average accelerations are bounded, and the unknown, instantaneous acceleration may still get very large.

I should probably hedge my statements and say "measured acceleration has a magnitude that is bounded above". My previous statement implies true acceleration is bounded, and I must admit, that would only apply if time and space are "truly" quantized.

In any case, I believe that the effect of "quantization" of space and time, in whatever sense, is that velocity and acceleration (at least as calculated) can only take on stepwise* values, and that by adding the further restriction of an upper bound on velocity, both velocity and acceleration become bounded.

Converge, I welcome any further insights you may have on this, whether they support my speculations or not.
---------------------
*The steps can vary in size, and the number of possible values for velocity and acceleration is countably infinite, not uncountably infinite as in calculus, which is defined over the real numbers. This "denumerable-discretizing" of velocity and acceleration isn't really an a priori quantization of them, but rather a consequence of the assumed "quantization" of distance and time.

[Mike Strand]

Another thought: As Julian Barbour shows in a recent essay (link below), time can be defined as a function of the change in position and the masses of each and every object in an isolated system of objects. This might imply that the quantization of time could be seen as a consequence of the quantization of distance or space -- my results would not have to assume both are quantized. This would also apply if the quantum of distance really only reflects inaccuracy in our measurement of distance -- it would also lead to inaccuracy in measuring time according to Barbour's formulas.

http://www.platonia.com/nature_of_time_essay.pdf

[Nassr]

Dear Colleagues,

Brazilian Physicist seems to have debunked the Hot Big Bang.

.....
Also, recently, with a persistent Heisenberg uncertainty, regarding the primordial position of a co-moving origin, a brazilian physicist obtained a solution of the Einstein field equations, within the cosmological context, giving an absolute zero temperature at the primordial universe: "On the Cold Big bang Cosmology".[9]

9. Assis, Armando V.D.B. On the Cold Big Bang Cosmology. Progress in Physics, 2011, v. 2, 58-63. [1]

http://www.ptep-online.com/index_files/ ... -25-14.PDF


Nassr ad perpetuam rei memoriam

[Cerveny]

...
I should probably hedge my statements and say "measured acceleration has a magnitude that is bounded above". My previous statement implies true acceleration is bounded, and I must admit, that would only apply if time and space are "truly" quantized.

In any case, I believe that the effect of "quantization" of space and time, in whatever sense, is that velocity and acceleration (at least as calculated) can only take on stepwise* values, and that by adding the further restriction of an upper bound on velocity, both velocity and acceleration become bounded.
...

Let us suppose unlimited fine structure of space. Let us suppose all (unlimited count) elements of space are liable to the causality. All elements act to each other. Causality certainly can work by limited speed only, thus (since certain complexity) a stable result of causality may not be possible to reach in limited time. Perhaps some oscillations are the solution. The pressure of the time is merciless. Then the causality is to release, to cast off semi-products, (Planck's) grains only. There is no time to learn more about structure of space. Time should have run more slowly...

[Mike Strand]

Nassr: The article by Assis is beyond my abilities to fully comprehend, but seems to be a new solution, using a new assumption, to Einstein's field equations. I'm not sure how this relates to the idea of time and space measured (or actually occurring) in discrete units.

Cerveny: You appear to be explaining a fundamental theory for the quantization of space and time. That would lead directly, I think, to bounded acceleration, given an upper limit on velocity. This in turn could have consequences for gravity, black holes, and other phenomena associated with acceleration -- namely, that singularities do not exist. Sorry if I misunderstand you.

[Mike Strand]

I've drawn three inferences from my work on "quantized" space and time:

(1) We won't be able directly to observe anomalies or singularities such as black holes. Due to the limitations in the precision of our measurements in astronomy of star motion (time and distance), we can only perceive average accelerations. Given light velocity as the upper limit of velocity, this implies that the magnitudes of those average accelerations are bounded above. Hence we could never directly observe the infinite accelerations (gravity) implied by the concept of a black hole.

(2) If time and distance are quantized as a law of physics, then black holes would be impossible, given such a law.

(3) The effect of "quantization" of space and time, in whatever sense, is that velocity and acceleration (at least as calculated) can only take on stepwise values. The steps can vary in size, and the number of possible values for velocity and acceleration is countably infinite, not uncountably infinite as in calculus, which is defined over the real numbers. This "denumerable-discretizing" of velocity and acceleration isn't really an a priori quantization of them, but rather a consequence of the assumed "quantization" of distance and time.

Furthermore, quantization of time may be a consequence of the quantization of distance. As Julian Barbour shows in an essay, http://www.platonia.com/nature_of_time_essay.pdf, time can be defined as a function of the masses and the changes in position of each and every object in an isolated system of objects. This would imply that the quantization of time is a consequence of the quantization of distance or space -- my results would not have to assume both are quantized. This would also apply if the quantum of distance really only reflects limited precision in our measurement of distance, since it would also lead to limited precision in the measurement of time, according to Barbour's formulas.

I've shared these "findings" with Dr. Lee Smolin, no reply yet, but I'll let you know if I get one.

In the meantime, I'd enjoy any comments from participants here.

[Cerveny]

I've drawn three inferences from my work on "quantized" space and time:

(1) We won't be able directly to observe anomalies or singularities such as black holes. Due to the limitations in the precision of our measurements in astronomy of star motion (time and distance), we can only perceive average accelerations. Given light velocity as the upper limit of velocity, this implies that the magnitudes of those average accelerations are bounded above. Hence we could never directly observe the infinite accelerations (gravity) implied by the concept of a black hole.

(2) If time and distance are quantized as a law of physics, then black holes would be impossible, given such a law.

(3) The effect of "quantization" of space and time, in whatever sense, is that velocity and acceleration (at least as calculated) can only take on stepwise values. The steps can vary in size, and the number of possible values for velocity and acceleration is countably infinite, not uncountably infinite as in calculus, which is defined over the real numbers. This "denumerable-discretizing" of velocity and acceleration isn't really an a priori quantization of them, but rather a consequence of the assumed "quantization" of distance and time.

Furthermore, quantization of time may be a consequence of the quantization of distance. As Julian Barbour shows in an essay, http://www.platonia.com/nature_of_time_essay.pdf, time can be defined as a function of the masses and the changes in position of each and every object in an isolated system of objects. This would imply that the quantization of time is a consequence of the quantization of distance or space -- my results would not have to assume both are quantized. This would also apply if the quantum of distance really only reflects limited precision in our measurement of distance, since it would also lead to limited precision in the measurement of time, according to Barbour's formulas.

I've shared these "findings" with Dr. Lee Smolin, no reply yet, but I'll let you know if I get one.

In the meantime, I'd enjoy any comments from participants here.

(1) There are not any singularities in the real physical world. The singularities are only thought, thought limits, the ideas... no real objects
(1) -> (2) There are not any singular black holes (perhaps some real defects in the physical space - why not just the real holes, cavities in the physical space structure…)
(3) The time is a real fourth dimension locally orthogonal to the space; it is the dimension the Universe locally grows along it. The time of "now" is instantaneous our 3-D space, the border, the last surface of the 4-D Universe. The “arrow” of the time is heading to the (uncaused yet) future... The Universe grows by a velocity somehow related to the "c". Thus m*c^2 is some kind of the kinetic energy... Nothing can move quicker....

In case we consider elementary particles as space defects it is possible to find out the information about the particles' (free) motion (information about its speed) in the difference (in the stress) of followed, of next time layers of the Universe...

http://www.soton.ac.uk/~engmats/xtal/de ... ation.html

Boys, forget TR, it is out of date