# Temperature, Velocity and Curved Space

## 24 posts in this topic

--- temperature ---

I was wondering if the speed of light implies a maximum temperature, and if so what that temperature is?

--- velocity ---

The maximum velocity 'c' is 3x10e8 m/s. The concept of velocity is defined as the change in distance divided by the change in time. For a particle to travel from point A to point B implies that the particle must occupy every single interveaning location along that line. Could the speed of light be viewed as the MINIMUM time that a particle can occupy a particular location?

If space and time were quantatized, there would be a time quantum (tq) and a space quantum (location). So the speed of light could be viewed as 1 tq/location. The next slowest speed would be 2 tq/location, and so on. Terrestial velocities would be large values like 123,456,789 tq/location. I wonder if incorporating this notion into current relativity theory would remove/clean up the lorentz ratio from the equations.

This idea requires that we only observe velocities such as c, c/2, c/3, c/4, c/5, etc... So it would never be possible to observe a particle going 90% the speed of light. It also requires that space consists of discrete locations filling space (an Ether-ish concept).

This is kind of a rationalistic desire on my part to give the speed of light a more interesting value (like '1 tq/loc') rather than 3x10e8 m/s. But on the other hand physics has always treated velocity as fundamental attribute of nature, perhaps we have been so used to defining velocity as delta distance/delta time, that it muddles our attempts to describe what it really going on in nature.

------- curved space ------

Finally, how can space be curved? For example, Earth is curved, but only when viewed by a flat euclidean mathematical model (which is not curved). So to claim space is curved and then offer a mathematical model that demonstrates this fact implies an undelying "meta-space" which is not curved. So perhaps this meta-space geometry is the real one, and not the curved space which it describes. (I smell a Godel/Turning self-referential issue here, and a little Kant too!)

PS. Betsy, Stephen: Thanks for this forum, its one of the best looking web forum's I have ever seen.

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1) By itself relativity alone places no upper bound on temperature; temperature is not a function of the particle velocity. There are other theories, such as string theory, in which a certain maximum possible temperature exists, but string theory is really just pure speculation, not physics per se.

2) Space and time are relational concepts, not things in themselves, and motion in reality is continuous, not discrete. So any notion of quantization can, at most, be just a mathematical device, not a description of reality. And the Lorentz transformation is the means by which special relativity arrives at an objective description of reality (in the context in which it is meant to be applied), so I see no reason at all to supplant it with a quantization scheme.

3) In relativity the curvature of spacetime is a mathematical abstraction, an attribute of a model that consists of an abstract mathematical manifold. As mentioned above, space and time are relational concepts, not entities, so the curvature of spacetime is not the curvature of a thing consisting of space and time.

Note, however, that Lewis Little's Theory of Elementary Waves (TEW) gives physical meaning to the mathematical abstraction of spacetime curvature. But this curvature is of a real physical existent -- the elementary wave -- and it is objects themselves that are actually curved as the particles they consist of follow along the curvature of the elementary wave.

4) Thanks for the kind words about THE FORUM. Coming from someone with your expertise, it is a compliment indeed.

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Physics is a difficult subject for the layman to dabble in, unlike darwinism/evolution in which the layman can do some armchair reasoning and have it actually bear fruit.

Physics requires a great deal of technical knowledge and mind boggling math. I miss the days when all I knew of physics was Newton. (Darn Reality for being so confusing!!!! <insert smiley here>). I appreciate being able to ask potentially naive questions about physics and get a respectful reply. Thanks Stephen

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Physics is a difficult subject for the layman to dabble in, unlike darwinism/evolution in which the layman can do some armchair reasoning and have it actually bear fruit.

Physics requires a great deal of technical knowledge and mind boggling math. I miss the days when all I knew of physics was Newton. (Darn Reality for being so confusing!!!! <insert smiley here>). I appreciate being able to ask potentially naive questions about physics and get a respectful reply. Thanks Stephen

You're welcome, Ken. Good questions are always appreciated.

p.s. About that "<insert smiley here>": we have a whole batch of them to select from, just to the left of the text box.

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Physics is a difficult subject for the layman to dabble in, unlike darwinism/evolution in which the layman can do some armchair reasoning and have it actually bear fruit.

Physics requires a great deal of technical knowledge and mind boggling math. I miss the days when all I knew of physics was Newton. (Darn Reality for being so confusing!!!! <insert smiley here>). I appreciate being able to ask potentially naive questions about physics and get a respectful reply. Thanks Stephen

No joke. But it's so interesting. I used to love science, then at a certain point got turned off to it. Once I started looking into physics I got interested all over again. I've been looking forward to this forum so I can ask potentially naive questions, as more and more arise the more I read, and I feel silly asking really, but I want to learn. In this and Objectivism I feel like a child again, which is good in the sense of wonderment and curiousity, but a little embarrassing

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kenstauffer: For a particle to travel from point A to point B implies that the particle must occupy every single interveaning location along that line. Could the speed of light be viewed as the MINIMUM time that a particle can occupy a particular location?
StephenSpeicher: Space and time are relational concepts, not things in themselves, and motion in reality is continuous, not discrete. So any notion of quantization can, at most, be just a mathematical device, not a description of reality.
In addition to realizing that space and time are relational concepts, not "things", the ideas of "occupying a particular location" and "motion in reality is continuous" have to be understood within the context of the principle that every measurement and every theory are valid only to some finite precision. In particular, a "location" at a point is meaningful only to some finite precision (a point is an abstraction with the "diameter" omitted; there is no infinitismal point in reality) and so is the specification of a change in position. (Likewise for measurements of time, which are based on physical changes.) "Continuous motion" is a valid abstraction, but the idea of moving "continuously" quantified to a greater precision than is applicable in any specific case is not scientifically meaningful.

But one cannot deduce from these principles of finite precision in measurement an actual "quantum of distance" (or of "time") corresponding to a kind of chunk of reality as some kinds of "things". Finite precision of measurement does not mean that something is somehow actually jumping in discrete transitions corresponding to the size of the truncated precision: Measurements, and the theories associated with them, are objective, not intrinsic. Whatever ultimate consistuents there may be, which in turn may place some limit on precision of measurements of distance or time, is beyond our current knowledge and is an entirely separate issue from a proper epistemological understanding of the nature of basic measurements and higher abstractions such as continuity.

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"Continuous motion" is a valid abstraction, but the idea of moving "continuously" quantified to a greater precision than is applicable in any specific case is not scientifically meaningful.

I enjoyed the care with which ewv's post was crafted, but I read this one sentence several times and I am still not sure of the intended point. Perhaps ewv can clarify.

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The abstract idea of the continuum entails an "infinite" subdivision, with the actual limit on "how small" left unspecified. In any particular case --- with measurements of finite precision -- it is not meaningful to specify and distinguish relative positions (differences) closer than the degree of precision to which any one of them can be specified. The same consideration applies for a theoretical understanding. Any theory can also be meaningful only to some finite degree of precision, both because it pertains to measurements and because it has some range of objective validity -- physical theories are objective, not intrinsic.

Even when we observe the continuous motion of a large object at the simple perceptual level, there is only a finite degree of precision to which we can claim it is "smooth" as we watch it go by (the lenses in our eyes have a finite resolving power, our nervous system works in finite impulses, etc.). The object does not move from an infinitesimal point to an adjacent infinitesimal point: These are mathematical abstractions -- there is no infinitesimal and we don't observe it. Neither is there any such thing as "perfect" smoothness -- whether in shape (like a circular arc) or motion -- just as there is no such thing as a "perfect", i.e., an infinitely small, point.

That doesn't mean that a large object moving across our field of view ceases to have a definite state beyond our realm of perception, only that we have to be careful not to extrapolate what it is and how it acts beyond the finite precision to which we can measure positions (including what we see through elementary direct observation).

I don't think there is anything profound or mysterious about this. It just emphasizes the finiteness of everything we observe and the form in which we observe it, measure it or theorize about it; distinguishes between the elementary motions that we observe as smooth versus the mathematical abstractions of points and continuity; and emphasizes that both our observations and our abstract understandings are objective, not intrinsic with some kind of infinite precision.

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With this distiction of a physical theory's being objective and not intrinsic, I can now understand why I should not jump on the String Theory bandwagon as quick as I had.

I suppose I have always thought of mathematics not just describing reality, but in a sense actually being the reality it describes. Almost in a Pythagorean sense now that I think about it. The way I was thinking in this regard was definitely intrinsic and not objective. I see that now.

That doesn't mean I am going to dismiss String Theory out of hand, yet; but I'm going to follow these discussions closely so that I may get a clearer picture of how I need to think and what objective criteria I need to use to analyze this theory (and the rest of physics) more closely.

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The abstract idea of the continuum entails an "infinite" subdivision, with the actual limit on "how small" left unspecified.

But the question I asked had to do with your comment on my expression that "motion in reality is continuous, not discrete." It is true that when we think of, say, real materials as being continuous, we are doing so with the understanding of the actual constituent nature of the material. It is only in that sense that actual -- not abstract -- objects can be thought of in terms of subdivisions. But when it comes to "motion in [metaphysical] reality [as] continuous" there, then, only an abstract, not an actual subdivision can be made. An object in motion is a continuous process of change in location, and it is only abstractly, mathematically, that we can even think of "subdivsions" in this regard.

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That doesn't mean I am going to dismiss String Theory out of hand, yet; but I'm going to follow these discussions closely so that I may get a clearer picture of how I need to think and what objective criteria I need to use to analyze this theory (and the rest of physics) more closely.

The most simple criteria to apply is to ask what are the facts of reality upon which the foundations of a theory are built. That is the easiest way to spot a rationalistic approach to anything, not just physics. Inevitably, rationalism in physics reflects, not the facts of metaphysical reality, but rather the treatment of the content of one's mind as if that were metaphysical reality. (Note that here I am using "rationalism" in its philosophical/epistemological sense, as deducing reality from concepts, rather than inducing from reality.)

As a p.s.: I know that "Rational_One" is studying physics and I can quite easily share his fascination with string theory. The mathematics of string theory is a beautiful thing to behold for those who can understand its various complexities. But not all things that men can dream up mathematically, reflect the way the world really is. Mathematics alone can create a universe that operates on principles different from the physical principles that govern the actual universe in which we live. This is fun, but it is not physics. (Note, however, that is not to say that some aspects of this approach may not eventually lead to something real. Even the most rationalistic can eventually be drawn back to reality as a matter of practical concern.)

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I should add that I have really enjoyed the very fine way that ewv has crafted his thoughts here. These are fascinating subjects and ewv has an interesting perpsective, and he expresses that perspective very well.

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An object in motion is a continuous process of change in location, and it is only abstractly, mathematically, that we can even think of "subdivsions" in this regard.

I agree.

First, however, since this is my first post to the new Forum, I have the happy obligation to offer many thanks to Betsy and Stephen for this wonderfully well thought out, well constructed new Forum. I'm rather awed by the sense they've made in arranging things the way they have, and lining up such powerhouse experts, and I know for a fact they're just the right people to carry it out. And I'm happy to see some friends I really respect (even a relative) as members.

To business: I find it helpful to distinguish between continuity (or continuum) as a physical concept and as a mathematical concept. I'm not at all saying that anyone in the Forum has mistakenly conflated them. I just want to draw the distinction explicitly if only for my own benefit. And seeing who some of the first members of the Forum are, if I misconceive or misstate anything I know I'll get some great feedback.

Continuity is first of all a physical idea. Perceptually there are no gaps or holes in physical reality, nor do instruments report them for our rational interpretation. The very notion is a contradiction. We do find perceptual objects, their grosser pieces, and ultimately their individual particles, as separated from each other; but that's no cause to deny the presence of entities everywhere. Non-existence means the lack of anything: it means non-identity and non-attributes -- including the non-attribute of not taking up space. The universe is everywhere a plenum, i.e. a fullness. Full of what, and how the various whats might act so as to produce the spatial extension in the universe that we perceive and measure, is for physics to answer. (And I always have to say at this point that Dr. Little has created a serious and beautiful answer with his TEW. In a sufficiently rational world of doing physics, this would count as a tremendous value, and indeed at this modern stage of physics it would count as a requirement in doing fundamental physical theory.)

Two notes. First, there can be no positing of infinite divisibility in existence, or of any physical analog of geometrical points. That notion comes only from the process of abstraction that produces the mathematical idea of continuity. To exist is to be finite. In every respect. Again, the universe consists of finite entities some of which form a plenum.

Second, there can be no positing of (fundamental) discontinuity in existence. A person can get carried away by a conception of quantum physics (or by some arbitrary metaphysical standpoint) that appears to suggest some minimum spatial size of objects and minimum temporal length of their actions. But this rips "space" and "time" from their context, implicitly picturing entities as embedded in a reified, pre-existing space and time that have an independent existence of their own (actually a Newtonian premise that Einstein rationally dispelled). Whatever we finally do discover and prove, fundamental entities (particles, waves, or otherwise) won't instantaneously cover finite distances with their motion, and neither will they stop to spend finite time unmoving before continuing their motion. Motion will be continuous.

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Stephen: ... the question I asked had to do with your comment on my expression that "motion in reality is continuous, not discrete."
"Continuous motion" is a valid abstraction, but the idea of moving "continuously" quantified to a greater precision than is applicable in any specific case is not scientifically meaningful.
This was not intended to deny that motion that we observe is continuous, only to qualify that we cannot infer from that, in the realm of scientific theory, what it is doing beyond the level of precision at which we observe it (or measure it at a more advanced level) -- where it may or may not be continuous in some respect.

There is a widespread tendency in physics to think of such things in an intrinsicist way, dropping the context of the precision and range of validity of a theory or observation. Every actual instance of the continuous is limited at some degree of precision. But whatever happens at some sub-microscopic level where motion may or may not be continuous in some particular way, it does not negate the knowledge we have at the level of precision at our perceptual level, which remains the base of all further abstractions. But if basic knowledge is regarded as intrinsic, improperly extrapolating beyond its limits of precision, it will appear to be inconsistent, leading to the common fallacies claiming "knowledge is never exact, only an approximation", etc. Our knowledge is "exact" -- in the context of some degree of precision.

Stephen: It is true that when we think of, say, real materials as being continuous, we are doing so with the understanding of the actual constituent nature of the material. It is only in that sense that actual -- not abstract -- objects can be thought of in terms of subdivisions. But when it comes to "motion in [metaphysical] reality [as] continuous" there, then, only an abstract, not an actual subdivision can be made. An object in motion is a continuous process of change in location, and it is only abstractly, mathematically, that we can even think of "subdivisions" in this regard.
To be clear, a number of distinctions must be kept straight:

- Actual vs. abstract subdivision of matter, space or time (where an actual subdivision may be considered as existing in reality or the result of a physical process like cutting, where this is possible or meaningful at all);

- Continuous spatial distribution vs. continuity across time for change or motion;

- Continuous motion in the sense of leaving no "gaps" in its spatial path vs. jumps in rate of progression;

- Motion regarded as a self-contained elementary process vs. analyzing its constituent components of change in space and change in time.

These are all relevant to the discussion, but I think Stephen is focusing mostly on a process of motion -- change in location of object -- itself as inherently continuous. I do not know what continuity as a valid metaphysical concept would mean apart from denying any kind of metaphysical void, but here we are talking about understanding the nature of things in more detail from a scientific perspective. My point is how to regard the conceptual status of continuity scientifically, and that one should be careful not to think of continuity or any other measurement as intrinsic.

Continuous motion is something directly observed at the perceptual level and also expressed at a more advanced level in terms of time rate of change using the concept of space as a relational concept (not a thing in itself). But that must be interpreted within the proper context of precision of measurement.

As an analogy, if you watch a television, the changes in the picture flow continuously; the motion does not jump (unless there is deliberate transition). Furthermore, there are no natural subdivisions of the flow; we can only (as Stephen described) conceive of subdividing it abstractly (or physically in an artificial editing process).

But knowing more about how the TV works, we know that a distinct image is flashed on the screen 30 times a second, something that we can measure but cannot know about one way or another only from our initial simple perception constrained by its limited precision. At that level of observation we see a continuous flow because we cannot resolve the image for time intervals small enough to detect the transitions: Discrete differences between successive images are too small to measure at a frequency we can keep up with. (Metaphysically we would still say that the 30Hz frame rate is "continuous" in the general sense that it leaves no voids in reality, but mathematically the transitions are discontinuous.)

That does not mean that we have been fooled and the TV image is not "really" continuous: continuity is at root a mathematical characteristic that only has meaning within some context of precision. In reality the image is in fact jumping at 30 Hz, but at our level of precision there are no discrete transitions; it is continuous at that level of precision. There is always some finite degree of precision for any kind of perceptual or conceptual knowledge.

Continuity, like anything else, can only be measured or observed within a finite precision. It should properly be regarded as an objective property -- the form in which we grasp an aspect of reality -- not something intrinsic. This is true whether we regard the continuous change as something observed directly or break it down into measurements of change in position and change of time, each of which also has to be measured within some finite degree of precision.

Stephen: I should add that I have really enjoyed the very fine way that ewv has crafted his thoughts here. These are fascinating subjects and ewv has an interesting perpsective, and he expresses that perspective very well.
This is very closely related to and dependent on Ayn Rand's insights under the heading "Exact Measurement and Continuity" in the Appendix on "Measurement, Unit, and Mathematics" in IOE.

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Thanks for the clarification, ewv. I was befuddled by that one sentence which did not seem to make sense, but your clarification makes your intended meaning clear. I think we are in agreement regarding "continuity," given what we currently know.

Note that this portion of the initial thread has evolved more into metaphysics than physics, so I would suggest that if anyone has any follow-up points or questions on this aspect of the discussion, that they start a new thread in the "Metaphysics & Epistemology" forum.

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1)  By itself relativity alone places no upper bound on temperature; temperature is not a function of the particle velocity

In a gas isn't the average particle energy equal to 3kT/2? Wouldn't this relate velocity to temperature?

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In a gas isn't the average particle energy equal to 3kT/2? Wouldn't this relate velocity to temperature?

Temperature is a measure of kinetic energy only in the rest frame of the gas.

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In a gas isn't the average particle energy equal to 3kT/2? Wouldn't this relate velocity to temperature?

I think that's a different context. For a gas in a confined space, the particles are bouncing around in all sorts of directions with a broad distribution of speeds.

That's different than looking at the maximum speed of a single particle traveling in one direction, and trying to infer from that whether there is some universal max temperature.

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I think that's a different context.  For a gas in a confined space, the particles are bouncing around in all sorts of directions with a broad distribution of speeds.

Right. In general, there are three energy modes; in additional to the translational mode, there is rotational and vibrational. The temperature is a property of the system, with a random distribution of motion in both magnitude and direction.

That's different than looking at the maximum speed of a single particle traveling in one direction, and trying to infer from that whether there is some universal max temperature.

Right again. A single particle does not have a temperature due to its speed. By itself, relativity sets no limit on temperature, other than energy. This is commonly misunderstood; acceleration can increase without bound, even though an object's speed will asymptotically approach c. But, in string theory, there is a maximum temperature: look up the superstring Hagedorn temperature. Hot stuff.

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This is commonly misunderstood; acceleration can increase without bound, even though an object's speed will asymptotically approach c.

Layman Physicsfan here.

Does this mean that a>c/s is possible?

On further thought, I guess so. The rate of change in velocity doesn't have to have anything to do with the instantaneous velocity.

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Layman Physicsfan here.

Does this mean that a>c/s is possible?

On further thought, I guess so. The rate of change in velocity doesn't have to have anything to do with the instantaneous velocity.

OK, I woke up this morning having remembered math. If acceleration can be greater than c/s, then a particle with velocity zero at time zero could accelerate at, say, 2c/s and at time 1s would have achieved a velocity of 2c. That can't be - acceleration must approach zero as velocity approaches c.

Or am I missing something?

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OK, I woke up this morning having remembered math. If acceleration can be greater than c/s, then a particle with velocity zero at time zero could accelerate at, say, 2c/s and at time 1s would have achieved a velocity of 2c. That can't be - acceleration must approach zero as velocity approaches c.

Or am I missing something?

You are thinking in terms of a Newtonian world. In such a world many strange things can occur; gravity can propagate instantaneously and velocity can increase without bound. But experimental fact shows this not to be the case, and relativity better describes the world in which we actually live.

Newtonian acceleration is expressed as a = dv/dt, the rate of change of velocity. But in special relativity,

a = dv/dt (1 - v^2/c^2)^(-3/2) [equ. 1]

where c is the speed of light.

If we integrate [equ. 1] for some constant acceleration in straight-line motion (assuming simple initial conditions), we arrive at,

x^2 - (ct)^2 = (c^4)/(a^2) [equ. 2]

(As an aside, note that [equ. 2] is the equation for an hyperbola, and the relativistic behavior is accordingly known as hyperbolic motion. By contrast, with the standard Newtonian formulation we arrive at parabolic motion. If you plot both functions together, for appreciable velocities the character of motion between Newtonian and relativistic motion is radically different.)

Now, note what happens in [equ. 2] when we allow the acceleration a to increase without bound. As a approaches infinity, the quantity on the right hand side of [equ. 2] approaches zero. This implies that,

x^2 = (ct)^2,

or,

x = +/- (ct) [equ. 3]

This demonstrates that as acceleration increases without bound the distance traveled approaches the speed of light times the time traveled. Hence, as I indicated earlier, "acceleration can increase without bound, even though an object's speed will asymptotically approach c."

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You are thinking in terms of a Newtonian world.

<snip>

This demonstrates that as acceleration increases without bound the distance traveled approaches the speed of light times the time traveled. Hence, as I indicated earlier, "acceleration can increase without bound, even though an object's speed will asymptotically approach c."

OK, I see now. Thanks!