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# Another way of looking at Special Relativity

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On 11/3/2019 at 6:42 PM, studiot said:

In a consolidated spacetime you do not have separate length and time meters, the basic measuring device measures spactime units (the spacetime interval, s),  in any direction in the spacetime continuum.

The Lorenz transformation becomes a (Euclidian) metric for this continuum.

Since all axes are now on an equal footing they fade into the background (where all good frames of reference belong) and what is important is the configuration of events (or points in the continuum if you will)

Hi studiot.  Just verifying ... The LTs are governed by the Minkowski Metric.  You said "Euclidean metric" above.  Did you mean "becomes a Euclidean-like metric", or were you thinking when relative velocity is zero, whereby the Minkowski metric reduces to the Euclidean metric (maybe since RAGORDON2010 had discussed 2 experiments of the same frame prior)?

Best Regards,

Celeritas

Edited by Celeritas

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1 hour ago, Celeritas said:

Hi studiot.  Just verifying ... The LTs are governed by the Minkowski Metric.  You said "Euclidean metric" above.  Did you mean "becomes a Euclidean-like metric", or were you thinking when relative velocity is zero, whereby the Minkowski metric reduces to the Euclidean metric (maybe since RAGORDON2010 had discussed 2 experiments of the same frame prior)?

Best Regards,

Celeritas

Sure thing.

A 'space' is a set of points with particular properties.
If one of those properties is a 'distance function' then it is called a metric space and the distance function is called the metric.
A distance function applies in an identical manner to all possible pairs of points in the set with a positive definite ( greater than zero) result - it can never be negative.
(but not necessarily an identical result this distance of London from Rome is not the same as the distance of London from New York, though each mile is the same as any other mile).
It is zero only in the case of the the 'distance' of a point from itself.

A Euclidian space is a metric space with the Euclidian metric which is

${d_{AB}} = \sqrt {\sum\limits_0^n {{{\left( {x_B^n - x_A^n} \right)}^2}} }$

for an n dimensional Euclidian space (conventionally counting dimensions as x0, x1,  x2.......etc
Note here that n is not a power but an indexing label for the dimension.

For Minkowski space with axes labelled  x,y,z and tau this becomes

$s = \sqrt {{{\left( {\Delta x} \right)}^2} + {{\left( {\Delta y} \right)}^2} + {{\left( {\Delta z} \right)}^2} + {{\left( {\Delta \tau } \right)}^2}}$

Which is the formula for the invariant spacetime interval s.

So 's' is the Euclian metric on spacetime from which we derive Lorenz.

A student of General Relativity would say that GR reduces to flat (= Euclidian) geometry when we consider local (infinitesimal) effects and SR is 'flat' spacetime.
This is good because GR is amenable to differential geometry because this is true.

Edited by studiot

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@studiot    is there a particular reason for using +(ict)^2 (or +τ^2 ) than -(ct)^2?

It is not just for the sake of "tidiness" ,is it?

Do you need all elements of the metric need to be positive for it to be called Euclidean?

Does this make the maths easier?

Edited by geordief

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7 hours ago, geordief said:

@studiot    is there a particular reason for using +(ict)^2 (or +τ^2 ) than -(ct)^2?

It is not just for the sake of "tidiness" ,is it?

Do you need all elements of the metric need to be positive for it to be called Euclidean?

Does this make the maths easier?

Thank you for the query I will expand the explanation, but first an apology to all.

I realised last night that a silly error crept into my first formula which should read

${d_{AB}} = \sqrt {\sum\limits_0^{\left( {n - 1} \right)} {{{\left( {x_B^n - x_A^n} \right)}^2}} }$

This is because for some reason the pundits chose to start counting axes and dimensions from zero not 1 so the sum should be from zero to (n-1) not to n.
So for 4 dimensions (Minkowski) we have x0, x1, x2 and x3

As to your second point, if some of the 'x's had a minus sign, how would you knowwithout writing out the formula in full?

For the space to be Euclidian, every axis has to have equal value or weight ie the same physical dimension.
A Euclidian space is a geometric space and you can't do geometry in a space where some axes measure length, some say time and some say something else eg temperature.

So it is perfectly acceptable to use x0, x1, x2 and x3 or x,y,z,ct or x,y,z,ict since all these spaces have the same underlying geometry, provided you write out the metric formula correctlyin full.
To do this you would need either the first or last to also use the general formula.

But it would not be correct to call x,y,z,t Euclidian.

The advantage of the general formula is that you can explore the effect different numbers of spatial and time dimensions.
Eddington considers this

You can also do something further.

Since when using x0, x1, x2 and x3 the axes are not only physically equivalent, but symbolically equivalent as well you can explore more complicated metric formulae.

Eddington does this as well. The Euclidian Minkowski metric is called the 'four squares metric'.
Squares imply quadratics and Eddington considers the general quadratic and the implications of reducing it to the four squares one for Relativity.

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On 11/8/2019 at 8:07 AM, Eise said:

I have: Abraham Pais, Subtle is the Lord, page 152. Einstein called Minkowski's 4-dimensional spacetime 'superfluous learnedness' (I just googled and only found a citation. I can look it up at home).

I have the German translation of Pais' Subtle is the Lord, so I will do my best. In the paragraph before the sentences I will try to translate, Abraham Pais discusses the new formalism of Minkowski, which obviously also included the use of tensors. Then Pais goes on, and says:

Quote

With that the tremendous formal simplifying of special relativity began. First, Einstein was not impressed, and saw the reforming of his theory in tensor form as 'superfluous learnedness'.

(Quotes also in the original). As source, Pais refers to a personal conversation with V. Bargmann, of which Wikipedia says:

Quote

At the Institute for Advanced Study in Princeton (1937–46) he worked as an assistant to Albert Einstein, publishing with him and Peter Bergmann on classical five-dimensional Kaluza–Klein theory (1941). He taught at Princeton University since 1946, to the rest of his career.

So it seems Einstein was referring to the tensor formulation of special relativity, not to the description of spacetime by Minkowski.

Edited by Eise

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41 minutes ago, Eise said:

I have the German translation of Pais' Subtle is the Lord, so I will do my best. In the paragraph before the sentences I will try to translate, Abraham Pais discusses the new formalism of Minkowski, which obviously also included the use of tensors. Then Pais goes on, and says:

(Quotes also in the original). As source, Pais refers to a personal conversation with V. Bargmann, of which Wikipedia says:

So it seems Einstein was referring to the tensor formulation of special relativity, not to the description of spacetime by Minkowski.

Is it quite easy to research this "tensor reformulation of SR" (googling) or might a few pointers be in order?

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39 minutes ago, geordief said:

Is it quite easy to research this "tensor reformulation of SR" (googling) or might a few pointers be in order?

1 hour ago, Eise said:

I have the German translation of Pais' Subtle is the Lord, so I will do my best. In the paragraph before the sentences I will try to translate, Abraham Pais discusses the new formalism of Minkowski, which obviously also included the use of tensors. Then Pais goes on, and says:

Doubtless a good book, and widely available in English.

As regards the tensor research I would counsel geordie not to go there.
Particularly as that would invoke the einstein summation convention. which is difficult to follow.
There is no gain to be had from the use of tensors in Special Relativity.
In General Relativity their use tends to obscure the Physics.

However I am with the great man himself when he wrote this book

This slim volume in my view "weighs an ounce, but contains a pound of wisdom"

In it Einstein explains, largely in words, the thinking and reasons behind his development and presentation of both SR and GR.
Originally written in 1916 I show the 15th (1952) edition in the attachment.
I don't think he mentions the word tensor once in the book although he does reproduce the GR tensor metric

$d{s^2} = {g_{ik}}d{x_i}d{x_k}$

and then points out that this seemingly innocous equation actually represets 44 equations in compact form.

But his development of the subject in an appendix called relativity of and the problem of space in words is the clearest I have seen.

Edited by studiot

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On 11/4/2019 at 12:19 AM, RAGORDON2010 said:

Just a cursory review of questions students ask about slow clocks and shrinking meter sticks illustrate the depth of the confusion.  In the related experiments approach, there are only one set of clocks, one set of meter sticks and, of course, one set of whatever additional laboratory apparatus is needed.  My clocks do not run fast or slow or whatever, my meter sticks do not shrink or grow or whatever, etc.  In my Monday/Tuesday imagery, I only flip a page on the calendar.

Hello. I've read through this interesting thread and have some follow up questions. First; alternative approaches and analogies for explaining relativity (or science in general) are good*. The concepts of no universal "now", time dilation and length contraction are AFAIK central parts of understanding SR. Why are students asking about them to understand more a problem? Why does it help to remove those concepts? If I try to put my self into the role of a beginner, studying relativity: length contraction / time dilation is not a point of view, it is what actually happens according to the theory and its supporting experimental results.

*) As long as such approaches and analogies are used within their respective area of applicability and not as replacement for models and math or contradict mainstream observations.

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1 hour ago, Ghideon said:

Hello. I've read through this interesting thread and have some follow up questions. First; alternative approaches and analogies for explaining relativity (or science in general) are good*. The concepts of no universal "now", time dilation and length contraction are AFAIK central parts of understanding SR. Why are students asking about them to understand more a problem? Why does it help to remove those concepts? If I try to put my self into the role of a beginner, studying relativity: length contraction / time dilation is not a point of view, it is what actually happens according to the theory and its supporting experimental results.

*) As long as such approaches and analogies are used within their respective area of applicability and not as replacement for models and math or contradict mainstream observations.

Good questions, why shouldn't students ask questions? +1

4 hours ago, Eise said:

I have the German translation of Pais' Subtle is the Lord, so I will do my best. In the paragraph before the sentences I will try to translate, Abraham Pais discusses the new formalism of Minkowski, which obviously also included the use of tensors. Then Pais goes on, and says:

Another point about Pais was that Wikipedia says he was uniquely placed to write these histories as he knew many if not most of the important player.

That remark shows just how polarised societies were back then. He knew none of the English speaking world at that time here is a comment from Ferreira that also mentions the imfamous

"superfluous erudition" but he goes on to show how Hilbert re-educated Einstein (circled at the end)

The extract also shows Eddington's position and contribution.

3 hours ago, studiot said:

I don't think he mentions the word tensor once in the book although he does reproduce the GR tensor metric

ds2=gikdxidxk

I think I should have made clear that ds2 is the Minkowski invariant.

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I'd like to add some details to Studiots posts in regards to the function of the metric. Though I will add the metric tensors. The tensors are useful for handling multiple unknowns.

The main purpose of the metric is to define the coordinate changes.

For example if you have Euclidean flat (Cartesian) coordinates the metric tensor takes the form.

$dx^2=(dx^0)^2+(dx^1)^2+(dx^2)^2(dx^3)^2$

In Cartesian coordinates the metric takes the form

$G_{\mu\nu}=\begin{pmatrix}g_{0,0}&g_{0,1}&g_{0,2}&g_{0,3}\\g_{1,0}&g_{1,1}&g_{1,2}&g_{1,3}\\g_{2,0}&g_{2,1}&g_{2,2}&g_{2,3}\\g_{3,0}&g_{3,1}&g_{3,2}&g_{3,3}\end{pmatrix}=\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

Now as Studiot mentioned infinitisimals to see the metric tensor in differential form which is used to handle infinitisimals.

$\frac{dx^\alpha}{dy^{\mu}}=\frac{dx^\beta}{dy^{\nu}}=\begin{pmatrix}\frac{dx^0}{dy^0}&\frac{dx^1}{dy^0}&\frac{dx^2}{dy^0}&\frac{dx^3}{dy^0}\\\frac{dx^0}{dy^1}&\frac{dx^1}{dy^1}&\frac{dx^2}{dy^1}&\frac{dx^3}{dy^1}\\\frac{dx^0}{dy^2}&\frac{dx^1}{dy^2}&\frac{dx^2}{dy^2}&\frac{dx^3}{dy^2}\\\frac{dx^0}{dy^3}&\frac{dx^1}{dy^3}&\frac{dx^2}{dy^3}&\frac{dx^3}{dy^3}\end{pmatrix}$

Now for Minkowskii metric the above will correspond to the following

$ds^2=-c^2dt^2+dx^2+dy^2+dz^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$

$\eta=\begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

Now when you have say spherical coordinates the metric changes.

$(x^0,x^1,x^2,x^3)=(\tau,r,\theta,\phi)$

$G_{a,b} =\begin{pmatrix}-1+\frac{2M}{r}& 0 & 0& 0 \\ 0 &1+\frac{2M}{r}^{-1}& 0 & 0 \\0 & 0& r^2 & 0 \\0 & 0 &0& r^2sin^2\theta\end{pmatrix}$

Now all the above simply described the metric changes. That is the primary purpose of the metric tensor.

Edited by Mordred

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59 minutes ago, Mordred said:

The tensors are useful for handling multiple unknowns

As the Man said, up to 44 of them.

Thanks for saving me a deal of wristache. +1

But please note that 'the metric' need not be a tensor its definition is much wider than that.

An acceptable metric on the real numbers is the modulus function of the difference between any two.

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7 hours ago, studiot said:

But please note that 'the metric' need not be a tensor its definition is much wider than that.

An acceptable metric on the real numbers is the modulus function of the difference between any two.

Absolutely correct +1. A side note those tensors in the format I placed them shows how the metric tensor has coordinate independance.

The line element of a metric is also incredibly useful. Any time you see ds^2 you are reading the metric in terms of

Seperation distance between two points. The line element describes the worldline of massless particles between to events. For Minkowskii the line element being

$ds^2=-c^2dt^2+dx^2+dy^2+dz^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$

The line element for the FLRW metric which includes curvature options positive and negative

d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]

$S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}$

Wiki has a decent list of the common 3d and 4d line elements (it also includes the Kronecker delta and Levi Civitta connections) which is a topic for its own thread lol.

Edited by Mordred

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When tensors are used is it the case that ,in electromagnetism the calculation is the same whether the FOR is that of the moving charge  or the "static" conductor?

That is what I seem to take from Markus Hanke's  blog

(about 1/3 of the way down that page)

Would seem to be a very interesting property of tensors (don't worry Studiot,I am not banging my head against that brick wall at the moment,although I did try to learn a bit  about them  some time ago)

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Yes in sense that the tensors employ  vectors and 1 forms to produce a scalar product. I should note that the EM tensors are not orthogonal where as the metric tensor is in the above examples. You can recognize an orthogonal tensor at a glance.

$eta=\begin{pmatrix}-c&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$

All non zero components are in the diagonal entries.

Markus has done an excellent job on that blog. However to fully understand it start with the inner product and cross product of vectors then look into one forms.

The product of a vector and a Covectors returns a scalar same way as the dot or inner product of two vectors.

Here is a quick overview in the earlier paragraphs.

For the Minkowskii tensor above were employing the inner (dot) product of two vectors $\mu \cdot \nu$ to produce a scalar this will correspond to your rows and columns of the tensor.

The tensor is orthogonal and symmetric when

$\mu \cdot \nu=\nu \cdot \mu$ all orthogonal matrix or tensors are symmetric.

I will also present the following MIT course not to scare you off but you provide a good guide into the topics to study as you go.

Edited by Mordred

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Perhaps Mordred has been reading this book.

This 1962 book is the only one I have ever seen that follows Mordred's route and also explains the difference between a component and a resolute (what you get when resolving a vector), and its importance.
Most applied tensors are cartesian tensors, which are othogonal. In this case there is no difference but in the case of skew axes (non orthognal) the difference is vitally important.
The book covers Maxwell rather well.

It is also interesting to note how far mathematical expectations have come since 1962 I have included the target audience on the left.

Other good easy expositions in applied science are

Cartesian Tensors in Engineering Science by Jaeger

Cartesian tensors by Harold Jeffreys

and a bit more advanced

Cartesian Tensors an introduction by Temple

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Would you   reccommend "Gravitation" by Misner,Thorne  et al?

I have read it has a good section on vectors/covectors.

I will definitely be ordering your last suggestion , E's "Popular Exposition" (embarrassed I haven't had anything like that in the house ever)

Edited by geordief

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2 hours ago, geordief said:

Would you   reccommend "Gravitation" by Misner,Thorne  et al?

I have read it has a good section on vectors/covectors.

I will definitely be ordering your last suggestion , E's "Popular Exposition" (embarrassed I haven't had anything like that in the house ever)

Sorry I don't know the book, perhaps Mordred ?

One thing though.
I would counsel caution using the explanation in any book written for another purpose.
Not that it will necessarily be wrong but it will be tailored to the main application of that book and may well be incomplete as a result.
We currently have a thread about boiling soup (water) due to someone applying (perfectly good) theory from somewhere else way outside the boundaries of its applicability with comic results.

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Yes Gravitation by Mister, Thorne is a good book. I haven't read the one Studiot posted. However another book with a good chapter in one forms is A First Course to General Relativity by Bernard Shultz.

One forms I consider an essential lesson to better understand the covariant and contravariant terms.

See here for example

You may note Misner is referenced in the vector component graph at top right.

Edited by Mordred

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When I joined this forum, I was hoping for the chance to engage in an intelligent conversation about issues in Special Relativity that have confused new students for decades.  Instead, I have found myself talking to walls. So I will be exiting from this forum, but I want to leave three parting thoughts:

Thought 1 -

Despite how many people say it for other people to hear,

despite how many people write it for other people to read,

despite how many people key it for other people to link to -

The Fact of Nature that the speed of light is the same in all inertial frames plays NO role in explaining the successful application of Special Relatively to solving physical problems!  It is a Red Herring! There is another Fact of Nature at work here.

What other Fact of Nature?  Well, I’ve hidden it in plain sight in Thought 2.

Thought 2 -

"Slip slidin' away.

Slip slidin' away.

The nearer your destination,

The more you're slip slidin' away."

- Paul Simon

I've been hoping to offer some thoughts on "relativistic mass" - the notion that mass increases with increasing speed.

It's common to come across the statement that accelerating a particle becomes more difficult as particle speed approaches c because "particle mass approaches infinity".  I prefer to state the issue differently.

I would say that accelerating a particle becomes more difficult as particle speed approaches c because the external field responsible for the acceleration loses effectiveness as the particle speed approaches the speed at which the field mechanisms function.

This, of course, offers an explanation for why light speed forms a limiting speed in nature.  An old boot can travel no faster through the water than the maximum speed at which the fisherman can reel in the line.

Whenever I think about this phenomenon, Paul Simon's song comes to mind. The speeding particle slips and slides away from the grasp of the external field.

Thought 3 -

I can’t leave this forum without saying something about time dilation.  It has always puzzled me that while the physics community easily accepts that time dilation effects in General Relativity relate in some way to the interaction between the time-keeping system and the surrounding gravitational field, the analogous time dilation effects in Special Relativity are viewed as “just so”.  Well, I have never cared much for a “just so” story.  But I do hold the view that Nature does not care at all for a “just so” story.  Something is going on out there!

In the most dominant example - the retarded decays of unstable particles moving at speeds close to light speed - I again must fall back on my belief that these effects are in some way a consequence, in ways not at all understood, of the rapid motion of the particles through surrounding electric and magnetic fields.

I hold (and this is where Special Relativity exhibits its most severe vulnerability as it is commonly described) that no physical effect can occur as a consequence of merely moving at a uniform speed in an inertial frame of reference.

And now, from sunny Alabama (a state in the USA, refer Rand McNally maps, circa 1934), I happily say GOODBYE, Y’ALL!!

ROLL TIDE!!

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54 minutes ago, RAGORDON2010 said:

When I joined this forum, I was hoping for the chance to engage in an intelligent conversation about issues in Special Relativity that have confused new students for decades.  Instead, I have found myself talking to walls.

Ok.I was hoping for a sincere discussion about helping new students, not speculations.

59 minutes ago, RAGORDON2010 said:

I hold (and this is where Special Relativity exhibits its most severe vulnerability as it is commonly described) that no physical effect can occur as a consequence of merely moving at a uniform speed in an inertial frame of reference.

Thanks for answering my question.

53 minutes ago, RAGORDON2010 said:

GOODBYE, Y’ALL!!

Understating present theories would be a good start if you wish to extend or replace something well supported like SR.
Why not stay around, ask questions, and "engage in an intelligent conversation about issues in Special Relativity"?

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On 11/8/2019 at 7:54 PM, studiot said:

For Minkowski space with axes labelled  x,y,z and tau this becomes

s=(Δx)2+(Δy)2+(Δz)2+(Δτ)2

Which is the formula for the invariant spacetime interval s.

So 's' is the Euclian metric on spacetime from which we derive Lorenz.

Studiot ... Did you mean ... s = √( ∆x² + ∆y² + ∆z² + ∆t²) ?

Best regards,

Celeritas

Edited by Celeritas

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3 hours ago, RAGORDON2010 said:

When I joined this forum, I was hoping for the chance to engage in an intelligent conversation about issues in Special Relativity that have confused new students for decades.  Instead, I have found myself talking to walls. So I will be exiting from this forum, but I want to leave three parting thoughts:

Thought 1 -

Despite how many people say it for other people to hear,

despite how many people write it for other people to read,

despite how many people key it for other people to link to -

The Fact of Nature that the speed of light is the same in all inertial frames plays NO role in explaining the successful application of Special Relatively to solving physical problems!  It is a Red Herring! There is another Fact of Nature at work here.

What other Fact of Nature?  Well, I’ve hidden it in plain sight in Thought 2.

Thought 2 -

"Slip slidin' away.

Slip slidin' away.

The nearer your destination,

The more you're slip slidin' away."

- Paul Simon

I've been hoping to offer some thoughts on "relativistic mass" - the notion that mass increases with increasing speed.

It's common to come across the statement that accelerating a particle becomes more difficult as particle speed approaches c because "particle mass approaches infinity".  I prefer to state the issue differently.

I would say that accelerating a particle becomes more difficult as particle speed approaches c because the external field responsible for the acceleration loses effectiveness as the particle speed approaches the speed at which the field mechanisms function.

This, of course, offers an explanation for why light speed forms a limiting speed in nature.  An old boot can travel no faster through the water than the maximum speed at which the fisherman can reel in the line.

Whenever I think about this phenomenon, Paul Simon's song comes to mind. The speeding particle slips and slides away from the grasp of the external field.

Thought 3 -

I can’t leave this forum without saying something about time dilation.  It has always puzzled me that while the physics community easily accepts that time dilation effects in General Relativity

Time dilation is well tested regardless of your beliefs so your doubts concerning time dilation is the issue not the theory.

You evidently have never connected the mass term. Resistance to inertia change to the numerous field coupling constants and the interval term described by the ct coordinate interval.

Even with EM signals propogation delays occur its fundamentally the same principle.

It's too bad you choose to leave instead of learning your loss

Edited by Mordred

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4 hours ago, RAGORDON2010 said:

When I joined this forum, I was hoping for the chance to engage in an intelligent conversation about issues in Special Relativity that have confused new students for decades.  Instead, I have found myself talking to walls.

And yet it is you that refuses to explain how your idea has any connection to SR. Not surprising as (a) it doesn’t and (b) you very obviously don’t have a clue.

4 hours ago, RAGORDON2010 said:

The Fact of Nature that the speed of light is the same in all inertial frames plays NO role in explaining the successful application of Special Relatively to solving physical problems!

Clearly, you are unable to understand the simple mathematics that shows this claim to be false.

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5 hours ago, RAGORDON2010 said:

When I joined this forum, I was hoping for the chance to engage in an intelligent conversation about issues in Special Relativity that have confused new students for decades.  Instead, I have found myself talking to walls. So I will be exiting from this forum, but I want to leave three parting thoughts:

Well the loss is entirely yours.

I would like you to thank you for introducing the subject since several other members have found something of interest to discuss in it.
So we have all gained from it, as you could have done.

2 hours ago, Celeritas said:

Studiot ... Did you mean ... s = √( ∆x² + ∆y² + ∆z² + ∆t²) ?

Best regards,

Celeritas

No I meant substitute a new variable tau.

It is conventional to introduce the new variable tau to distinguish it from time.
tau has the units of length as do the three spatial coordinates.
This is the necessary and sufficient for treating them all equally.
It is necessary to introduce 'i' to make the coefficient of the square negative.

τ = ict

Does this help ?

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6 hours ago, RAGORDON2010 said:

The Fact of Nature that the speed of light is the same in all inertial frames plays NO role in explaining the successful application of Special Relatively to solving physical problems!  It is a Red Herring! There is another Fact of Nature at work here.

And yet you cannot define nor describe this mysterious fact of nature.

Claims without support is one most common Speculative theory poster's we receive on forums. It is a rare poster that can back up his against mainstream physics regardless of topic.

You don't  need the speed of light to prove the speed limit set by the constant c. The addition of relativistic velocities can show that even with massive objects.

51 minutes ago, studiot said:

Well the loss is entirely yours.

I would like you to thank you for introducing the subject since several other members have found something of interest to discuss in it.
So we have all gained from it, as you could have done.

No I meant substitute a new variable tau.

It is conventional to introduce the new variable tau to distinguish it from time.
tau has the units of length as do the three spatial coordinates.
This is the necessary and sufficient for treating them all equally.
It is necessary to introduce 'i' to make the coefficient of the square negative.

τ = ict

Does this help ?

To provide better clarity. You need the new viriable to distinguish coordinate time from proper time.

A helpful hint past the oft over complicated distinction between the two.

Coordinate time is the time at a specific coordinate event. While proper time is any location along the worldline between any two events. (Emitter,observer). Where  [math]\tau[\math] is the proper time. The further qualification is that proper time is the invariant time where all observers can agree upon. This is set by the Einstein synchronization rules.

FYI edit side note. I always find it amusing in regards to time dilation. The real secret to understand it has been interaction causality/signal delays. You simply have to correlate it to the signal delays of all fields at once in a given volume. It is the sum of all field interactions that make  the dynamics of spacetime. The Standard model comprises of 18 coupling constants that correlate to mass terms and hence the signal delays.

Edited by Mordred

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