
#109
Jan2206, 04:47 PM

P: 1,444

Randall  you are obviously going to have a difficult time undertanding the Gravity" book you borrowed  you are having a difficult time with my several statements to the effect that the resolution of the question doesn't involve anything but SR  first you accuse me of saying Relativity is Broken  title of another post  they you allege I am making a claim about GR  my reference to GR is strictly an analogy  nothing to do with it other than the fact that SR has in common with GR the fact that in some experiments things are not reciprocal.




#110
Jan2206, 05:19 PM

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Let's first consider the related concept of "Galileiinvariance", which is a bit simpler mathematically. The Galilei transform for transforming between different frames in Newtonian mechanics looks like this: [tex]x' = x  vt[/tex] [tex]y' = y[/tex] [tex]z' = z[/tex] [tex]t' = t[/tex] and [tex]x = x' + vt'[/tex] [tex]y = y'[/tex] [tex]z = z'[/tex] [tex]t = t'[/tex] To say a certain physical equation is "Galileiinvariant" just means the form of the equation is unchanged if you make these substitutions. For example, suppose at time t you have a mass [tex]m_1[/tex] at position [tex](x_1 , y_1 , z_1)[/tex] and another mass [tex]m_2[/tex] at position [tex](x_2 , y_2 , z_2 )[/tex] in your reference frame. Then the Newtonian equation for the gravitational force between them would be: [tex]F = \frac{G m_1 m_2}{(x_1  x_2 )^2 + (y_1  y_2 )^2 + (z_1  z_2 )^2} [/tex] Now, suppose we want to transform into a new coordinate system moving at velocity v along the xaxis of the first one. In this coordinate system, at time t' the mass [tex]m_1[/tex] has coordinates [tex](x'_1 , y'_1 , z'_1)[/tex] and the mass [tex]m_2[/tex] has coordinates [tex](x'_2 , y'_2 , z'_2 )[/tex]. Using the Galilei transformation, we can figure how the force would look in this new coordinate system, by substituting in [tex]x_1 = x'_1 + v t'[/tex], [tex]x_2 = x'_2 + v t'[/tex], [tex]y_1 = y'_1[/tex], [tex]y_2 = y'_2[/tex], and so forth. With these substitutions, the above equation becomes: [tex]F = \frac{G m_1 m_2 }{(x'_1 + vt'  (x'_2 + vt'))^2 + (y'_1  y'_2 )^2 + (z'_1  z'_2 )^2}[/tex] and you can see that this simplifies to: [tex]F = \frac{G m_1 m_2 }{(x'_1  x'_2 )^2 + (y'_1  y'_2 )^2 + (z'_1  z'_2 )^2}[/tex] Comparing this with the original equation, you can see the equation has exactly the same form in the primed coordinate system as in the unprimed coordinate system. This is what it means to be "Galilei invariant". More generally, if you have any physical equation which computes some quantity (say, force) as a function of various space and time coordinates, like [tex]f(x,y,z,t)[/tex] [of course it may have more than one of each coordinate, like the [tex]x_1[/tex] and [tex]x_2[/tex] above, and it may be a function of additional variables as well, like [tex]m_1[/tex] and [tex]m_2[/tex] above] then for this equation to be "Galilei invariant", it must satisfy: [tex]f(x'+vt',y',z',t') = f(x',y',z',t') [/tex] So in the same way, if we look at the Lorentz transform: [tex]x' = \gamma (x  vt)[/tex] [tex]y' = y[/tex] [tex]z' = z[/tex] [tex]t' = \gamma (t  vx/c^2)[/tex] where [tex]\gamma = 1/\sqrt{1  v^2/c^2}[/tex] and [tex]x = \gamma (x' + vt')[/tex] [tex]y = y'[/tex] [tex]z = z'[/tex] [tex]t = \gamma (t' + vx'/c^2)[/tex] Then all that is required for an equation to be "Lorentzinvariant" is that it satisfies: [tex]f( \gamma (x' + vt' ), y' , z', \gamma (t' + vx' /c^2 ) ) = f(x' ,y' ,z' , t')[/tex] There may be some more sophisticated way of stating the meaning of Lorentzinvariance in terms of group theory or something, but if an equation is Lorentzinvariant, then it should certainly satisfy the condition above. Maxwell's laws of electromagnetism would satisfy it, for example. And it's pretty easy to see that if it satisfies this mathematical condition, then the equation must have the same form when you transform into a different inertial frame using the Lorentz transform. So this is enough to show beyond a shadow of a doubt that given Lorentzinvariant fundamental laws, all the fundamental laws must work the same in any inertial reference frame, and if you know the equation for a given law as expressed in some particular inertial frame (the rest frame of the center of the earth, for example) then it is a straightforward mathematical question as to whether or not this equation is Lorentzinvariant, it's not an experimental issue (the only experimental issue is whether that equation makes correct predictions in the first place). Do you disagree with any of this? Also, the GPS clocks are programmed to adjust themselves so that they tick at a constant rate in the frame of the earth. My other point was that this is a completely arbitrary choice made by the designers, you could just as well design the orbiting GPS clocks to adjust themselves so that they tick at a constant rate in the frame of an inertial observer moving at 0.99c relative to the earth. Would you then say it is not "permissible" to analyze these clocks in the rest frame of the earth, since they would not be running at a uniform rate in the earth's frame? 



#111
Jan2206, 06:22 PM

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If this roundtrip time is always constant, we can always compare the rate of two clocks unambiguously. We may need to demand that the roundtrip time is constant for both A and B before we can compare rates, but I think it is true that if A's round trip time is constant, so is B's. I'm relying on my intuition a bit here, though. Of course A and B don't necessarily have to agree on the value of the roundtrip time (and in general they won't)  they just have to agree that it's constant. 



#112
Jan2206, 06:39 PM

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PF Gold
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#113
Jan2306, 09:05 AM

P: 1,545

What I’d wish you’d do tell us what your trying to do in your posts, they seem to flip from one objective to a the other. Are you: A) trying to convince others that the standard view of SR (and extending into GR as well) is somehow wrong or incomplete. And your 'LRlike' view or something else must be better. OR B) sincerely trying to learn SR completely, to filling the gaps of information about it, that leave you unable to see concepts in books you’ve had for 10 years. For the sake of all those that are trying to respond to you, be clear on this; are you arguing a point of view; or trying to learn something. And please refrain on saying “yes & no” or “both”, do one or the other. 



#114
Jan2306, 10:38 PM

P: 1,444

I have no agenda Randall  my interest in SR goes back many years  likely before you were born  I pose questions that come to mind  if those questions lead to a different view ...one I have overlooked  great. Usually what happens on these boards is an attack  either I don't understand it or I am not qualified to question it  but for a few posters, the attitude is always one of condesention.




#115
Jan2306, 10:43 PM

P: 1,444

If anyone thinks it is easy to synchronize GPS satellite clocks in some frame than the nonrotating earth centered system I would like to see how you would go about doing it.




#116
Jan2306, 10:53 PM

P: 1,444

Hurkyl  The transforms relate time and distances  they are not abstract mathematical artifact  these are physical things  my point is with you and Jesse  the mathematical relationships (LT) have been confirmed in certain experiments  but those experiments do not test the fundamental premise upon which Einstein's derivation was based  perhaps w/o complete justification, I do have a stong conviction that the spacetime interval is invarient.




#117
Jan2306, 11:04 PM

P: 1,444

pervect  i would agree that any interrogation must depend upon the constancy of the round trip time  and since the GPS clock will always be found to be running at a uniform rate relative to the ground station  we have a convincing demonstration of the constancy of c  at least in the earth centered frame




#118
Jan2306, 11:48 PM

P: 1,444

Jesse  your post 110  last Paragraph: Transforming to another frame  what I have been trying to discuss was the non reciprocal reality of time dilation in certain experiments  if we have a GPS clock that was originally at the top of the tower and launched into orbit from there with no velocity correction  we have a situation where the ground clocks run consistently fast wrt to the GPS satellite clock and the GPS satellite clock always runs slow wrt to the ground clocks and we can verify this non symmetrical situation by interrogating the GPS clock with radio signals  that is the experiment. To inquire as to transforming to a frame in high speed motion wrt to the earth simply subverts the objective. Such a transformation of course is possible, but you have missed the whole point we are now back in a situation where whatever is measured is apparent  the two frames have not been synchronized  Einstein only makes a prediction about real time dilation when one of two originally synchronized clocks is accelerated to a uniform velocity wrt the other. This is the subject of the thread  which i would like to explore further.




#119
Jan2406, 12:57 AM

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#120
Jan2406, 01:20 AM

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Do you agree that if the GPS clock is orbiting the earth at a constant speed in the centeroftheearth frame, that means that in other inertial frames the speed of the GPS clock is not constant? (This would be just as true in Newtonian mechanics as in relativity, of course.) Do you agree that if each inertial frame assumes the same relationship between instantaneous speed in that frame and instantaneous rate of ticking (ie that if the clock is moving at speed v in that frame it will be slowed down by a factor of [tex]\sqrt{1  v^2/c^2}[/tex]), then different inertial frames will disagree about the relative rate of the tower clock and the orbiting clock at a given moment (with 'given moment' meaning something different in different frames too, due to different definitions of simultaneity), yet they will all make the same prediction about how far behind the orbiting clock will be at the moment it completes an orbit and reunites with the tower clock at a single point in space? Please, please give me direct answers to this question, when I ask you questions in my posts they are not meant to be rhetorical, and it's incredibly frustrating when I ask you questions that I hope will help pin down your nebulous comments and you just ignore them and comment on a single statement in my post. 



#121
Jan2406, 08:09 AM

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PF Gold
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The transforms relate (inertial) coordinate systems, which most certainly are "abstract mathematical artifact". E.g., IIRC, Einstein made a big deal about showing how to construct what he called an inertial reference frame, via a hypothetical network of clocks, each of them "at rest" with a given inertial observer, and "synchronized" with his clock. I put those terms in quotes because those are also terms in need of construction. If coordinate systems were physical things, Einstein would have just said "measure it". 



#122
Jan2406, 09:37 AM

P: 1,545

I understand I don’t have the “pro” tag, banner or ribbon of the mentors & advisors on this board  so you don’t have to take my encouragements to heart. As to reciprocal views being relative – Based on how you pose those questions as conclusions, from the frame of reference of many trying to respond, it can often be seen as attacks on SR as they know it to be. So that part seems to be the same from both views. 



#123
Jan2406, 08:25 PM

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PF Gold
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The transforms do relate coordiante position, and coordinate time, so in some sense they can be said to relate times and distances. But my point still stands  coordinate position and coordinate time are derived from coordinates, which are not physical things. Case in point: the Julian and Gregorian calendars are not the same  they^{1} assign different time coordinates to events. Would you say that the transformation between these calendars is a physical thing? 1: I'm assuming any usual^{2} method of identifying spatial position. A calendar, by itself, is unable to assign time coordinates to almost every event! 2: Notice I said usual, and nothing such as "intrinsic" or "determined by reality"  it's a convention we use as humans. And it's even changed over time, such as when time zones were instituted, and when calendars were changed! 



#124
Jan2406, 10:51 PM

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Hopefully the point I wanted to make has been made, though. 



#125
Jan2406, 11:49 PM

P: 1,444

Pervect  I appreciate your comments and frequently read your posts although a cannot always make a sensible response. See Below




#126
Jan2506, 12:19 AM

P: 1,444

Pervect  I appreciate your comments and frequently read your posts although often cannot make a worthwhile response.
I do recall at one time in another thread you made a statement that the time discrepancy is physically explainable  but then the direction of the thread veered off along other lines as is often the case  so i wasn't able to get back to the subject In my previous posts on this thread I have tried to target the situation Einstein created when he combined the relationships that were obtained from observations in another frame  specifically to my way of thinking there is a quantum jump from the notion of two observers in relative motion drawing identical conclusions about lengths and clock rates in the other frame  as opposed to the idea that one of two synchronized clocks put in motion will run slower than the one that was not moved. There is a real loss of time between the clocks when they are later compared  in other words, did Einstein get the right answer for the wrong reason  or by intuition. He does not attempt to justify or even rationalize this  it works to explain the experiments  but it doesn't follow from the initial postulates of SR  in one sense it seems to require its own special postulate and  i guess what i am saying is that something is missing. 


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