How Do Points M and O' Move Relative to Point O in Space and Time?

[ilasus]

TL;DR Summary

Physical interpretation of a relations

I consider three material points O, O', M, in uniform rectilinear motion in a common direction, so that in relation to the point O, the points O' and M move in the same direction with the constant velocities v and u (u>v>0). Assuming that at the initial moment (t₀=0), the points O, O', M were in the same initial place (x₀=0), the distances traveled by the points M and O' in relation to O, denoted by x and x₁, respectively , are given by the laws of motion

(*) x = ut, x₁ = vt

where time t is measured from the initial moment. As it turns out, the point O' divides the trajectory OM described by M in relation to O into two parts, OO' and O'M, which the point M travels in different time intervals, t₁ and t₂, respectively. Given that the point M travels the distance x in time

t = x/u

respectively the distance

x₁ = (v/u)x

during

t₁ = x₁/u = (v/u²)x

we can conclude that in relation to point O, point M travels the distance x in time t according to the relations

(1) x = ut, t = (1/u)x

between points O and O', point M travels the distance x₁ during t₁ expressed by the relations

(2) x₁ = vt, t₁ = (v/u²)x

and in relation to the point O', the point M moves on the distance x₂ during t₂ given by the relations

(3) x₂ = x - vt, t₂ = t - (v/u²)x

As a personal opinion, I believe that relations (1), (2), (3) describe the motion in space and time of points M and O' in relation to point O. For example, relations (*) describe the motion in space of points M and O' in relation to O during t, and the relations

(**) t = (1/u)x, t₁ = (v/u²)x

describe the movement in time of the points M and O’ in relation to the point O on the distance x. In this case, point O is considered to be at relative rest in both space and time: in space, point O is at relative rest because it has traveled the distance x₀=0 during time t, and in time, point O is at rest relative because a time t₀=0 has traveled the distance x. In relation to point O, however, point M is moving, according to (1), both in space and time. In space, the point M is moving with speed u with respect to O, because in a unit of time, the distance x between points O and M extends by u units of space, and in time, the point M is moving with speed 1/u relative to point O, because on a unit of space, the time t between points O and M extends by 1/u units of time. At the same time, on the trajectory described by the motion of point M in relation to O we can read the space and time coordinates associated with point O', respectively the distance x₁ and time t₁ expressed by the relations (2), where v is the speed which extends the distance x₁ between the points O and O', v being the distance traveled by the point O' in relation to the point O in the unit of time, and v/u² is the speed with which the time t₁ extends between the points O and O', v/u² being the time interval in which the point O’ moves in relation to the point O on a unit of space.

In the hypothesis presented above, we look at the OM trajectory not only as a trajectory in space, but also as a trajectory in time. For example, if we identify the trajectory described by the motion of the point M in relation to O with the abscissa axis of a reference S with origin O, then we could look at the trajectory OM both as a spatial axis, if on the abscissa axis of the reference S are marked divisions measure for space, as well as as a time axis, if on the axis of the abscissas of the referential S are marked divisions that represent units of measure for time. For example, we could assume that the units of measurement for space and time, marked on the abscissa axis of the referential S, are defined by the motion of the point M with respect to O, so that the unit of space is defined by the distance traveled by the point M with respect to point O during 1/u, and the unit of time is defined by the time interval in which the point M moves with respect to the point O on the distance u.

What do you think about this physical interpretation of relations (1), (2) and (3)?

 

[Ibix]

PeroK was correct in your last thread. You shouldn't introduce different time parameters for the different objects. $t_1$ should be equal to $t$ and your relations should be $x=vt$, $x_1=ut$, and $x_2=(u-v)t$. I think introducing independent time variables has just confused you. You end up talking about different objects at different positions, moving at different speeds, for different durations and you just end up with confusion.

In this case, point O is considered to be at relative rest in both space and time:

This is nonsense. At rest in time? So it's always Monday morning for point O?

 

[PeroK]

The correct equation of motion in this case is: $$x = R_0e^{-\lambda t} \cos \omega t, \ y = R_0e^{-\lambda t} \sin \omega t \ \ \ (R_0, \lambda > 0)$$

 

[ilasus]

This is nonsense. At rest in time? So it's always Monday morning for point O?

In space, point O is at relative rest because it has traveled the distance x₀=0 during time t. In time, point O is at rest relative because a time t₀=0 has traveled the distance x. That’s by definition. Is not correct?

 

[Ibix]

Is not correct?

It's meaningless.

 

[ilasus]

PeroK was correct in your last thread. You shouldn't introduce different time parameters for the different objects. $t_1$ should be equal to $t$

But if t=t₁, then in your opinion (and PeroK's) the point M traveled at the same time t and at the same speed u the distances x₁ and x, that is, we can write x=ut and x₁=ut and so it follows that x=x₁. Correct?

 

[Ibix]

But if t=t₁, then in your opinion (and PeroK's) the point M traveled at the same time t and at the same speed u the distances x₁ and x, that is, we can write x=ut and x₁=ut and so it follows that x=x₁. Correct?

You previously defined $x_1=vt$ as the distance O' traveled in time $t$ and explicitly stated that $u>v$. So $x=ut$, the distance M travels in time $t$, cannot be equal to $x_1=vt$.

 

[ilasus]

You previously defined $x_1=vt$ as the distance O' traveled in time $t$ and explicitly stated that $u>v$. So $x=ut$, the distance M travels in time $t$, cannot be equal to $x_1=vt$.

OK, but if M traverses the distance x during t=x/u and the distance x₁ during t₁=x₁/u, and x₁=(v/u)x, then it follows that t₁=(v/u)t, i.e. that t₁<t and so it does not turn out that t₁=t, as you say (and PeroK). Correct?

 

[PeroK]

OK, but if M traverses the distance x during t=x/u and the distance x₁ during t₁=x₁/u, and x₁=(v/u)x, then it follows that t₁=(v/u)t, i.e. that t₁<t and so it does not turn out that t₁=t, as you say (and PeroK). Correct?

Technically, you are confusing a free parameter $t$ with specific times $t_1, t_2$ etc. For a given reference frame, there is only one coordinate time: normally denoted by $t$ or $t'$.

The root of your problem is mixing up coordinates with specific times and distances between points. For example, where you define $x_2 = x - x_1$, this means that $x_2$ is no longer a coordinate in that reference frame, but the difference of two coordinates.

This is where you start to get into a tangle. What you should have is something like $d = x_2 - x_1$ to indicate that this is a distance, not a coordinate.

In general, specific events take place at a given time. A clear approach would be:

The event "M is at the origin" has time coordinate $t = 0$. The event "M is at location $x = x_1$" has time coordinate $t = t_1$ and "M is at location $x = x_2$" has time coordinate $t = t_2$. And, you would use $x$ and '$t$ as your general coordinates for this frame of reference; and $x_1, t_1; x_2, t_2$ as the coordinate values for specific events.

 

[kuruman]

Equation (3) is the Lorentz transformation with $\gamma =1$ and $u=c$. That $\gamma= 1$ is easily understood. That $u=c$ (I think) assumes that $u$ is the same in the reference frames of O' and M which it clearly isn't.

 

[Ibix]

Equation (3) is the Lorentz transformation with $\gamma =1$ and $u=c$. That $\gamma= 1$ is easily understood. That $u=c$ (I think) assumes that $u$ is the same in the reference frames of O' and M which it clearly isn't.

But $\gamma=1$ implies $v=0$, so I think this is not significant of anything.

 

[Ibix]

As it turns out, the point O' divides the trajectory OM described by M in relation to O into two parts, OO' and O'M, which the point M travels in different time intervals, t1 and t2, respectively. Given that the point M travels the distance x in time

So I've tried to draw a displacement-time diagram based on this. My software for this uses the relativistic convention of time going up the page and space across it, but this is otherwise exactly the kind of diagram one learns at school. I've drawn the trajectory of O in red, O' in green, and M in blue, and marked the equations of motion.

I've also marked the time $t_1+t_2$ from which we can deduce $x_1$, the distance traveled by O' in this time, and $x_1+x_2$, the distance traveled by M in this time. Note, though, that although the distance traveled by M between time 0 and $t_1+t_2$ (measured by the horizontal grey line) is divided into two sections where the line crosses the trajectory of O', there is no similar division of the vertical line - it does not cross the green line at all. That is, there is no natural division of $t_1+t_2$ into $t_1$ and $t_2$. It is, of course, possible to construct $t_1$ by noting where the dotted grey line crosses M's trajectory and dropping a perpendicular. But this is not anything that can be measured in any way - there are no markers that will let you do it in practice. Of course you can take the elapsed time $t_1+t_2$ and multiply by $x_1/(x_1+x_2)$, which is what you are doing, but what's the point?

 

[berkeman]

@ilasus -- Since this is a restart of a previously closed thread (which is against the PF rules), please try to finish up this thread before it is closed. Do you have any final confusions, or do you understand all of the good replies you have received now?

 

[ilasus]

@Ibix (and PeroK), I was going to send a drawing and clarify the problem with that time t₁<t. But I will have to give up because the thread closes. I don't understand this rush to close the threads - does it cost the organizers money if we talk on the forum? I will definitely give up this forum with problems. If you want to talk more, we can do it by e-mail [personal e-mail deleted by the Mentors]. Good.

 

[berkeman]

The thread is not closed yet. But since it appears that both of your thread starts are due to simple math fundamentals issues rather than anything related to Relativity, that is why your 2nd try at the thread start is subject to another closure. Hopefully somebody involved in the thread can summarize the issue(s) and we can post final responses to those and close this. Thanks.

 

[Delta2]

I have to say I agree with post #14, why close a thread (even if the thread keeps talking nonsense or keeps recycling), now days hard disks are cheap, and the text has a really high compression ratio regardless if text is nonsense or not LOL.
(Btw , I don't mean that this thread is nonsense, I just say about even in that extreme case that a thread is nonsense).
EDIT: OK well I guess I can understand, we want to keep some quality standards in these forums. PF is indeed one of the highest quality forums in science and engineering.

 

[PeroK]

@Ibix (and PeroK), I was going to send a drawing and clarify the problem with that time t₁<t. But I will have to give up because the thread closes. I don't understand this rush to close the threads - does it cost the organizers money if we talk on the forum? I will definitely give up this forum with problems. If you want to talk more, we can do it by e-mail [personal e-mail deleted by the Mentors]. Good.

If you are mistaken about some fundamental issues, then we can try to teach you. But, if you simply persist in posting the same errors again and again, and show no sign of being able and willing to learn, then the thread becomes pointless.

This thread was changed from an A level (postgraduate) to a B (basic) level. If you believe you are working at the forefront of physics (the equivalent of a PhD student), whereas, in fact, you are struggling with the basics, then this is part of the reason that no progress can be made.

I'll happy to withdraw at this point and let others try to help you.

 

[Ibix]

Hopefully somebody involved in the thread can summarize the issue(s) and we can post final responses to those and close this.

I would say the issues are fourfold. First, as @PeroK laid out in #9, OP confuses coordinates with what we think are supposed to be fixed values of those coordinates. Second, as I attempted to illustrate in #12, no clear relationship between the description and the reality. Third, a tendency to invent ill-defined terms (real and virtual speeds in the other thread, "at relative rest in time" in #1 here). Fourth, in the other thread, no apparent realisation that velocities (with a few honourable exceptions) cannot be equal in different frames.

I was hoping to see a response to my diagram, which separates times and distances where ilasus' diagram in the other thread conflates them. I thought that might move the conversation in a helpful direction. Obviously not if, as seems to be the case from #14, OP is gone.

 

[ilasus]

I would say the issues are fourfold. First, as @PeroK laid out in #9, OP confuses coordinates with what we think are supposed to be fixed values of those coordinates. Second, as I attempted to illustrate in #12, no clear relationship between the description and the reality. Third, a tendency to invent ill-defined terms (real and virtual speeds in the other thread, "at relative rest in time" in #1 here). Fourth, in the other thread, no apparent realisation that velocities (with a few honourable exceptions) cannot be equal in different frames.

I was hoping to see a response to my diagram, which separates times and distances where ilasus' diagram in the other thread conflates them. I thought that might move the conversation in a helpful direction. Obviously not if, as seems to be the case from #14, OP is gone.

Sometimes we consider mistake what we cannot understand. The “fifth error” is present in the attached graph, in which the formulas (1), (2), (3) are visualized, in which O' and M are mobile moving with constant speeds v=3km/h and respectively u=5km/h, on a straight road on which a landmark O is fixed in the initial place x₀=0km and respectively in the initial moment t₀=0h - vertically the distances x₁ and x₂ are seen at the moments t=1h, t=2h, t=3h, and horizontally you can see the time intervals t₁ and t₂ at distances x=5km, x=10km, x=15km - colored in green and red respectively.

 

[Ibix]

The line at approximately 45° represents M. The line below it represents O'. What's the line above it?

Edit: note that ilasus' diagram is flipped with respect to mine, with the time and displacement axes switched.

 

[ilasus]

The line at approximately 45° represents M. The line below it represents O'. What's the line above it?

Edit: note that ilasus' diagram is flipped with respect to mine, with the time and displacement axes switched.

In the diagram presented by me also takes into account the time travel of M (with speed 1/u=(1/5)h/km) and O' (with speed v/u²=(3/25)h/km) in relation to O considered at rest relatively in time.

 

[Ibix]

In the diagram presented by me also takes into account the time travel of M (with speed 1/u=(1/5)h/km) and O' (with speed v/u²=(3/25)h/km) in relation to O considered at rest relatively in time.

In other words, you've added a line that connects some points you want to be connected but doesn't represent anything that's actually there in reality. That is exactly the criticism I was making in #14 #12.

 

[ilasus]

In other words, you've added a line that connects some points you want to be connected but doesn't represent anything that's actually there in reality. That is exactly the criticism I was making in #14.

In #18.

 

[Ibix]

In #18.

#12, in fact. Apologies.

 

[ilasus]

#12, in fact. Apologies.

bye

 

[berkeman]

Thread is closed now, and OP is reminded not to try yet again to restart it. Thank you to all the folks who patiently replied trying to help the OP with his misunderstandings.