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Gravity waves and the aether


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Waves , signals , Bouncing back is the same way hat signals down a transmission line . Can bounce back or be reflected from an open circuit or short circuit .

 

Mike

That requires that they reach a boundary.

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That requires that they reach a boundary.

.

That is precisely what I am suggesting was happening in those early years of the universe . Dark matter was present in the medium , together with the gravitational waves , created at the Big Bang.

 

These waves reached the boundary , at that time not so far away . They hit the boundary and were reflected in the way I describe with a standing wave in a transmission line. This had the effect of causing voids and concentrated threads.. In the dark matter , existing in the medium . From these threads the newly forming stars and galaxies formed . This the current state we see much expanded, in the galaxy structure today.

.post-33514-0-87549600-1469035948_thumb.jpeg.post-33514-0-50390200-1469036271_thumb.jpeg

 

Mike

Edited by Mike Smith Cosmos
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That is precisely what I am suggesting was happening in those early years of the universe . Dark matter was present in the medium , together with the gravitational waves , created at the Big Bang.

 

These waves reached the boundary , at that time not so far away . They hit the boundary and were reflected in the way I describe with a standing wave in a transmission line. This had the effect of causing voids and concentrated threads.. In the dark matter , existing in the medium . From these threads the newly forming stars and galaxies formed . This the current state we see much expanded, in the galaxy structure today.

.attachicon.gifimage.jpeg.attachicon.gifimage.jpeg

 

Mike

How do you reach a boundary if you are limited to traveling at c, and the boundary is receding at c (before taking expansion into account)?

 

The universe doesn't have walls, like a box does. In the 2 dimensions of a surface, where is the edge of a ballon that's expanding?

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In the 2 dimensions of a surface, where is the edge of a ballon that's expanding?

I am never sure if that balloon is intended just as an analogy..

 

If it can be taken literally is the universe to be found in the skin of the balloon?

 

Does that skin have thickness?

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How do you reach a boundary if you are limited to traveling at c, and the boundary is receding at c (before taking expansion into account)?The universe doesn't have walls, like a box does. In the 2 dimensions of a surface, where is the edge of a ballon that's expanding?

I am not so sure all your figures are necessarily right, about those very early days of the expansion of the universe. There was an awfull, lot of stuff about , in a very small space at that early time . After all it was this expansion of dark matter , ordinary matter ,light , dust gas , medium and a whole lot of stuff in a small space expanding to make space and the classical universe . The walls would not be a surface of 2 dimensions but a sphere of three dimensions .

 

As I have previously shown pressure waves do travel in three dimensions in a medium even like water .let alone space filled to bursting with ' things

 

'post-33514-0-74788300-1469041147_thumb.jpeg. post-33514-0-67120600-1469041207_thumb.jpeg

 

post-33514-0-00673200-1469041239_thumb.jpeg

 

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As regards . Consiousness and the medium .

Quote "

Mike Smith Cosmos, on 19 Jul 2016 - 9:08 PM, said:

. Carlo Ruvelli , calls on our human Consiousness , as a final source of understanding of the Universe . As human observers , he says we stand at a very important position , in understanding the Universe by our observation of it . He recommends both Spinoza the Dutch philosopher and Lucretius as esteemed philosophers , who help us understand the importance of human Consiousness in beeing the important observer of the Universe. As such we command an important postition in the understanding of the universe .

 

He recommends both Sinoza and Lucresius as philosophers of repute in this endeavour .

Unquote "

 

. Ruvelli takes up the story with our early ancestors sittin around a camp fire . Thinking how the signs worked with broken twigs etc.

He explains how observation of the universe and thinking about it philosophising, guessing what signs they can find and what they mean , has led us to a greater scale of Consiousness. His thought on this is .....the universe is only of any point if it is observed. And we above all others that we know if our building up a Consiousness , which has a close correlation to the number of stars in the known universe. So his reasoning is ( if I am representing his thinking process correctly ) that although the universe is a vaste place, with trillions upon trillions of stars , so also is our brain of comparable ( memory for star proportion ) although not exact by any means , we do have a brain that can comprehend the universe and

Metaphorically sit by the camp fire and contemplate our understanding of the universe and how it works and exists.

 

What has this to do with " aether " or as I would have it Medium ( aether has a bad press ) .

We can , we must contemplate " what is what " and what is it all about . We have the brain , go for it ! Like our ancestors by the fire, like the philosiphers of today and the thinkers of tomorrow .

 

We have the brain, we have the Cosmos .....

 

Mike

Edited by Mike Smith Cosmos
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I am not so sure all your figures are necessarily right, about those very early days of the expansion of the universe. There was an awfull, lot of stuff about , in a very small space at that early time . After all it was this expansion of dark matter , ordinary matter ,light , dust gas , medium and a whole lot of stuff in a small space expanding to make space and the classical universe .

How did anything "catch up" to the expansion?

 

 

The walls would not be a surface of 2 dimensions but a sphere of three dimensions .

 

It's a 2D example.

I am never sure if that balloon is intended just as an analogy..

 

If it can be taken literally is the universe to be found in the skin of the balloon?

 

Does that skin have thickness?

 

It's a 2D example. The thickness is irrelevant; you only care about the surface. There is no boundary, and there is no part of the 2D space that it expands into.

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How did anything "catch up" to the expansion?

 

expands into.

Who said anything about a " balloon " ?

 

And who said anything about a 2 dimensional surface?

 

There is nothing beyond the surface of the expansion . That edge is " open circuit " . Any wave reaching an open circuit , will be reflected , like as in a transmission line . The boundary will be the extent of what is inside, which is the sum total of all the expanding contents , as listed previously . Any waves travelling in that medium , will reach the boundary ,as I presume the boundary is moving out at a less than light speed . You can't have all the contents , moving at light speed surely . Any photons bussing about at light speed will surely act like they do in the sun . Colliding and bouncing about ? Surely .

 

Mike

What is this boundary composed of, that it can reflect these 'waves' ?

( and why isn't it part of the universe ? )

 

And what's on the other side of the boundary Mike ?

?

 

What's on the other side of the boundary .? Nothing , absolutely nothing , you could not even put your hand through. There is no stuff , no medium for anything , no space , as we know it ,Nothing

 

The boundary is just the front of everything expanding , and occupying more of the nothingness and becoming something ! But this edge is indeed the edge of everything in the universe?

 

But these waves would have been happening somewhere on the way up. Working there way through the medium of everything reflecting from the boundary and possibly setting up some form of standing wave , much as in a wire or in simple terms rope . The dark matter and energy , might well have been part of this cockophony , being as how the dark matter ended up shaping the galaxies and universe . Perhaps !

 

To produce :-

post-33514-0-77539900-1469081873_thumb.jpeg

 

Mike

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The mistake is assuming that paradigm shifts occur from without science whereas they are always driven from within science.

Blue sky thinking and out of the box imagineering are marketing speak - but they create nothing but copy. Scientists must be able to say but how do we test that... and therein lies the reason that why questions are not investigated - because as soon as the empirical method is used the question becomes "what model can I create that explains not only that which we know, does not contradict the other models we are sure of, but also explains a tiny bit of what we are asking 'why?' about?"

 

Ok, I am not sure if you are addressing me or someone else or just making a general statement. I, for one, am quite familiar with Ayer, Kuhn, and Popper and therefore would not suggest that untestable hypotheses should be excessively entertained in the first place.

 

'Why?' questions are infinitely regressive - 'How can I show?' questions admit to a real-world answer.

 

I would suggest that we we are entering into the area of semantics a bit here. In practice I would suggest that one can take a 'why statement' and turn into a "how-do-I-show statement, though one is still essentially asking the same question. Hence, I don't necessarily agree that why questions always involve an infinite regression, though if one listens to many a 5-year-old, it does seem like that. But when adults ask 'why', they are usually looking for the cause of something.

 

So, "Why is the sky blue" can easily be transposed into "What causes the sky to be blue," and once we come up with a hypothesis about the cause of the sky's blueness, we then can ask "How can I show" that the cause that I have identified is indeed the cause of the sky's blueness.

 

The answer to why the sky is blue and the answer to what causes the sky to be blue is the same for each: "Blue is scattered more than other colors because it travels as shorter, smaller waves. "

 

Of course, one can then ask why blue is scattered more than other colors, or one can then ask what causes blue to be scattered more than other colors, etc., ad infinitum. In either case, we can say at some point that we don't know what the cause is, or that's just the way things are and we don't know why, or there are several causes, or we don't know if there is a cause, pr we don't know 'how to show' what causes something, or whatever. But there is no reason to suggest that "why" questions are any worse than "what causes" questions in the sense of being prone to infinite regression.

 

In most cases, we can replicate what happens in nature in the laboratory, thereby answering the "How can I show" question and sometimes not. Work done at Cern for example, as well as experiments done to confirm predictions made by relativity help establish the validity of scientific claims by answering the "How can I show" question.

 

But idle speculation is not entirely worthless even though we can't think of ways to answer the "How can I show" question. I am sure that Einstein did not know how scientists in the future, for example, would be able to "show" that some of his theories were correct, or made correct predictions. Perhaps he did not know we would detect the gravitational waves he predicted from the collision of black holes.

 

Some scientists have more of a speculative/philosophical streak in them than others, who are more pragmatic, e.g., some physicists such as Schrodinger labor over the ultimate meaning of quantum effects, while others, suggest that we just "shut up and calculate":

 

The 'shut up and calculate' approach is "the most popular of interpretations. It recognizes that the important content of QM is the mathematical models and the ability to apply those models to real experiments.As long as we understand the models and their application we do not need an interpretation. Advocates of this position like to argue that the existing framework allows us to solve all real problems and that is all that is important. Franson's analysis of Aspect's experiment shows this is not entirely true. Because there is no objective criterion in QM for determining when a measurement is complete (and hence irreversible) there is no objective criterion for measuring the delays in a test of Bell's inequality. If the demise of Schrodinger's cat may not be determined until someone looks in the box, how are we to know when a measurement in tests of Bells inequality is irreversible and thus measure the critical timing in these experiments?" http://www.mtnmath.com/faq/meas-qm-6.html


The other trouble - especially manifest in fora like this - is that the 'How can I show?' requires a background of many years previous study, serious hard graft, and an engagement with current knowledge and research colleagues. "Why?" questions most often merely demonstrate a reluctance to "make the hard yards" and learn what we currently know.

 

Lay persons in the public are in the habit of asking "why" type questions, and scientists might assist them by helping them rephrase their questions and direct their research in a manner that is likely to be most productive, rather than to just tell them to go away until they have studied the subject matter in depth.

 

If someone, for example, is looking for some physical nature in space, or some underlying medium of spacetime, it does not hurt to entertain such an idea, even though it does not seem to be compatible with relativity as we now know it. Indeed, the idea of an "aether," as I mentioned above is seen by some as being such a medium.

 

So if some "why" questions seem pointless, one might respond by

  • rephrasing it in a more scientific manner
  • suggesting avenues for research
  • showing that it is or is not compatible with present theories/paradigms
  • explaining that there is no way to verify or falsify it
  • suggesting similar alternative questions that can be tested
  • examining the motives for asking the question
  • acknowledging that science has no answers and never will, or, alternatively might someday given certain developments
  • identifying any false assumptions nested within the question
  • etc.

 

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In either case, we can say at some point that we don't know what the cause is, or that's just the way things are and we don't know why, or there are several causes, or we don't know if there is a cause, pr we don't know 'how to show' what causes something, or whatever.

 

 

Please tell Mike that.

 

And point out that making up your own answers (which are not only NOT supported by evidence but are, in fact, contradicted by evidence) is not a valid response.

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Please tell Mike that.

 

And point out that making up your own answers (which are not only NOT supported by evidence but are, in fact, contradicted by evidence) is not a valid response.

 

I am not in the habit of making personal observations, but I would note that astronomy/astrology, astrophysics, quantum physics, and relativity deal with such grand narratives that they often strike a mystic chord with those trying to peer into the depths of chemicals, cosmos, and creation, in search of ultimate and arcane meanings from the lips of any wandering modern day Delphic oracle. Apparently such alchemania can afflict even the most intelligent, Newton being a case in point, not to mention the tons of quasi-scientific books on the spiritual significance of quantum theory and the like that one finds on the shelves of any New Age bookstore.

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Who said anything about a " balloon " ?

 

And who said anything about a 2 dimensional surface?

I did:

 

In the 2 dimensions of a surface, where is the edge of a ballon that's expanding?

 

If you aren't going to read my posts, why bother responding to them.

 

There is nothing beyond the surface of the expansion . That edge is " open circuit " . Any wave reaching an open circuit , will be reflected , like as in a transmission line .

 

That works for a static universe of finite size. How does it work for an expanding one? (it also assumes a change in index. What is the index of refraction of your aether? How do we measure it?)

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It's a 2D example. The thickness is irrelevant; you only care about the surface. There is no boundary, and there is no part of the 2D space that it expands into.

Is there any way that that example allows us to visualize the (expansion) of the universe? Or is it meant to be understood in a mathematical way only?

 

I have heard about the expanding cake with raisins in it . Is that the closest analogy we can get? And is that analogy also imperfect?

 

Are we condemned to partial analogies by virtue of being ourselves a part of the thing we (well I am anyway) are trying to visualize?

 

Or is it the visualization process itself that is condemned to only partially represent the universe .?

 

To me the phrase" seeing is believing" has always been a touchstone of my understandings. Have I been mislead by this "false friend"?

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Is there any way that that example allows us to visualize the (expansion) of the universe? Or is it meant to be understood in a mathematical way only?

 

I have heard about the expanding cake with raisins in it . Is that the closest analogy we can get? And is that analogy also imperfect?

 

Are we condemned to partial analogies by virtue of being ourselves a part of the thing we (well I am anyway) are trying to visualize?

 

Or is it the visualization process itself that is condemned to only partially represent the universe .?

 

To me the phrase" seeing is believing" has always been a touchstone of my understandings. Have I been mislead by this "false friend"?

 

 

 

The problem is that people can't easily visualize the 3D expansion. The raisin cake works to show how everything can move away from everything else, but it's expanding into existing space and has edges, so it's imperfect. The balloon example shows expansion without edges or expanding into existing space, but it's 2D.

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The problem is that people can't easily visualize the 3D expansion. The raisin cake works to show how everything can move away from everything else, but it's expanding into existing space and has edges, so it's imperfect. The balloon example shows expansion without edges or expanding into existing space, but it's 2D.

Is it an argument to say that it is "logically" (intuitively?) impossible to model "everything"?

 

That a model ,by definition almost models parts of things ?

 

So ,if we cannot model ( or visualize) the entirety of the universe, then this is not a failure but a realization (that realization itself being a holding position too) ?

 

Is knowledge of this subject always going to be (in part ) like using a flashlight in the dark and we just need to get used to it ?

 

Is the itch to understand how the universe in its entirety a valuable motivation but also a hangup we have to get past?

Edited by geordief
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So ,if we cannot model ( or visualize) the entirety of the universe, then this is not a failure but a realization (that realization itself being a holding position too) ?

 

A question here might be, for example, do we need a unified theory of everything....that is, would such a theory be of much use or something that would just satisfy our Laplacean or Faustian desire to be, for lack of a better word, omniscient? ...Apart from making a personalized license plate out of some definitive equation.

 

Is knowledge of this subject always going to be (in part ) like using a flashlight in the dark and we just need to get used to it ?

 

Am reminded here of the Garden of Eden narrative for some reason! Science is doing pretty dang well and makes progress at a pretty good rate, though there are areas apparently in which science has reached certain impasses...but who knows?

 

Is the itch to understand how the universe in its entirety a valuable motivation but also a hangup we have to get past?

 

Supposedly this perennial itch is a male thing....books have been written (by women) accusing physicists of flooded the field of physics with the male urge to understand everything as an offshoot of their desire to dominate, explore, and conquer everything. (Am reluctant to refer to Amazon, and can't recall authors off the top of my head, but can find them if need be).

 

As for the Mike's water experiments and the aether or physical medium with regards to gravitational waves, a concise article entitled "Why Does Gravity Wave and Why Does it Matter" begins with a discussion of ripples on water and concludes by noting that "in the late 1800s it was found that the speed of light was always the same regardless of ones motion through the aether. This is a deeply un-wavelike behavior. If light really moved through the aether at a particular speed, the Earth’s motion through the aether should make the speed of light appear faster or slower at different times of the year. An unchanging speed of light meant our assumption about the aether must be wrong."

Edited by disarray
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geordief, on 21 Jul 2016 - 11:35 AM, said:


So ,if we cannot model ( or visualize) the entirety of the universe, then this is not a failure but a realization (that realization itself being a holding position too) ?


disarray: A question here might be, for example, do we need a unified theory of everything....that is, would such a theory be of much use or something that would just satisfy our Laplacean or Faustian desire to be, for lack of a better word, omniscient? ...Apart from making a personalized license plate out of some definitive equation.



Just on that point. Such an understanding (unachievable in my eyes but...) would replace flawed (well they would then seem flawed although they may have been correct for their time and place) theories.


On that basis it is laudable to look for more and more complete theories (apart from the possible material benefits such a theory might bring.


But I believe that it may well be that a theory that encompasses "everything" is a tautology and the supposed TOE was ,I am guessing so named tongue in cheek (TIC?)
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geordief, on 21 Jul 2016 - 11:35 AM, said:
So ,if we cannot model ( or visualize) the entirety of the universe, then this is not a failure but a realization (that realization itself being a holding position too) ?
disarray: A question here might be, for example, do we need a unified theory of everything....that is, would such a theory be of much use or something that would just satisfy our Laplacean or Faustian desire to be, for lack of a better word, omniscient? ...Apart from making a personalized license plate out of some definitive equation.
Just on that point. Such an understanding (unachievable in my eyes but...) would replace flawed (well they would then seem flawed although they may have been correct for their time and place) theories.
On that basis it is laudable to look for more and more complete theories (apart from the possible material benefits such a theory might bring.
But I believe that it may well be that a theory that encompasses "everything" is a tautology and the supposed TOE was ,I am guessing so named tongue in cheek (TIC?)

 

 

i agree, that any hope that science will ever know everything about absolutely everything or about the entire universe is a rather fanciful idea to say the least, but the term ToE is, I think, seriously used, though,perhaps, in a somewhat different manner.

 

I don't usually refer to Wiki, but its article on ToE illustrates that this term is often taken seriously. The acronym typically refers to efforts of mathematics, e.g., with reference to Godel, or to efforts to unify quantum theory and relativity; "In parallel to the intense search for a ToE, various scholars have seriously debated the possibility of its discovery" (followed by list of scientists and their efforts):

 

'Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable: "Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind.'

 

 

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To paraphrase George E.P. Box, all models are imperfect. Some are useful.

Great name. If I hadn't checked I would have said it was made up. :)

 

The description seems to fit.

 

Can there be such a (useful) thing as an inverse model where you start with a reality and create sub realities? Is that perhaps what art is ?

 

i agree, that any hope that science will ever know everything about absolutely everything or about the entire universe is a rather fanciful idea to say the least, but the term ToE is, I think, seriously used, though,perhaps, in a somewhat different manner.

 

I don't usually refer to Wiki, but its article on ToE illustrates that this term is often taken seriously. The acronym typically refers to efforts of mathematics, e.g., with reference to Godel, or to efforts to unify quantum theory and relativity; "In parallel to the intense search for a ToE, various scholars have seriously debated the possibility of its discovery" (followed by list of scientists and their efforts):

 

'Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable: "Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind.'

If Quantum mechanics and Relativity could be unified ,I would be happy to see it called a "Theory of Everything" (even if it was literally hyperbole) since ,at this point of time it would probably feel worthy of the name. :)

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Geordief asked: Can there be such a (useful) thing as an inverse model where you start with a reality and create sub realities? Is that perhaps what art is ? Geo

 

Perhaps the term you might be seeking is "sub-representations." By "real" we perhaps mean that there is something that corresponds to our (representational) mental image, or something that corresponds to Gogh's (representational) pool table, or some physical medium that corresponds to Mike's (representational) water photos.

 

In that sense one might say that the models (mental images, equations, photos, paintings, etc.) are sub realities of what we encounter apart from our own images, art pieces, and artifacts. We can link up any model with anything else really, e.g., Debussey Claire de Lune with the moon. (The representational theory of art is not universally taken as entirely valid). But in a larger sense, all of these things are "real" in that they exist and take part in reality. To mimic Gertrude Stein, a reality is a reality is a reality. The problem is that it is non-reality or 'nothingness' that is hard to imagine (or perhaps is just meaningless).

 

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Can there be such a (useful) thing as an inverse model where you start with a reality and create sub realities? Is that perhaps what art is ?

 

 

Not sure what you mean here. A model is a model; it doesn't matter what path was used to make it. Models are often created based on observation.

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Not sure what you mean here. A model is a model; it doesn't matter what path was used to make it. Models are often created based on observation.

No, I didn't have a clear idea myself of what I was getting at.

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I have just returned from A Visit to a Water Engineer , residing near Lake Trasimeno nr Perugia .

 

He has a perfectly round pool of 5 ft deep . I provided waves emanating from 4 sides of this round pool .

 

Note all show the sort Of waves that I am suggesting illustrate by a model form , movement in a medium of waves reflected from the border of a container of medium . Toward each other producing by interference , the sort of shapes found in the 3D computer representation of the Universe .

 

Judge for yourself , whether these have a remarkable , similarity ?

 

 

 

post-33514-0-07923500-1469134468_thumb.jpegpost-33514-0-76432100-1469134495_thumb.jpeg

post-33514-0-12311400-1469134524_thumb.jpegpost-33514-0-75793300-1469134742_thumb.jpeg.post-33514-0-45144500-1469135858_thumb.jpeg

 

Mike

.

 

 

That works for a static universe of finite size. How does it work for an expanding one?

 

(it also assumes a change in index. What is the index of refraction of your aether?

 

How do we measure it?)

..

 

Well it all depends , what is driving the expansion ?

 

If the expansion is caused by the sheer , mass of products in their entirety , of the expanding universe, then surely everything will just grow bigger in its own time like 'fire prevention foam ' . ( or rather the equivalent , in the ingredients of the expanding universe ) . Having established itself , over some finite period of time , conditions may arise , as illustrated , in previous pictures . When the nature of the universe is such , conditions of oscillations or waves may start to arise. These could grow to become reflected , standing waves in the whole fabric of space time , in the way a Tibetan gong starts ringing harmonically when struck . All and any of these analogies or models , can work in an environment of 'MEDIUM ' , but not without it ? Resulting in the illustrated concentration patterns , shown in the models.

 

I do not see that a refractive index is needed as the waves are not going through a change in medium. Possibly some form of reflective constant may be required . Although.i am not sure about that ?

 

Mike

Edited by Mike Smith Cosmos
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Mike:

Yes, I can see the similarities. But I too was under the impression that the universe somehow curved back on itself so that there are no edges.

 

Perhaps one is putting the cart before the horse. It might help to know a little more about the basis of that which is causing said waves before speculating about the medium, e.g., the elusive gravitons.

 

I understand that it seems like every other type of wave seems to need a medium, so it seems logical that gravity would follow suit. But I wonder whether that is true:

 

light/electromagnetic waves can travel through a vacuum, and do not require a medium. In empty space, the wave does not dissipate (grow smaller) no matter how far it travels, because the wave is not interacting with anything else.

 

I gather that gravitational waves also not only travel at the speed of light, but can travel through a vacuum. I presume that gravitational waves (re gravitons?) per se also have no mass.

 

One explanation for the fact that light does not need a physical medium, is that light, and electric/magnetic waves in general travel through their own all-pervasive fields, which keep oscillating and propagating themselves through space. The electric and magnetic waves don't require a medium (other than spacetime and electromagnetic fields) because they aren't transmitted via massive particles. (An massive aether is unlikely given that there is, for one thing, no detectable drag on objects through space.)

 

Apparently one can say that there are higher probabilities of detecting photons at certain places in beams of light and that the pattern of these probabilities has a formal resemblance to physical waves. So it should be of no overwhelming surprise or significance that we find resemblances between massive physical waves (e.g., water) and those in space/time.

 

Perhaps a key point, then,regarding the necessity of a physical medium is the issue of whether or not a wave has mass. Somebody please correct me if I am off track, but it seems that massless waves do not require a medium because they, in some sense, provide the ubiquitous field through which they travel, while massive waves do require massive fields.

 

I agree that I need a lot more math in order to really know what I am talking about, but I can often follow the abstract paragraph in a quantum theory study, while the rest is mostly gibberish to me. So at this point I am just looking for the abstract paragraph here, and will wade into the math with less of a chill when I have it.

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