Aeschylus

Why dko you need to approximate anyway? the area of a circle is pi*r^2, you shpould always keep the pi in, unless for some reason you need to know where it approximately lies on the real number line, but generally there's no need to know.

time for another crappy diagram. how can we, who travel at a velocity less than c, beat light, which travels at c, from point a to b?

The simple answer is we can't.

The CMBR is not the after glow of the big bang per se but it is from 300,00 years after the big bang when the unievrse's density became low enoguh that any photon emitted wasn't instantly reabsorbed, during a short period of time all areas of the univere (because at this time the universe was very, very, very homogenous) emitted alot of light within a very short period of time. Nothing to do with antimatter.

The CMBR does pass us and it doesn't come back due to the effects of gravity, BUT there is always more CMBR coming to us.

From our postion, the CMBR that we see at any insatnt is like viewing a giant spherical shell with a radius of about ~13 billion light years taht we are at the centre of, but this is only due to the effects of the unievrse we are exapniodn as by looking at the CMBR we are really looking at an area of space, much, much smaller than a spherical shell of radius 13 billion light years. If you've ever performed an inversion on a circle you might understand what I mean.

Fair enough, but I've seen [math]\phi[/math'] used a lot, especially on my courses.

Then that may be the case, I don't know beacuse I haven't done alot of the really pure maths.

Basically can not be a member of the field of real numbers. I'll show ou if you want, but you might like to try and construct the proof yourself by looking at the field axioms.

I don't think [math]\emptyset[/math] or [math]\varnothing[/math] are supposed to be a [math]\Phi[/math], or a [math]\phi[/math'], simply a separate symbol.

The empty Set symbol is a capital letter phi, I looked it up.

Divison by zero is simply not defined, so your proof is worthless.

I guess the reasoning is somewhat similar to why people use phi ([math]\phi[/math']) to designate the empty set.

You know I never relaized that it was phi as capital phi is usually printed like this: [math]\Phi[/math].

All numbers exist as only abstract entities, maths is an abstarct subject which has no intent to describe 'physical reality' (if your interested in why maths does seem to be useful in describing physical relaity look up Wigner's essay on the subject).

Aeschylus:

Please be patient with this initial' date=' rudimentary review; it leads to more worthily debateable turf, as you may or not agree...

 

Metric Mathematics

Presuming you are familiar with metric and non-metric space. That the mathematics of physical scientists is of the former definition; where metric math is mandtorily responsive to and descriptive of physical conditions or dynamics that exist, with or without persons and mathematical formulae to respond to and describe physical realities. E=MC squared, for example, describes an ongoing phenomenon, with or without people or equations to describe same...

Practitioners of metric mathematics are required to be confined to mathematically describing and otherwise accounting for real - existential - physical dynamics and/or conditions.[/quote']

 

I do not know where you ghot this idea from, but you are completely wrong. A metric space is a (nonempty) set and a function (the metric) which maps two members of the set onto a number and which also satsifies sevral other axioms. The concept of a metric space is completely abstract (for example the real number line is an example of a metric space which I can assure is a complete abstarction), though like nearly all maths it has uses in physics. A vector space has a metric when the scalar product is defined.

 

Non-Metric Mathematics

Whereas the latter - non-metric - definitions of whatever real or imagined conditions or dynamics of the physical universe: are not required to respond or conform to physical conditions or dynamics.

That is, for example, two non-metrical mathematical formulas - both equally and independently correct - can be and not infrequently are mutually contradictory - cancelling one another out.

Perhaps a superfluous qualification of the obvious; whereas it may be that you (and many others) improvised each, as though they were and are compatibly interchangeable... There is no argument in the potentially infinite number of dimensions, if and when every direction in space is considered, from every point in space, ad infinitum...

 

Again you are wrong (besides which there is no great division between using a metric and not using a metric, it dopepnds entirely on context), (vector) spaces without metrics are used in physics too.

 

 

As far as I can presently determine:

You are interchanging two different standards of (metric & non-metric) mathematics; while making a perhaps token reference to the physical geometry of functional (metrically standardized and measured) space-time.

 

No I am specifically talking about vector spaces.

 

Please bear with the following abbreviated window of turn of the century evolutions of scientific history; hopefully with this review, we may agree (approximately?) on where 'we' are (and are not) and how and why we did (and didn't) get there (altogether here)...

 

Riemann's ground breaking non-Euclidean geometry far preceded Einstein, who had an improvisationally gifted way of finding applications of earlier achievements (Riemann-Minkowski) or independently established contemporary systems and methods (Lorentz), to standardize what had been or was was previously considered, 'unrelated' and/or 'unconventional' (if not heretic).

 

Although Minkowski benchmarked the modernized term of 'space-time', and Einstein further refined it - it was Galileo who originated it - with a concept of 'simultanaeity'...

 

No, spacetime comes specifically from Minkowski, nothing to do with Gallileo.

 

Later questioned but not resolved by Newton; finally resolved in Einstein's application of Minkowski's immortal proclamation:

"Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." - H. Minkowski

 

What Minkowski pointed out basically it would be rather useful in the context of Einstein's STR to use to combine space and time into spacetime, due to the invariance of the interval.

 

*Einstein and Minkowski co-jointly and increasingly leaned on Maxwell's electromagnetic equations as a variable yardstick representing and determining the the non-absoluteness of space, time and (Galileo's formerly 'simultaneous') simultaneity (of events in space).

 

Lorentz formulated his transformations, Eibstein then formalised the theory, recognizing the failure of simulatenity, Minkowski then came up with the iodea of spacetime as a useful tool.

 

*They tandemly formulated a 'light cone' of electromagnetic emissions from a center source, the upward - 45 dg - conical portion of which represented (in the expansion of the inverse square) the distance light traveled in a second (186,282 m.p.s.), whereas the 45 degree conical shape beneath the light source was perceived as past time; it was from this conical model of light that the concept of non-absolute space, non absolute time (space-time) and non absolute simultaneity emerged; signalling the departure from Galileo's space-time, to the proposed space-time model proffered by Maxwell - who more comprehensively based his theories on what he discovered from the structural characteristics - and the speed - of light. (A platform of knowledge unavailable to Galileo.)

 

The discovery of the failure of simukataneity porecedes the use of lightcones.

 

 

The (quasi-flat planed) Gaussian co-ordinate system is inspirationally derived from the Cartesian (2-D mapping and chartering) co-ordinate system. but applicable only to non-Euclidean systems; only when applied to relatively smaller distances and sizes (of the Euclidean continuum).

 

Curvilinear coordinate systems can be used in non-Euclidean spaces.

 

Non Euclidean space of several definitions (Cartesian, Gaussean, Riemannian) introduced considerations of a 'Finite but unbounded' universe pioneered for the most and original parts by, Helmholtz and Poincare, while once again, further and more recently refined by Einstein.

 

I'm not sure where your getting this from, but a finite and unbounded universe is a specifc FRW cosmology.

 

As you may well know, much more can easily and importantly be added to the succession of these historical developments and transient perspectives, whereas, my point is that your mathematically proposed, mathematically standardized, 'innumerable number of points in space' do not fullfill the arrow of historical time, or the issued geometric definitions for observed, measured and sensorially experienced dimensions; specifically:

 

Geometric Point 0, moves to generate One Dimensional Straight Line 0 - A. Moves at right angles to itself generating 2-D Plane A - B. Two dimensional Plane moves at right angles to itself generating 3-D Solid (occupied or unoccupied by matter) B - C.

 

Einstein discovers a 4th dimension, closely related to time and motion, inherent but previously unrecognized in all microcosmic and macrocosmic three dimensional entities, consequently obliging the recognition and acknowledgement of continuous right angle motion of all three dimensional matter (growing ever smaller or ever larger. n either of the two exclusively alternative cases), at right angles to/from the three recognized dimensions.

 

It is colloquially and scientifically maintained that the 4th dimension of time and motion is inherent - but 'unrecognzable' and even non-mathematically 'unimaginable' and 'incomprehensible' - within the three recognized spatial dimensions of width, breadth and depth (of space); whereas the 4th dimension inherent to the preceding three dimensions it resides in, previously unrecognized 4th coordinate (uniting space & time to 'space-time') is a dimension of duration - time and motion (are synonymous)...

 

Albeit, matter is acknowledged (as well as directly experienced and observed in any falling or parabolically trajectoried object or missile - even audibly heard, in the accelerating sound of a spinning coin, plate, or automobile hubcap, settling down - while resounding ever more swiftly - on a hard surface. The (accelerating) sound of gravity - is the 4th dimension - is also frequently and distinctively audible in the rhythmically increasing rocking motions of any number of rigid objects, settling down to a quick stop on a hard surface <often in or around a kitchen sink or sideboard> ); while unrecognized as being 4 dimensional manifestations of gravity (re: http://einstein.periphery.cc/ )

 

Moreover, by the (processional) right angle law of physical dimensions, electricity is observed, measured and sensorially experienced as (constantly) moving at right angles out of (4-D) matter ('particles', 'charges', 'planets', 'stars', etc.);

consequently identifying electricity as the 5th dimension.

 

Summarily (until further notice?), magnetism is observed, measured and sensorially experienced as (constantly) moving at right angles to electricity; therefore identifying magnetism as the 6th dimension.

......................................

 

If and whatever may be measured as moving - in processional sequence - at right angles to magnetism, will self identify as the 7th dimension (Possibly heat, as it may somehow be distinguished from the inherent heat <motion> in matter and electromagnetism; though this consideration of a 7th dimension is only conjecture on the part of this record...)

.......................................

 

Concluding the similarities - and the differences - between metric and non metric space; mathematics and geometry...

 

The application of non metric mathematics to describe metric geometry is - however popularly employed - a non sequitur, with regard to the definition and recognition of physically manifest, measurable, observable and sensorially experienced, finitely presiding dimensions...

 

I respectfully stand by for any counterpoints, or concurrences, from yourself (Aeschylus), or any other sincere reader & writer.

 

 

 

Sincerely, KBR (Aka etceteras...)

 

I think you need to review exactly what relativity says about mtter gravity and other dimensions, because staem,ents like 'electricity is the 5th diemnsion' really don't make any sense in the context of relativity.

***********************************

 

Aeschylus:

By your leave' date=' sir.

It may be better for both and each of us to abstain from proclaiming either is *'playing semantic (or any other form of capricious) games'.

 

Note that you allude (presumably from my preceding post- THE LAW OF PHYSICAL DIMENSIONS: Revisited' - review of the geometric definition for physical dimensions) to:

*'A dimension and a direction are not the same thing... doesn't change their actual mathematical definition'...

 

With no semantic (or other inappropriate) games intended:

 

I do not see how the 'actual mathematical definition' of physical dimensions, overrules the (singular and stringent) geometric definition for physical dimensions.

(Do you question the geometric definition for physical dimensions as I <merely> reviewed them?)

 

Should not any such two (comparable) definitions for a very specific issue (definition for physical dimensions) - geometric and mathematical - complement/parallel one another, rather than brachiate to different - especially 'conflicting' - meanings?

In your repeated use of *'vector' - reference to *magnitude and direction - how does this gainsay my qualified premise that dimensions do indeed - very much - have to do with direction; in contrast to your qualified statement that 'a dimension and a direction are not the same thing'...

 

Whereas: the right angle projection (direction) of each succeeding physical dimension is patently defined and determined - by the law of physical (functionally metric, spatial) dimensional geometry - by the 'directional' (if not magnitudinal) definition of 'vector(s)'...

 

Fairly confident that you are familiar with the more recent works of Ouspensky (A NEW MODEL OF THE UNIVERSE: On The 4th Dimension), and other geometricians, all the way back to antiquity, regarding the meaning and (invariably identical) definition of (physical, as distinguished from any number of 'semantic') dimensions, one through three, and what is (since Einstein proved a 4th Dimension, closely related to time and motion, causing, among other definitional transitions, the revision of 'space and time', to 'space-time') called a 'hyper'-cube, or 'super-cube'.

 

That is, a three dimensional cube (or for that matter, any 3-D physical object or entity of whatever shape or density) projecting itself at right angles from it's three recognized dimensions, thereby fullfilling it's (Einsteinian) requirement to be 4-dimensional.

 

It is clarified throughout physical geometry that each given physical dimension proceeds at right angles from the dimension preceding it.

 

Do you question this definition?

 

Are you suggesting (proclaiming?) that whatever mathematical definition you allude to (which you have not specified), has precedence over the singular (unambiguous, non-ephemeral, non-anachronistic, time tested) geometric definition for physical dimensions?

 

What is your proposed, unrevealed, mathematical definition of physical dimensions?

 

How does it conflict with the (singular) geometric definition provided?

 

 

Sincerely, KBR.[/quote']

 

What your defintion essientially notes is that n-1 dimensional space can be a hypersurface in n-dimensional space, but this really doesn't define a dimensionality of a space.

 

However in my last post I did define the number dimensions of a vector space:

 

the cardinality of the maximal set of linearly independent vectors

 

(Interestingly this needn't be finite or even countable).

 

 

A simpler less formal defintion would be the number of numbers needed to describe every point in space.

 

Lets say we take our familair 3 dimensional Euclidean space and define each radius unit vector as a 'direction', how many directons are there? The answer is there are an infinite amount of them. However from this subspace of unit vectors we can pick an orthonormal basis of three unit vectors (and no more) that are linearly independent, this is why we say the space is has 3 dimensions (of course again it's worth saying that the number of orthonormal bases is infinite so each 'dimension' is not related to any specific one direction). So the concept of 'dimension' and 'direction', whilst being related are most certainly not the same thing.

 

Minkowskian spacetime is an interesting, example of a space that is flat but non-Euclidean, as it has a pseudo-Riemmanian metric. Of course spacetime, needen't be and isn't flat.

A dimension and direction are not the same thing, by playing semantic games, doesn't change their actual mathematical definition.

The number of dimensions of a (vector) space is the number of lineraly indpendent 'directions' if you like (or more formally the cardinality of the maximal set of lineraly independent vectors).

What we call 'Arabic numerals' are actually callled 'Indian numerals', by the Arabs.

since it`s widely accepted that the oldest Earth civilisation was Mesopotamia (arabic) I`de be more inclined to agree with Dave on this one.

The Sumerians were not Arabic (the descendnats of the sumerians almost certainly are though), infact they used a (broadly speaking base-60 number system). The Arabs rose as a force thousands of years later in the 7th century AD. Arabian traders imporetd the numerals of the Inidans into Arabian civilisation, this syatem was further developed by the Arabs with such inovations as decimal fractions.

What about the EPR paradox? This has been shown to exist but I have never seen a rationalisation as to why or how .

EPR paradox does not violate causality or relativty as no information is transmitted faster than c. What it does illustrate is the non-local nature of quantum mechanics.

But the Arabs got their numerals originally from India.

Like all our other numerals, it comes from India via the Arabs.

FTL is possible!

 

I think this proves a majority of Einstein's theories wrong!

 

Read this article:

http://www.newscientist.com/news/news.jsp?id=ns99992796

 

Encrypted

No it doesn't.

it's just an example of quanutm tunneling, a phenoumenum that's been known abou for mnay years.

You have to ask what is specifically travelling faster than the speed of light? Is causality violatd?

A direction and a dimension are not the same thing.

From the point of view of classical wave equations, the use of complex numbers is simply convenient.

From the point of view of the wavefunctions of quantum mechanics the wavefunction is fundamentally complex. The wavefunction though doesn't have any direct physical significance, so Im(Psi) and Re(Psi) don't have any direct physical signifcance either.

All circles are similar' date=' no angle change and there would be an angular speed change or w, since the circumference will get either bigger or smaller (given).

If the circumference gets smaller w increases if the linear speed remains the same, since a bigger percentage and thus a greater angle will be covered per t, the time. Vice versa if C increases.

 

The linear speed of a circle is just arclength/time. The fact that C becomes bigger doesn't have an effect. I see what your confusion might be.

The linear speed doesn't change because the arclength is the length m of the arc/second or t time, not an actual arc percentage of C, which would make a cirlce have a linear speed of only one speed which is ridiculous.

 

So linear speed is not proportional to the circumference, angular speed increases with decrease in C, and decreases with increases in C.[/quote']

 

I did not say the angle changes, what I said is that the way that youdagonapogos has decided to define the angle the angle changes, because he has defined it in a way that is dependent on the ratio of the diamter and the circumefrence of an arbitary circle which is not a constant in non-Euclidian spaces

 

The actual defintion of an angle theta between two vectors in a vector space where the scalar product (<x|y>) has been defined is:

 

[math]\cos{\theta} = \frac{\langle x|y\rangle}{\sqrt{\langle x|x\rangle}\sqrt{\langle y|y\rangle}}[/math]

Differential geometry deals with this.

What happens when you change the size of the circle that you use to measure the angle? The answer is according to the way you've defined the angle, the size of the angle changes! So you can't define an angle from any old circle as on such a surface the ratio depends on the size of the circle (and possibly where the circle is). However as the radius of the circle tends to zero the ratio always tends to pi, so you can deine the angle from the point where the two lines intersect.

What I mean is that it's like talking about 'north' of the North pole, by the way we define the north and the North pole there is nothing north of the North pole, simlairly by the way we define reference frames and photons, photons don't have reference frames. It's not a gap in our knowledge, it's just that our knowledge tells us that such questions are meaningless.

The question becomes menaingless in SR, as you cannot define reference frame for the photon so you cannot talk about whetehr or not time passes for a photon. The problem is that peole use theb formula for time dialation and input the value 'c' and think the answer is meaningful as it tells you nothing about the photon's reference frame.

I would even go a step futher and restate what I already did. I believe massless particles do not obey special relativity. I might be wrong as this is more of a hunch than a confirmed scientific principle but it would eliminate the need to talk about such frames. How can an object see the whole universe collapsed to a point and have time stop for it if outside observers clearly see it traversing from one region to another. Its beyond contradiction, its impossible.

Massless particles do obey special relatvity, but they also produce some reuslts that may be counterintutive. For exmaple the 4-velpcity is not defiend for a massless particle and they also give rise to null vectors (a null vector is not necessarily a zero vector in SR; a null vector is a vector whose square norm is zero and a zero vector is a vector whose components are all zero), which by the defintion of orthogonality are orthogonal to themselves (of course zero vectors which are the only kind of null vectors in basic 3-vector algebra can also be considered to be orthogonal to themselves anyway)! But none of these are actually problems becasue photons don't have rest frames.