Time is NOT the 4th dimension...

Amrit has been a poster at Physforum for quite a while.

 

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Um, I wasn't trying to be condiscending I was actually trying to help. My appologies if it came off that way :shrug:

 

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Not sure what you mean.
To my way of thinking if they admit that time DOES exist in the brain but not in the rest of the universe then they have problems, no?

 

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The guy noticed a weight the couldn't be accounted for... how is that unscientific to explore and wonder about?

 

Because that experiment has been done MANY times by scientists without an agenda.
Those other experiments also didn't rely on specious double-talk and unfounded claims.

 

In regards to the thread title, there appears to be a misunderstanding of what is meant by 4 dimensions. Simply put, four coordinates are required to localize an event in time-and-space if that coordinate system covers all the time-and-space which physics experimentally recognizes.
http://en.wikipedia.org/wiki/Dimension

So if the poster is arguing that space is not three-dimensional ("Space is that which prevents everything from happening in the same place") or that the time experienced by a person isn't one-dimensional ("Time is that which prevents everything from happening at once") then some empirical basis for this claim stronger than the combined observations of humanity over thousands of years is needed. Even the ancient Greeks recognized these. Even Newton worked in the three dimensions of space and the one dimension of time.

That's not to say that time doesn't have a different nature from space, but that difference in flavor does not matter to the definition of "dimension" as it is used to talk about 4-dimensional space-time. Nor is it to say that Newton's concept of time was the only mathematically self-consistent description, but for one to understand the empirically preferred alternative to Newton one needs to master relativity, both Galilean and Special.

In the language of Fox News, "some people" is usually a cowardly way to hide when one would like the fallacious authority of popular thought on one's side but couldn't find such authority. In the language of Wikipedia, a "citation" is "required" to ensure that the forces of justice and truth are on one's side and that one isn't just some aggressively ignorant anti-science troll incapable of presenting a contemporaneous source without your own biases swamping the signal. English translations of the two 1905 papers by Einstein are readily available on the Internet, but require that the reader be familiar with the experimental basis for electrodynamics and applications of Maxwell's equations.
http://www.fourmilab.ch/etexts/einstein/specrel/www/
http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

As should be obvious, \(E = mc^2\) is not the whole, nor even the greater part of special relativity. In fact, the more general equations of motion for a free particle are:
\(\begin{eqnarray}E^2 & = & \left( \mathbf{p} c \right)^2 + \left( m_0 c^2 \right)^2 \\ \mathbf{v} & = & \frac{ \mathbf{p} c^2 }{E} \end{eqnarray}\)

I agree it is difficult to provide meaningful instruction to the aggressively ignorant. But my job is complete when I provide useful and accessible instruction, and the reader's job doesn't end until the reader understand it.

Einstein, as it is clear from the translated papers, used the experimental confirmation of Maxwell's equations as they relate to the phenomenon of electrodynamics to show that the laws of physics don't have a preferred inertial coordinate system (i.e. any coordinate system where Newton's laws of inertia hold is equally valid) and were incompatible with the older notion of Galilean relativity in a specific manner that replaces the space + time of Newtonian physics where there is no theoretical limit on speed, with Minkowski space-time, where not only is there an upper limit to speed, but space and time can substitute for each other just like a rotation of the Euclidean plane lets x and y substitute for each other.

Understanding relativity is tantamount to mathematically placing oneself in the shoes of another and working out the consequences, and not only is it vital to understand this for physics, but in my opinion a failure to understand it diminishes one's ability to enjoy the humanity of others and the nature of mathematical thought.

http://www.sciforums.com/showthread.php?p=2540723

The following should look familiar... (since it is based on my 2008 post which suffered formatting problems when the forum changed processing engines).

What's this Theory of Relativity?
It's called relativity because all of the laws of physics look the same to the observer who is moving at the same inertial velocity as the watched experiment as to the observer who is in relative motion. It has rules that say that the precise manner in which the experiments (not the laws) look different when you and the experiment are not moving at the same velocity and the way it looks when you are at relative rest only depends on the relative motion between you and the experiment.

This is the principle of relativity. The modern theory of relativity differs from the Galilean theory of relativity in that clocks in relative motion and rulers parallel to their direction of relative motion cannot be made to agree with one another.

But that's crazy!
I would say "counter-intuitive" because our intuition is shaped by evolution over billions of years at scales and speeds where relativistic effects are far from obvious. Besides, if raw intuition was all that was needed to study the universe, Plato and Aristotle would have gotten around to finishing it.

I don't think that how intuitive a physical theory is should be a criteria in its value, because the universe is under no obligation to be intuitively comprehensible. In fact, the original Newtonian theory of Universal Gravitation contradicts over 1000 years of intuition by resting on the axiom that terrestrial and celestial laws of motion are identical.

In the same way, Special Relativity is the well-tested assertion, in contradiction to Newton's conflict with Maxwell's equations, that the laws of physics are the same for light as for more familiar objects. It turns out that some violence has to be done to the assumptions that Newton, following Descartes, was working from, but Special Relativity is highly successful in being one set of laws for fast and slow.

It's still crazy -- it has to be mathematically complicated and inconsistent
Almost immediately, it was demonstrated to be not only mathematically consistent but simple by Minkowski, but somewhat simpler, as space-time, than Newtonian separate concepts of space and time. While the mathematics of the Poincaré group will never be introductory mathematics, most of the tools to understand special relativity, geometrically, algebraically and logically are developed in common high school courses.

Well, anyone could be wrong and Einstein is just one man
You will note at the end are some references and that many of these references are to "review articles" which are summaries of hundreds of research papers. Review articles are like textbooks of recent developments in science, and are frequently written to a wider audience than the source articles which they cite.

It's not "just Einstein" but every serious investigation of high speed objects which supports this model of the physical world.

That's just lies made up by the modern physics conspiracy. No evidence before Einstein exists for Special Relativity
While Einstein (1879-1955) was fortunate to be living at a time when classical physics was suffering under the load of experiments inconsistent with Newtonian mechanics, clear-cut evidence which favors Einstein over Newton was published before Einstein was born.

There are conflicts between Newton and experiment which was increasingly apparent in the nineteenth century. In fact, by 1859 we had enough experimental evidence to favor special relativity over Galilean relativity and Newtonian absolute space and time.

By 1859, Hippolyte Fizeau's experiment to measure the Augustin Fresnel's hypothetical ether drag was exactly consistent with a velocity addition formula of \(v_3 = {{v_1 + v_2}\over{1 + K v_1 v_2}}\) with \(K = \frac{1}{c^2}\) while Newton and Galileo would predict \(K = 0\).

Lorentz and FitzGerald came up with equations which correctly relate observables like elapsed time and relative position for two distinct inertial observers, but didn't have physics understanding beyond phenomenology. Einstein proposed new candidate axioms of physics and demonstrated that not only were such axioms consistent with existing experimental results, but than they provided a basis to derive previous phenomenological results like the Lorentz-FitzGerald contraction or the Fresnel drag coefficients as first-class physical results. Finally, Minkowski demonstrated that there was a mathematical beauty behind the physical results, such that we no longer speak of Euclidean and absolute space and time, but only of Minkowski space-time.

While astonishing to some, these results were shown by von Ignatowsky and others to be very natural descriptions, provided we were willing to let Nature be our guide and not just rely on the authority of Newton, Aristotle and Euclid.

Wait, wait, wait -- not just any velocity addition formula can work! You have to start with physics, not just math created from thin air.
Well, you need math to create a framework in which to place your observations, but let us see.

On Time Dilation
Time Dilation, as was shown shortly after Einstein's 1905 papers, is a natural result which arises from testable intuitive statements of the nature of space and time and physics.

First of all, assuming God doesn't write a physics textbook, man will be forever ignorant of the actual mechanisms of the universe. Even if Professor Y comes up with the mechanistic ONE TRUE THEORY OF EVERYTHING, there will be no way to distinguish the universe of the mechanistic theory from another universe where everything conspires to act just like Y's mechanistic model. That's why physicists (as opposed to philosophers) use mathematical models to avoid talking about the mechanism and only the behavior. Our everyday experience of the universe has taught us a lot of everyday assumptions. The following 4 should be non-controversial when you neglect gravity. They are all statements about observed symmetries of the universe -- so all of them are falsifiable if you found a counter-example.

Four everyday assumptions

Let us assume the laws of physics are translationally invariant in space. Then it follows a statement about a experiment happening in an arbitrary place will work the same if we center our coordinate basis with it. This also implies that we can calculate what's happening in an arbitrary place and apply a translation transform to it, and the physics is the same.
\(\begin{array}{rclcr}x' & = & \mathbf{T}_x & + & x \\ y' & = & \mathbf{T}_y & + & y \\ z' & = & \mathbf{T}_z & + & z \end{array} \)

Let us assume the laws of physics are translationally invariant in time. Then it follows a statement about a experiment happening in an arbitrary time will work the same if we center our basis of "now" with it. This also implies that we can calculate what's happening in an arbitrary time and apply a translation transform to it, and the physics is the same.
\(t' = \mathbf{T}_t + t\)

Shorthand: \( \left( { t' \\ \mathbf{x}' } \right) = \mathbf{T} + \left( { t \\ \mathbf{x} } \right) \)

Let us assume the laws of physics are rotationally invariant. Then it follows a statement about a experiment oriented in an arbitrary direction will work the same if rotate our coordinate basis to be aligned with it. This also implies that we can calculate what's happening in an arbitrary aligned experiment and apply a rotation transform to it, and the physics is the same.
\(\begin{array}{rcllcrlcrl} x' & = & \mathbf{R}_{xx} & x & + & \mathbf{R}_{xy} & y & + & \mathbf{R}_{xz} & z \\ y' & = & \mathbf{R}_{yx} & x & + & \mathbf{R}_{yy} & y & + & \mathbf{R}_{yz} & z \\ z' & = & \mathbf{R}_{zx} & x & + & \mathbf{R}_{zy} & y & + & \mathbf{R}_{zz} & z \end{array}\) Where R is a proper orthogonal matrix, which can be parameterized in various ways by 3 rotation angles.

Shorthand: \( \mathbf{x}' = \mathbf{R} \mathbf{x} \)

Let us assume the laws of physics are invariant with respect to inertial frame. Then it follows a statement about a experiment with a freely moving center of mass moving in an arbitrary direction will work the same if set up our coordinate basis to be co-moving with it with it. But clearly any corresponding change-of-frame transform must tie velocity, time and space together. Since we already assumed we are rotationally invariant and translationally invariant, let us work with v in the x direction and just coordinate differences.

\(\begin{array}{rcllcrlcrlcrlcrl} \Delta x' & = & \mathbf{F}_{xx} & \Delta x & + & \mathbf{F}_{xy} & \Delta y & + & \mathbf{F}_{xz} & \Delta z & + & \mathbf{F}_{xt} & \Delta t & + & \mathbf{F}_{x1} \\ \Delta y' & = & \mathbf{F}_{yx} & \Delta x & + & \mathbf{F}_{yy} & \Delta y & + & \mathbf{F}_{yz} & \Delta z & + & \mathbf{F}_{yt} & \Delta t & + & \mathbf{F}_{y1} \\ \Delta z' & = & \mathbf{F}_{zx} & \Delta x & + & \mathbf{F}_{zy} & \Delta y & + & \mathbf{F}_{zz} & \Delta z & + & \mathbf{F}_{zt} & \Delta t & + & \mathbf{F}_{z1} \\ \Delta t' & = & \mathbf{F}_{tx} & \Delta x & + & \mathbf{F}_{ty} & \Delta y & + & \mathbf{F}_{tz} & \Delta z & + & \mathbf{F}_{tt} & \Delta t & + & \mathbf{F}_{t1} \end{array}\) where F is a function of v, which we have agreed to consider in the x direction.

Shorthand: \( \left( { \Delta t' \\ \Delta \mathbf{x}' } \right) = \mathbf{F}_{\mathrm{inhomogeneous}} + \mathbf{F} \left( { \Delta t \\ \Delta \mathbf{x} } \right) \)


Limiting the form of the velocity transform

Since it makes no sense to talk about \(( \Delta x, \, \Delta y, \, \Delta z, \, \Delta t) = ( 0, \, 0, \, 0, \, 0 )\) which says that the two events happened in the same time and place in one frame transforming into other than \( ( 0, \, 0, \, 0, \, 0 )\) in the primed frame, it follows that \(( \mathbf{F}_{x1} , \, \mathbf{F}_{y1} , \, \mathbf{F}_{z1} , \, \mathbf{F}_{t1} ) = ( 0 , \, 0 , \, 0 , \, 0 )\). Thus \(\mathbf{F}(\mathbf{v})\) represents a homogeneous transform.

If you think you know of a reason why a v in the x direction should involve displacements in the y or z direction, please let me know. I think that the rotational invariance we assumed earlier means that if v is in the +x direction, then it cannot have a reason to prefer +y or -y, and so the effect on y must be zero, and vice-versa, and the same for z.

Then \(\mathbf{F}_{yy} = \mathbf{F}_{zz} = 1\) and \(\mathbf{F}_{xy} = \mathbf{F}_{xz} = \mathbf{F}_{yx} = \mathbf{F}_{yz} = \mathbf{F}_{yt} = \mathbf{F}_{zx} = \mathbf{F}_{zy} = \mathbf{F}_{zt} = \mathbf{F}_{ty} = \mathbf{F}_{tz} = 0\). So we are over half done.

\(\begin{array}{rcllcrlcrlcrlcrl} \Delta x' & = & \mathbf{F}_{xx} & \Delta x & & & & & & & + & \mathbf{F}_{xt} & \Delta t & & \\ \Delta y' & = & & & & & \Delta y & & & & & & & & \\ \Delta z' & = & & & & & & & & \Delta z & & & & & \\ \Delta t' & = & \mathbf{F}_{tx} & \Delta x & & & & & & & + & \mathbf{F}_{tt} & \Delta t & & \end{array}\)

Since two events one frame which don't move at all have \(\Delta x = 0\), but in the other frame \(\frac{\Delta x'}{\Delta t'} = v\), then \(\mathbf{F}_{xt} = v \mathbf{F}_{tt}\). Since if two events in one frame are connected by a particle moving at speed \(-v\), and not moving in the other frame then \(\Delta x = -v \Delta t \; \Rightarrow \; \Delta x' = 0 = -\mathbf{F}_{xx} v \Delta t + \mathbf{F}_{xt} \Delta t \; \Rightarrow \; \mathbf{F}_{xt} = v \mathbf{F}_{xx} \; \Rightarrow \; \mathbf{F}_{xx} = \mathbf{F}_{tt}\). Let's call that \(A(v)\).

\(\begin{array}{rcllcrlcrlcrlcrl} \Delta x' & = & A(v) & \Delta x & & & & & & & + & v A(v) & \Delta t & & \\ \Delta y' & = & & & & & \Delta y & & & & & & & & \\ \Delta z' & = & & & & & & & & \Delta z & & & & & \\ \Delta t' & = & \mathbf{F}_{tx} & \Delta x & & & & & & & + & A(v) & \Delta t & & \end{array}\)

For the same reason that length-contraction must be in the direction of movement, we expect two observers to experience the same relative time dilation. Since there are no preferred directions, then nothing but convention distinguished -x from +x and so nothing distinguished -v from +v and so we expect that the time dilation to be the same for two observers in relative motion, if there is any time dilation.

Consider a motionless clock. Two tick of the clock are separated by \( \Delta t \ne 0\), but \(\Delta x = 0\).
so \(\Delta t' = A(v) \Delta t\) . Now let's move the clock at -v so it is motionless for \(\Delta x' = 0\). So we want to solve \(\Delta t = A(v) \Delta t'\), \(\Delta x = - v \Delta t\), and \(\Delta t'= \mathbf{F}_{tx} \Delta x + A(v) \Delta t\)
So \(\Delta x = - v A(v) \Delta t', \; \Delta t' = \mathbf{F}_{tx} \Delta x + A(v) A(v) \Delta t'\) and so
\(\Delta t' = - v \mathbf{F}_{tx} A(v) \Delta t' + A(v) A(v) \Delta t'\) and so
\(\mathbf{F}_{tx} = \frac{ A(v)A(v) - 1 }{ v A(v) } = \frac{1}{v} \left( A(v) - \frac{1}{A(v)} \right)\)

\(\begin{array}{rcllcrlcrlcrlcrl} \Delta x' & = & A(v) & \Delta x & & & & & & & + & v A(v) & \Delta t & & \\ \Delta y' & = & & & & & \Delta y & & & & & & & & \\ \Delta z' & = & & & & & & & & \Delta z & & & & & \\ \Delta t' & = & {{1}\over{v}} \left( A(v) - {{1}\over{A(v)}}\right) & \Delta x & & & & & & & + & A(v) & \Delta t & & \end{array}\)

At this point both the Newtonian and the Relativist should be happy. The Newtonian assumes that \(A(v)\) is a constant with value 1, while the Relativist sees that our four initial assumptions do not yet force that choice. \(A(v)\), based on our four assumptions, is just a number and may yet turn out to be a non-constant function of v.

Working with the velocity transformation

Now with translations or rotations, they form a group. (A group is a mathematical way of talking about symmetries.) So that if you apply T1 and then T2, you get T3 which is also in the form of a translations. (Same for rotations.) This should be the same for two transforms related to velocity.

\(\begin{eqnarray} \Delta x' & = & A(v_1) \Delta x + v A(v_1) \Delta t \\ \Delta y' & = & \Delta y \\ \Delta z' & = & \Delta z \\ \Delta t' & = & {{1}\over{v_1}} \left( A(v_1) - {{1}\over{A(v_1)}}\right) \Delta x + A(v_1) \Delta t \end{eqnarray}\)

\(\begin{eqnarray} \Delta x'' & = & A(v_2) \Delta x' + v A(v_2) \Delta t' \\ \Delta y'' & = & \Delta y' \\ \Delta z'' & = & \Delta z' \\ \Delta t'' & = & {{1}\over{v_2}} \left( A(v_2) - {{1}\over{A(v_2)}}\right) \Delta x' + A(v_2) \Delta t' \end{eqnarray}\)

\(\begin{eqnarray} \Delta x'' & = & A(v_3) \Delta x + v A(v_3) \Delta t \\ \, & = & A(v_2) \left( A(v_1) \Delta x + v1 A(v_1) \Delta t \right) + v_2 A(v_2) \left( {{1}\over{v_1}} \left( A(v_1) - {{1}\over{A(v_1)}} \right) \Delta x + A(v_1) \Delta t \right) \\ \Delta y'' & = & \Delta y \\ \Delta z'' & = & \Delta z \\ \Delta t'' & = & {{1}\over{v_3}} \left( A(v_3) - {{1}\over{A(v_3)}}\right) \Delta x + A(v_3) \Delta t \\ \, & = & {{1}\over{v_2}} \left( A(v_2) - {{1}\over{A(v_2)}} \right) \left( A(v_1) \Delta x + v_1 A(v_1) \Delta t \right) + A(v_2) \left( {{1}\over{v1}} \left( A(v_1) - {{1}\over{A(v_1)}} \right) \Delta x + A(v_1) \Delta t \right) \end{eqnarray}\)

When we equate our expressions of the double-primed coordinates in terms of the unprimed coordinates, we have the following relations in \(v\) and \(A(v)\) :

1. \(A(v_3) = A(v_2) A(v_1) + {{v_2}\over{v1}} A(v_2) A(v_1) - {{v_2}\over{v1}} {{A(v_2)}\over{A(v_1)}} \)
2. \(v_3 A(v_3) = A(v_2) v_1 A(v_1) + v_2 A(v_2) A(v_1) = (v2 + v1) A(v_2) A(v_1) \)
3. \( {{1}\over{v3}} \left( A(v_3) - {{1}\over{A(v_3)}} \right) = {{1}\over{v_2}} \left( A(v_2) - {{1}\over{A(v_2)}} \right) A(v_1) + A(v_2) {{1}\over{v1}} \left( A(v_1) - {{1}\over{A(v_1)}} \right) \)
4. \(A(v_3) = {{v_1}\over{v_2}} A(v_1) A(v_2) - {{v_1}\over{v_2}} {{A(v_1)}\over{A(v_2)}} + A(v_2) A(v_1) \)

From equations 1 and 4, we have the important equality:
\(\begin{eqnarray} & & A(v_3) = A(v_2) A(v_1) + {{v_2}\over{v1}} A(v_2) A(v_1) - {{v_2}\over{v1}} {{A(v_2)}\over{A(v_1)}} \\ & = & A(v_3) = {{v_1}\over{v_2}} A(v_1) A(v_2) - {{v_1}\over{v_2}} {{A(v_1)}\over{A(v_2)}} + A(v_2) A(v_1) \end{eqnarray}\)
or
\({{v_2}\over{v_1}} A(v_2) A(v_1) - {{v_2}\over{v_1}} {{A(v_2)}\over{A(v_1)}} = {{v_1}\over{v_2}} A(v_1) A(v_2) - {{v_1}\over{v_2}} {{A(v_1)}\over{A(v_2)}} \)
or
\({{v_2 v_1}\over{v_1^2}} A(v_2) A(v_1) - {{v_2 v_1}\over{v_1^2}} {{A(v_2) A(v_1)}\over{A(v_1)^2}} = {{v_1 v_2}\over{v_2^2}} A(v_1) A(v_2) - {{v_1 v_2}\over{v_2^2}} {{A(v_1) A(v_2)}\over{A(v_2)^2}}\)
or, for generic v1 and v2,
\({{1}\over{v_1^2}} \left( 1 - {{1}\over{A(v_1)^2}} \right) = {{1}\over{v_2^2}} \left( 1 - {{1}\over{A(v_2)^2}} \right)\)
But since this is true for any v, then there is some constant \(K \equiv {{1}\over{v^2}} \left( 1 - {{1}\over{A(v)^2}} \right)\) for all v. This means \(A(v)\) can be written in the form \(A(v) = {{1}\over{\sqrt{1 - K v^2}}}\) ; Using the binomial theorem, we can show that when \(K v^2 << 1\) \(A(v)\) is approximately: \(1 + \frac{1}{2} K v^2 + \frac{3}{8} \left( K v^2 \right)^2 + \frac{5}{16} \left( K v^2 \right)^3 + \frac{35}{128} \left( K v^2 \right)^4 + \frac{63}{256} \left( K v^2 \right)^5 + \frac{231}{1024} \left( K v^2 \right)^6 + ... \) so the Newtonian will always appear correct as long as |v| is "small," or \(K << \frac{1}{v^2}\).

Only at high speed would there be evidence that \(K\) is not zero. (If \(K\) is zero, then \(A(v) = 1\), just like the Newtonian assumed.)

(See Pal)

The generic velocity addition law
From equation 2 we see something that with a little algebraic reworking can become our velocity addition law from our four assumptions.
\(v_3 A(v_3) = (v_2 + v_1) A(v_2) A(v_1)\)
or
\( {{v_3}\over{ \sqrt{1 - K v_3^2} }} = {{v_2 + v_1}\over{\sqrt{1 - K v_2^2} \sqrt{1 - K v_1^2}}} \)
or
\( {{v_3^2}\over{ 1 - K v_3^2 }} = {{(v_2 + v_1)^2}\over{(1 - K v_2^2)(1 - K v_1^2)}} \)
or
\( v_3^2 = {{(v_2 + v_1)^2}\over{(1 - K v_2^2)(1 - K v_1^2)}} {{(1 - K v_2^2)(1 - K v_1^2)}\over{K (v_2 + v_1)^2 + (1 - K v_2^2)(1 - K v_1^2)}} \)
or
\( v_3^2 = {{(v_2 + v_1)^2}\over{K (v_2 + v_1)^2 + (1 - K v_2^2)(1 - K v_1^2)}} \)
or
\( v_3^2 = {{(v_2 + v_1)^2}\over{K v_2^2 + 2 K v_1 v_2 + K v_1^2 + 1 - K v_2^2 - K v_1^2 + K^2 v_1^2 v_2^2}} \)
or
\( v_3^2 = {{(v_2 + v_1)^2}\over{1 + 2 K v_1 v_2 + K^2 v_1^2 v_2^2}} \)
or
\( v_3 = {{v_2 + v_1}\over{1 + K v_1 v_2}} \)

Clearly if \(K\) is measured to be zero, then \(A(v) = 1\) and there is no time dilation. However the results of an 1859 experiment (among thousands of others) are inconsistent with \(K = 0\).

Fresnel and Fizeau's measured value of K
In the discredited dragged-ether theory of Fresnel, light is "slowed" and "dragged" by a transparent dielectric with dielectric constant \(n\). It is slowed to \(V=\frac{c}{n}\), but if the medium is moving at speed \(v\), it is dragged and the measured speed is about \(U = \frac{c}{n} + v\left(1 - \frac{1}{n^2} \right)\). The amount of "ether dragging" by a moving dielectric is measured as the unexplained Frensel drag coefficient, \(1 - \frac{1}{n^2}\). The result eventually helped cause the downfall of the dragged-ether model, for where a physical medium can support a wide variety of waves, the phenomenon of dispersion shows that \(n\) is a function of wavelength, and so the Fresnel drag coefficient must also be a function of wavelength, and therefore there must be a different ether to drag for every wavelength of light.

But let's just take \(V= \frac{c}{n}\) the experimental velocity of light in stationary medium, and apply our generic velocity addition law to it and see how it predicts an observer moving relative to the medium measure its speed at.

\(v_3 = {{v_1 + v_2 }\over{1 + K v_1 v_2}} = {{v + V}\over{1 + K v V}} = {{v + { {c} \over {n} } } \over { 1 + K v c/n } } \)

if we assume \(K v \frac{c}{n} << 1\), then we can use the binomial theorem to approximate \(v_3\) as \((v + {{c}\over{n}}) - (K v {{c}\over{n}}) (v + {{c}\over{n}}) + (K v {{c}\over{n}})^2 (v + {{c}\over{n}}) + ... \\ = {{c}\over{n}} + v - K v^2 {{c}\over{n}} - K v {{c^2}\over{n^2}} + K^2 v^3 {{c^2}\over{n^2}} + K^2 v^2 {{c^3}\over{n^3}} ... \) or, if you drop the terms which aren't linear in v, \(v_3 = {{c}\over{n}} + v (1 - K {{c^2}\over{n^2}})\)

If \(v_3\) is close to the observed value \(U\), then \(K c^2 = 1\), or \(K = \frac{1}{c^2}\)

Epilogue
This was written to show that time dilation, by which I mean that observers who differ in velocity must have \(A(v)\) different than 1, is the only physical result if you accept the four assumptions. The other consequences of this idea have been well-developed. \(K\) is very close to zero in ordinary units, but we have thousands of experimental results which suggest that it is much closer to \(\frac{1}{c^2}\) than to zero.

This should show that if the four assumptions are good, then even experiments not involving light will show that c is an physically important speed in our universe. Further \(E = m_0 A(v) c^2\) gives us \(E = m_0 c^2 + \frac{1}{2} m_0 v^2 + \cdots\) which has been used to relate Newton's approximate formula for kinetic energy to Einstein's Relativity.
See Mattingly.

According to Silagadze people have shown this to be true with various degrees of rigor since at least 1910.

Following Einstein, Minkowski (1908) showed that algebraically this was the same as saying space and time were not separate things, which is how all physicists work today. Both Length contraction and time dilation arise from treating space and time as separate things with separate meanings, which is the Newtonian and intuitionist view. In Minkowski space-time, these are trivial (boring!) effects.

In fact, the velocity transform is identical to the hyperbolic analogue of Euclidean rotation, since it's equivalent to:
\(\Delta x' = ( \cosh \, \tanh^{-1} \, \frac{v}{c} ) \Delta x + ( \sinh \, \tanh^{-1} \, \frac{v}{c} ) c \Delta t , \; \Delta t' = ( \sinh \, \tanh^{-1} \, \frac{v}{c} ) \frac{1}{c} \Delta x + ( \cosh \, \tanh^{-1} \, \frac{v}{c} ) \Delta t\).

It's still crazy -- besides you can't show that this works with anything other than light

Well it works everywhere we test it including for decaying muons and in Thomas precession. It's used in the design of modern electronics, the GPS system and of course particle accelerators.

What is Thomas precession?

It is the observation that a boost in the X-direction and a boost in the Y-direction do not combine into a boost in some third direction, but a boost in a third direction AND a rotation. And the rotation depends on the order of the boosts.

This could (and has been) expressed as a statement relating to the commutators of the Poincaré group (inhomogeneous Lorentz group, i.e. Special Relativity). See page 55 of Kim and Noz or any comparable text.

Historically, Thomas precession was observed as a mismatch between the predictions of non-relativistic electromagnetics and physical experiment, where non-relativistic electromagnetics predicted a precession rate twice what was measured. Thomas precession (which doesn't depend on the charge of the electron or the magnetic field of the nucleus) predicts a value -1/2 of the non-relativistic electromagnetic prediction, so since 1 - 1/2 = 1/2 this was Thomas' 1927 explanation of the experimental result.

I don't see E=mc², so you are pulling a fast one!
\(E=mc^2\) is not synonymous with the theory of special relativity, but it is an important result of the theory.

You don't even mention energy or mass in those equations, so how can you get E=mc²?
Lets start with proper time, \(\Delta \tau\), between two events which are seen be an observer to happen at different times, \(\Delta t\) and different places, \(\Delta \mathbf{x}\). (Let us assume that these two events are on the world line of a massive particle moving slower than light, so we have finite proper time...)
\(c^2( \Delta \tau )^2 = c^2(\Delta t)^2 - (\Delta \mathbf{x})^2\) (Equation 5 which is justified by the velocity transformation of a moving clock into the unique frame where it is not moving)
or
\(c^2\left( \frac{\Delta \tau}{\Delta t} \right)^2 = c^2 - \left( \frac{\Delta \mathbf{x}}{\Delta t} \right)^2 = c^2 - \mathbf{v}^2\)
or
\(\Delta t = \Delta \tau \frac{1}{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}} = \gamma \Delta \tau\)

Where for the first time we use the shorthand \(\gamma = \frac{1}{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}}\)

Now, lets say that this \(\Delta t\) and \(\Delta \mathbf{x}\) were associated with a massive body, with invariant mass \(m_0\). Then it follows from equation 5 that
\(m_0^2 c^2( \Delta \tau )^2 = m_0^2 c^2(\Delta t)^2 - m_0^2 (\Delta \mathbf{x})^2\)
or
\(m_0^2 c^2 = m_0^2 c^2(\Delta t / \Delta \tau)^2 - m_0^2 (\Delta \mathbf{x} /\Delta \tau )^2\)
or
\(m_0^2 c^2 = m_0^2 c^2(\Delta t / \Delta \tau)^2 - m_0^2 (\Delta \mathbf{x} / \Delta t)^2 (\Delta t / \Delta \tau )^2\)
or since we introduced the shorthand, \(\gamma\),
\(m_0^2 c^2 = (m_0 c \gamma)^2 - (m_0 \mathbf{v} \gamma)^2\)
or
\((m_0 c^2)^2 = (m_0 c^2 \gamma)^2 - (m_0 \mathbf{v} c \gamma)^2\)

Now,
\(m_0 c^2 \gamma = \frac{m_0 c^2}{\sqrt{1 - \frac{\mathbf{v}^2}{c^2}}} \approx m_0 c^2 \left( 1 + \frac{1}{2}\frac{\mathbf{v}^2}{c^2} + \cdots \right) = m_0 c^2 + \frac{1}{2}m_0 \mathbf{v}^2 + \cdots\)
which has units of energy and the second term exactly corresponds to the Newtonian kinetic energy of a particle of mass m and moving at speed v, so because it changes pretty much like the Newtonian Energy lets say
\(E = \gamma m_0 c^2\)
and
\(\mathbf{p} = \gamma m_0 \mathbf{v}\) (or \(\mathbf{p} = \gamma \frac{E}{c^2} \mathbf{v}\) which works better in the case of \( \left| \mathbf{v} \right| = c\) and \(m_0 = 0\))
then our last result could be written as
\(\left( m_0 c^2 \right)^2 = E^2 - (\mathbf{p} c)^2\)
which for a non-moving particle is \(E = m_0 c^2\).

So because \(\left( c \Delta t, \, \Delta \mathbf{x} \right)\) can be described by all observers in all frames as being associated with a relativistic invariant \(c \Delta \tau\), so must \(\left( E = \gamma m_0 c^2 ,\, \mathbf{p}c = \gamma m_0 \mathbf{v} c \right)\) be associated with a relativistic invariant \(m_0 c^2\).

Which is just high school algebra. The interesting physics comes from the cases when \(E\) actually acts like energy and \(\mathbf{p}\) acts like a momentum, which is to say they are conserved. Proving this requires Noether's theorem and a physically tested Lagrangian. Basically, if the physics don't change over time, then the Energy is expected to be conserved, and if the the physics don't care where in the universe you are, then the momentum is conserved.

This bites us a second time in quantum physics where canonical conjugates like time and energy or position and momentum are related by the uncertainty principle.

According to Lev Okun, who I have stolen from before, all you need for the physics of a free particle are:

\(E^2 - \mathbf{p}^2c^2 = m_0^2c^4\)
and
\(\mathbf{p} = \frac{E \mathbf{v}}{c^2}\)

So Relativity gives us Minkowski space-time, and space-time gives us relations between \(E\) and \(\mathbf{p}\) and \(m_0\) and \(\mathbf{v}\). And we call \(E\) and \(\mathbf{p}\), Energy and Momentum because relativistic Lagrangians indicate they are conserved in the cases where Newtonian Lagrangians indicate that the Newtonian quantities \(\frac{1}{2} m_0 \mathbf{v}^2\) and \(m_0 \mathbf{v}\) are conserved.

Getting tired now... what about gravity?

If Newton was wrong about kinematics of fast-moving things, then logically, the original description of Universal Gravitation is suspect. Not only did Einstein correctly describe the action of gravity upon fast-moving objects, but he removed the feature Newton found philosophically objectionable about his own theory: Einstein's General Relativity is both in better agreement with experiment than Universal Gravitation and it is a local theory with no action-at-a-distance.

Will's famous reference article also states in axiomatic form what are the principles of General Relativity in simple terms. Let me simplify them one step more:

• The trajectory of a freely falling “test” body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition.
• The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed.
• The outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed.
• The Einstein tensor, which is the description of the trace of the curvature of space-time such that setting the derivative equal to zero is equivalent to saying the curvature of space-time obeys the Bianchi identities, is proportional to the local energy-stress tensor, for which setting the derivative equal to zero implies local conservation of energy. Thus local conservation of energy implies that the Bianchi identities hold (or vice-versa).

References

H. Fizeau "Sur les hypothèses relatives à l'éther lumineux" Annales de chimie et de physique 57 385-403 (1859) http://gallica.bnf.fr/ark:/12148/bpt6k347981/f381.table
W. A. von Ignatowsky, “Einige allgemeine Bemerkungen zum Relativitatsprinzip,” Phys. Z. 11, 972-976 (1910).
Y.S. Kim, M.E. Noz Theory and Applications of the Poincaré Group (1986) http://books.google.com/books?id=Ok6dPuqc8XMC
L.B. Okun "The Concept of Mass" Physics Today 42(6) 31-36 (1989) http://www.physicstoday.org/vol-42/iss-6/vol42no6p31_36.pdf
N Ashby "Relativity in the Global Positioning System" Living Rev. Relativity 6, 1 (2003) http://www.livingreviews.org/lrr-2003-1
P.B. Pal "Nothing but Relativity" Eur. J. Phys. 24 315-319 (2003) http://arxiv.org/abs/physics/0302045
R.C. Henry "Teaching Special Relativity: Minkowski trumps Einstein" http://henry.pha.jhu.edu/henryMinkowski.pdf
D. Mattingly, "Modern Tests of Lorentz Invariance", Living Rev. Relativity 8, 5 (2005) http://www.livingreviews.org/lrr-2005-5
M. Montesinos, E. Flores "Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem" Rev. Mex. Fis. 52, 29-36 (2006) http://arxiv.org/abs/hep-th/0602190
K. Brown "2.12 Thomas Precession" in Reflections on Relativity http://www.mathpages.com/rr/s2-11/2-11.htm
C.M. Will, "The Confrontation between General Relativity and Experiment", Living Rev. Relativity 9, 3 (2006) http://www.livingreviews.org/lrr-2006-3
Z.K. Silagadze "Relativity without tears" http://arxiv.org/abs/0708.0929

http://en.wikipedia.org/wiki/Aether_drag_hypothesis
http://en.wikipedia.org/wiki/Fizeau_experiment
http://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_energy-momentum_equation
http://en.wikipedia.org/wiki/Noether's_theorem

 

@rpenner

Wow, awesome! Did you copy that from something that you have pre-made or did you start writing your post when the tread started!!

 

Many, many many, thanks rpenner. Sorry to drag you in.

@Believe: I left a message on rpenner's notice board, and he jumped in. I wasn't expecting such a lengthy (or comprehensive) post

 

@rpenner - there's no question you know your physics! Great!! you're still a condescending douche that dwells on shit that was resolved for the chance to hear yourself think.

I can imagine the inner dialogue that brought you here... "oh look! There's a guy that has an interest in the logical and philosophical implications of physics! Rather than take part in a mutual interest and possibly show him an angle he might have missed or humor his idea (because after all he seems like a good ol' chap) I think I'll bludgeon him relentlessly with numbers that are meaningless to him so as to make up for how much I utterly hate myself!"

BRAVO!!! CONGRATULATIONS!!! You win the heads of two sycophants shoved firmly up your ass! :bravo::worship::bravo:

 

Dumb pillock. Rpenner posted exactly what YOU asked for.

Post #9.

Or are you so far gone in duplicity you don't even realise that?

 

It reeks of condescension the same way all of your posts do. If he had detailed the cure for aids my response would have only been slightly less caustic.

 

Ah, you asked to be condescended to?
Never mind, that turns out to be a rhetorical question given your "performance" in this thread.

 

First off form the limited portion that I read I am not sure they were saying that time is limited to human perception. But I admit I did not read the whole paper. What struck me initially was the connect between time and our experience of time or change. If what they were promoting is that time is imaginary and does not exist apart from our conception of it, I would say, "I'm pretty sure their cheese has slid off their cracker!" (that phrase makes me laugh every time, I can't help it)

Continuing and at the risk of over talking the issue a few thoughts...

Time as we experience it is a perceived event or events, just as everything else we see, hear, touch... And then assemble into a virtually reconstructed whole experience.

It does not mean that any of it, is not also a representation of the world outside of that process of perception and recognition.

I was just pointing out that at least in some sense all of this around us does exist within our minds. Our brains essentially assemble the ultimate 3-D virtual representation of the world, for is to play or try to survive in, which also allows us to have discussions like this one.

This was and is a comment or perspective drawn in part from the science of the mind and work of Ramachabdran, Zeman and Burton to name a few and a bit of philosophy of consciousness.

I did not mean to affirm anyone's assertion that time does not exist outside of experience and that inner virtual world, or that it is not valid to conceptualize it as a dimension, only to point out that time is a perception of change. We have chosen to label how we define the rate of change as time.

In the second paragraph of rpenner's above post he says, "the time experienced by a person", sometimes we forget when we are dealing with these discussions that most of what we are talking about is a description of experience, or an attempt to project some experience beyond the localized limitations of our ability to perceive the world.

I think I should probably just stop at that...

 

Ah, got you. Thanks for the clarification.
And may your cheese stay firmly on your cracker.

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BTW rpenner, that was a great post.

I was sad to find the first reference was not in English, but there are several references there that will keep me busy for a bit.

 

Actually, I was making a subtle dig at the Newtonian concept of absolute time when I referred to "the time experienced by a person" since both a Newtonian Physicist and a proponent of Relativity may sit down in the same place and experience to very high precision the same flow of time, but Special Relativity is about time dilation between moving clocks (or people) and General Relativity is about time dilation in different places (like different floors of a building). So should they sit down to chat they can agree that they spend an hour together. It's only if they are in radically different states of motion or gravitational potentials that the effects are significant -- to the other party. But to the person himself, nothing seems odd about his personal time or the clocks he keeps with him -- it's always the other guy's clocks that seem weird.

So both Newton and Einstein agree on the particular amount of "time experienced by a person" when that person is themselves or someone nearby and in the same state of motion -- it is the far-away person or the person with the different state of motion than oneself who has the interesting clocks -- clocks that Newtonian conceptions based on absolute time fail to explain.

It's this requirement putting oneself in the shoes of the other party that distinguishes the physics of Newton and his absolute time -- as if kept by some omnipresent authority everywhere in the universe -- from the local and velocity limited causality of Relativity. Thus in Relativity, one does not need to know the position of every particle of the universe to make a prediction -- just the particles in the past which have had the opportunity to communicate with the experiment in the here and now. For Newton, theoretically, today's stock market news on Antares IV could influence the growth of this spring's crops -- for the Relativist there can be no such causal connection since the universe is connected differently. Signals and influences only travel at the speed of light or slower.

 

I understood that I was using the quote out of context. I did not get the depth of the dig. But I do understand the relationship Newtonian vs relativistic.

Once again it was a very good post.

 

WOW, can you not even see when someone it trying to help you????? You made a premise and you have been shown otherwise. The person was not mean about it and in fact provided you with more information then you would ever need.

 

Argument Invalid man!

 

This is pretty typical of your posts so far. Care to elaborate?