Neutral simultaneity for two frames.

[Halc]

 This two-year old topic started with the below comment.  I've not read almost any of it, but I see a lot of word salad that seems unrelated to this original question.

On 12/9/2022 at 2:01 PM, DimaMazin said:

If we want to accelerate very long train,

instantly accelerating each railway carryage into a moving frame at v, then...

1) No rigid object can instantly change its velocity without breaking, per Bell's spaceship scenario.  You can accelerate it over time (finite proper acceleration), and how much time that takes depends on what clock is used to do it.

There are limits. I worked out the minimum time it takes to move a 100 LY rigid train a distance of one light day, being stopped at beginning and end of trip. It takes almost 2 months and cannot be done faster without violating rigidity. It cannot be done at all without applying force to all parts of the object instead of say pulling it with an engine up front.

 

Now regarding this latest post, forgive me if I am unaware of any context that might help make sense of any of it.

1 hour ago, DimaMazin said:

Let's use velocity u as distance

How can a velocity be treated as a distance? This seems meaningless.

1 hour ago, DimaMazin said:

addition of any velocity, even when it is as nonsimultaneity bigger than c

How does a simultaneity or a nonsimultaneity have a size?  Two events not simultaneous in some frame would have a time difference in time, but that difference would be a time, not a speed. You are seemingly comparing a time to a speed, which is total nonsense.

How is any of the posts (in any of 2024) relevant to the topic?  Is this just one of those blogs left open to keep the forum from filling up with dozens of crazy topics from one user?

[Mordred]

45 minutes ago, Halc said:

 

How is any of the posts (in any of 2024) relevant to the topic?  Is this just one of those blogs left open to keep the forum from filling up with dozens of crazy topics from one user?

It's also a thread I gave up on getting the notion the convection of simultaneity loses any practicality in curved space-time. So it's essentially only applicable to Maximally symmetric spacetimes such as Minkowsii. You can consider the local vs global ramifications on that.

At the OP the norm of the length is not the same as the norm of a vector 

Length alone won't include changes in direction 

Ie rotations acceleration due to change in direction as opposed to boosts change in magnitude of velocity.

 

Edited November 2 by Mordred

[Mordred]

@DimaMazin I would like you to consider the following scenario

now in the above signals sent by A and signals sent by B will be received at point P simultaneous as point P is the center between A and B. When you have a change in the magnitude component of velocity although you have length contraction this will not change. Treat this as 3 observers/emitters A,B,P

However if that train goes around a corner which is also acceleration this is no longer true.

If you were to draw a line between A and B Observer P will not fall on that line. It would not receive that signal. Instead P will only receive signals along the line from A to P and B to P but not the signal between A and B. 

So you are no longer dealing with one geodesic path of light but now three separate geodesic light paths. The geodesic path between A and B of the second case no longer goes through P but follows the hypotenuse connecting A and B  

If you were to say add another point in the second graph along the line connecting A and B and place another observer there. say Observer P_2. Then the events seen by P and P_2 observing A and B will not be seen simultaneous to each other by Observer P and P_2.

This is a simple example where the notion of simultaneity breaks down. You have the same issue with curved spacetime.

That issue is handled via rotations of the Lorentz transformations via a rotation matrix. Whereas is the first case change in magnitude of velocity it is a Boost not a rotation however in curved spacetime the light follows the curvature.

These graphs ae simply a simple way to show the distinction between Lorentz transforms involving the two types of acceleration.

A way to help understand why Proper time lies on a point of a geodesic path. That point is described by placing a tangent line to the surface of the curvature and have a parallel transported vector 90 degrees to that point connecting the Tangent vector to the curvature vector along with the perpendicular to the tangent connecting at the same point. 

Your convectors used under GR

see figure 1.2

https://amslaurea.unibo.it/18755/1/Raychaudhuri.pdf

figure 1.2 can be used to describe an Einstein train following a geodesic path not curvature of a physical object but rather the signal via photons. It show parallel transport along a geodesic path and the related mathematics that not only  describe the observers but also how one can measure curvature using parallel transport which when you get down to is a form of train Einstein train

PS In parallel transport the length connecting the parallel transported covectors is the affine connection in essence each section of the train following the geodesic

please study the article and note how it leads ups to 

Principle of General Covariance: “The laws of physics in a general reference frame are obtained from the laws of Special Relativity by replacing tensor quantities of the Lorentz group with tensor quantities of the space-time manifold.”

Edited November 5 by Mordred

[DimaMazin]

Let's consider hypotheses.

When velocity U is bigger than c, then we use velocity of MD65536   V=v/(1-1/gamma))

U=u/(1-1/gamma))

Then   u=2Uc²/(U²+c²)

Then we should define u'

u'=(v+u) / (1+vu/c²)

Then we can define U' using again formula of MD65536

U'=u'/(1-1/gamma)

U'=u'/(1-(1-u'²/c²)^1/2)😛

[Mordred]

Ok obviously any previous help hasn't worked for you in that your still not really making the connection that any acceleration is not a straight line path

A path through the space-time diagram is called a world line. The world line for the acceleration motion is described by a curve, but not a straight line.

as trying to tell you using boosts and rotations nor instantaneous velocity treatments done previously albeit  there was a ton of cross argument that had absolutely nothing to do with you and should never have occurred to begin with. That is not your fault in any form.

The above BOLDED statement is one of the primary reasons why hyperbolic geometry is needed. That hyperbolic geometry is applied for the Minkowskii diagrams.

This article has the most straightforward examination of all the applicable mathematics without using a single tensor or matrix. It has the train included in the article including simultaneity

https://bingweb.binghamton.edu/~suzuki/ModernPhysics/2_Minkowski_spacetime_diagram.pdf

please study this and ask questions if needed.

for rapidity which isn't needed but useful the relation to velocity is the inverse hyperbolic function between v and c.

\[w=\tanh(\frac{v}{c})\]

The article uses a scaling factor k which is one method. One could also apply rapidity as that Tanh function can give a Natural log function for scaling as part of its ladder operation.

seen here under inverse hyperbolic functions.

https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions

 

Events that occur at two separate places at the same time in the S' frame do not happen at the same time, as viewed in the S frame.

previous to that statement the article has a mathematical expression

\[\Delta \acute{t}=\frac{\Delta(t)\frac{v}{c^2}\Delta(x)}{\sqrt{1-\frac{v^2}{c^2}}}\]

see previous mathematics for where the numerator and denominator gets derived from this is the section under relativity of simultaneity.

the next sections will apply that to your train as you can see figure 27 is quite complex as a result with repetitious signals being figure 28

for the record the very reasons I kept mentioning velocity vs acceleration and referred to rapidity was to indicate that with acceleration you are no longer dealing with linear equations of motion but non linear equations of motions which uses hyperbolic geometry to describe hence the use of the tangent in the earlier sections of the article Tangent to the slope which is how one can linearize a non linear graph.

The proper time is the point where the tangent intersects the worldline. Anywhere else its coordinate time and it is defined by

\[\Delta(\tau)^2=(t_1-t_2)^2+(x_1-x_2)^2+(y_1-x_2)+(z_1-z_2)^2\]

or simply

\[d\tau^2=dt^2-(x^2+y^2+z^2)\]

 

The article link the following video this can also be applied to the train.

I will have to wait a bit till I can get past post merging as any  images or videos will force any latex to Rich text format screwing it up.

However I was trying to describe a similar example previous post to the observers on the train where I used the curved path to indicate that the worldline null geodesic does not always follow the path of the train to every observer on the train or other observers not on the train.

 

 

Edited November 9 by Mordred

[Mordred]

for the above and my previous post notice the usage of an observer watching the two spaceships now consider an observer not on the train.

 

here is 4 coordinate systems reference

3982882af388dae3407906357a419cba_coordsproptime.pdf

notice what occurs in each case

last case is using rapidity.

first two cases are two sperate reference frames under SR. Third and 4th case is using  complex conjugations of the first 2 graphs ie dual vectors, co-vectors and contra-vectors are examples of those complex conjugates

 

Edited November 10 by Mordred