B Newtonian vs Relativistic Mechanics

[Ibix]

What space-time diagram are you referring to? The only space-time diagram I see in this entire thread is in the video referenced in post 49.

The diagram in #83 has axes labelled as if it were a space-time diagram but looks more like a diagram in the x-y plane. Both Dale and vanhees71 are expressing confusion about exactly what is being shown. You seem to be interpreting it as the latter, but it isn't clear that that is what Grimble intends. You are assuming that you know which mistake Grimble has made - which may or may not help him correct whatever his misunderstanding is depending on whether your assumption is correct.

Edit: beaten to it by vanhees, I see.

 

[the_emi_guy]

The diagram in #83 has axes labelled as if it were a space-time diagram but looks more like a diagram in the x-y plane. Both Dale and vanhees71 are expressing confusion about exactly what is being shown. You seem to be interpreting it as the latter, but it isn't clear that that is what Grimble intends. You are assuming that you know which mistake Grimble has made - which may or may not help him correct whatever his misunderstanding is depending on whether your assumption is correct.

Edit: beaten to it by vanhees, I see.

The OP, who indicated a high school level of education, has derived the Lorentz transformation from first principals in a manner typically employed in college physics classes and this seems to have gone completely unnoticed.

Have you gone back and read posts 47 and 48? I suggested this just 5 posts ago to clarify where the OPs diagrams originated.

 

[Grimble]

OK. Let us examine the diagram in #83 that is causing such confusion and I will explain exactly what it is intended to shew.

First things first, this is not a Minkowski Spacetime diagram. It is no more than a diagram to demonstrate how, by making the invariance of 'c' central there is no need to use hyperbolae. That is NOT to say there is anything wrong with Minkowski’s Spacetime; it having stood the test of time very robustly.

In both traditional Newtonian and in spacetime diagrams - as I understand them - the time axis is vertical because time is being measured by the distance light would have traveled along the y axis.

In the clock diagrams time for each observer, in clock A and in clock B, is measured parallel to the y/ct axis vertically for each clock at rest. Which due to the construction of said clocks is also the path of the light in each clock, as observed by an observer at rest with each clock. So time measured by either observer is the distance light traveled in their own clock, it reaches the mirrors, 1 light second away in 1 second. Identical time for identical clocks, in their own frames.

Yet for time in each clock measured from the other clock, we know that in one second that light will have traveled for one light second, in a clock that has has traveled vt light seconds, at 0.6c that will be 0.6 light seconds.

So measured from the stationary clock, the light in the moving clock will have two components to its motion, the movement of the light in the clock and the movement of the clock itself. Hence at 0.6c the light in the moving clock must reach point (0.6,0.8) in the frame of the rest clock after 1 second. Less time will be measured by the resting observer as the speed of the light has two components, the speed of light in the clock and the speed of the clock. So measured from the resting clock the time measured must be less that in the resting clock(?).

Now to me that is plain simple Newtonian Mechanics, working from what is known rather than assumed, that the speed of the light will be traveling at c. As that is a constraint placed upon this scenario by relativity.

Time is measured by virtual light emitted at the initial event, traveling at c in every direction - so an expanding sphere of virtual light, centred on that initial event; any radius of that sphere will be a measure of the interval from that initial event. (in this two dimensional view it will, of course, be a circle)

For a body at rest - such as the observer whose frame we are drawing it will be vertical as there is no displacement.

For a moving body, the line along which time is measured will be rotated because of the lateral displacement. In Newtonian mechanics that displacement is measured after the interval measured on the observer's time axis - the vertical axis of the diagram. In the mechanics of relativity - because of the invariance of c - the displacement means that time in the moving clock has to be measured where the displacement (0.6 light seconds) intersect the 1 light second circle of time because those are the two constraints which we know are true: the displacement of the clock (0,6 light seconds in my diagrams) and the one light second the light must have travelled.

The time interval is measured on the moving body's rotated time axis. In the clock diagram the light in the clock will have traveled 1 light second - to the virtual time sphere while moving clock has traveled vt from the observer in the stationary clock. That is why the time axis for the moving clock is rotated through angle β (sin β = ^v/_c rather than angle α (tan α = ^v/_c because the 1 second coordinate constitutes the hypotenuse rather than the adjacent side of the triangle; it has to be that way because of the invariance of c.

This is no more than simple euclidean geometry.

We are intending to measure the time between two "events" by measuring the distance, horizontal and vertical, that light must travel between the events. (The first event is the flashlight being switched on, the second event is the light reaching the mirror).

On the other hand, the space-time diagram (video referenced in post 49), has time as the vertical axis, thus the hyperbolas and 45 degree "speed of light" line.

This is a diagram showing three views of the mechanics of relativity

1. First, classical Newtonian Mechanics with no account taken of the second postulate. Resulting in a speed greater than 'c'.
2. Second, Minkowski's great Spacetime where taking heed of the hyperbolic function caters for the invariance of 'c'. Yet it is truly a mathematicians solution that leaves those of us with a less mathematical background, difficulties in appreciation. There is nothing wrong with it - it has been accepted for more than 100 years!
3. Thirdly, is but a simple view of the mechanics, centred on the invariance of 'c' and trying with the best will in the world to apply Occam's razor. Keeping to the facts that we are certain of: that light will travel at 'c', that the moving clock will travel the distance vt, and how that rotation affects the measurement by the resting observer.

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/3%20mechanics%20compared.png?attachauth=ANoY7coaMPUDTIO6hy2MrLS6UA0jDfN3g7ZAN_d_dYGxxFrLFD1F93wbzGyClgdYgDCPnXBpWs34Sj6gnWmMjSkd5bfAlV9Lj7jKMPV5opPLquyBEphh_th791WMA0n1AUE4SZxL33Lm-03eT2SI2nsHIXYMlMt5Y7BzfaILnrBs8eGMUiXDo3bI54IMJsB-9kDI1GAscn0uKYnRBjAnALgAxoinCixSOdGZdQHeSYWXx70nhmaL7j9pT5wUaMdey5By8-N87nu99VB0xO7wRrq8cCvEWDNi2h7iLWIGsz7OQ-4-_ci1rOk%3D&attredirects=0
The best thing that can be said for this last attempt to understand what happens in simple terms is that time dilation and the Lorentz factor fall out of it without any effort.

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Lorentz%20factor.png?attachauth=ANoY7cp8JpFJWuip-BS5Zt2Bxq3gX06yEMV6SAXi_uGKqO3cv3bOgFtwPl9xMQBeFcuc7nXX5Fk4KZ49UpdrCKak0k-LvYZaSQQ0gY29LbmSRByGzGTFrkhNGjCtR5SCH_v-SP_PE1Y-hiuF-TgZA2NfIjGaVHVW3dGAFXbpBoS2Isknzam8bhAoocoEEszL4knxcneXi8ANdiVi9JWvQYR73UokJPY6dZt43u7fzQp_2htoojq4E6sbSo0Iw5WPtMMqmnaqYPWdWyIwF3GPBTdsJtVQalLdvw%3D%3D&attredirects=0

 

[vanhees71]

I don't understand your diagrams. At least the $(ct,x)$-plane cannot be a Euclidean plane. This wouldn't enable us to define a causality structure. The correct construction of the Lorentz transformation can be found in

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

pp. 8 and 9.

The Lorentz transformation implies all kinematic phenomena like time dilation and length contraction as is worked out in this manuscript too.

 

[bahamagreen]

"Nor must we overlook the fact that arguments which proceed from fundamental principles are different from arguments that lead up to them."
Aristotle

No doubt Einstein, Minkowski, et al knew this; and the difficulty when a change in the fundamental principles is itself the object of argument.

 

[Dale]

this is not a Minkowski Spacetime diagram

So then the vertical axis should not be labeled "ct", it should be labeled "y"

by making the invariance of 'c' central there is no need to use hyperbolae

There is still always going to be the need to use hyperbolas. The equation $\Delta s^2=-c^2 \Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2$ is a hyperboloid.

 

[Grimble]

I don't understand your diagrams. At least the $(ct,x)$-plane cannot be a Euclidean plane. This wouldn't enable us to define a causality structure. The correct construction of the Lorentz transformation can be found in

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

pp. 8 and 9.

The Lorentz transformation implies all kinematic phenomena like time dilation and length contraction as is worked out in this manuscript too.

Thank you, Vanhees71, I do understand that you are trying hard to help me, but with my level of High School Education (1960s) I am struggling with the first sentence of p8 and totally lost by sentence 2. Not by the wording but by the mathematical terms used, which I am unfamiliar with, in those terms at least.

On the other hand I am at a loss as to how you can fail to understand my diagrams!

I start with a simple diagram of two light clocks moving apart at 0.6c (to keep the diagram and any calculations simple. Perhaps if I take you through it again; starting with the difference between Newtonian mechanics with and without the invariance of 'c'.
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Mechanics%20compared.png?attachauth=ANoY7cp9GaHYoo-aiMlFyVhXmgpAWLw7Y4ZXy-MbINOoEK7NI6A2GzRvClzDgb_ZLF0H9cw9RzRFyiqXrOLPuoLcvDGafLnGPtswJdjScrSzAjyeiZ2YxTnI_vNHz5uBfVDlONZZCaokzsr7MLFKjBSKJEcgYp2A-Z-dtLnkAZLt1Rw3XBHscpycCyeSJP-taYPoccpRQYImLwDqZ8QvofhyrXNW6doxkt4STBS-nk5nIkTz1Fde-dNpL8g4TNChWLGA9gR5peECVEhVa9AWxctK_KrdCDdpQg%3D%3D&attredirects=0
In both diagrams the vertical axis can be read as both/either a ct (time) axis or as a y (spatial) axis. It is the path of a flash of light normal to the x axis. Being the passage of light it is possible to use it as a time axis for the observer in the clock at 0,0.

In the first diagram speed is unlimited and the moving light travels 1.166 light seconds in one second measured on Clock A's time axis.

In the second diagram, observing the 2nd postulate, the moving light of clock A (in the frame of Clock B, which is at rest at rest) travels a rotated path, but still only 1 light second (The curving red line) from 0,0 while clock A also travels 0.6 light years in that same 1 second. Hence the light will arrive at (-0.6,0.8) after traveling for one second, measured along Clock A's rotated time axis, as measured by Observer B.

I drew the second diagram that way round to emphasise the reciprocality of relativity. Either clock can take either role.

The diagrams in Post #93 reduce those diagrams to the fundamental items, time measured on the observers time line, time measured on the moving clock's time line and the distance traveled by the moving clock.

The new part I introduced was the second part which I included to shew how the invariant spacetime interval function can be seen as a hyperbola.

Is there any point in particular that is causing you difficulty?

 

[vanhees71]

Well, perhaps I'm just too used to the usual Minkowski diagrams. I cannot make sense of your "Euclidean" space-time plane. It's contradicting any intuition we have about the relativistic space-time structure.

 

[DrGreg]

In both diagrams the vertical axis can be read as both/either a ct (time) axis or as a y (spatial) axis. It is the path of a flash of light normal to the x axis. Being the passage of light it is possible to use it as a time axis for the observer in the clock at 0,0.

I see two problems.

1. The vertical axis is the y axis only. You can't treat it as the ct axis as well, because, although $y = ct$ is true for one path in each diagram, it's not true for other paths in the same diagram.
2. In each diagram the red parts and the green parts refer to different observers, i.e. different coodinate systems. It's misleading to superimpose both in the same diagram, so really you ought to split each diagram into separate green and red diagrams. And use different names for the red and green coordinates. Traditionally the two coordinate systems are written as $(ct, x, y)$ and $(ct', x', y')$.

 

[Dale]

In both diagrams the vertical axis can be read as both/either a ct (time) axis or as a y (spatial) axis.

This is wrong. Time is orthogonal to space so it cannot be represented by the same axis.

It is the path of a flash of light normal to the x axis. Being the passage of light it is possible to use it as a time axis for the observer in the clock at 0,0.

This would be a different axis, neither time nor space. It would be called a null axis. Something like this is used in radar coordinates, but usually with two null axes.

 

[Grimble]

Thank you, Gentlemen. I can see what you mean and how my diagrams can be confusing. I have tried to include too much information in a (each) single diagram.
I will heed your comments and advice and redraw them, as separate diagrams. That makes a lot of sense. Thank you.

 

[the_emi_guy]

Thank you, Gentlemen. I can see what you mean and how my diagrams can be confusing. I have tried to include too much information in a (each) single diagram.
I will heed your comments and advice and redraw them, as separate diagrams. That makes a lot of sense. Thank you.

Grimble,
One thing to keep in mind is that one of the premises of this thought experiment is that each observer agrees about the vertical distance traveled by the light beam.
In relativity, observers will generally disagree about lengths (Lorentz contraction), but in this case, because there is no vertical motion, both will agree about this vertical distance. Since they are agreeing about this distance, we might as well make that distance something simple such as a meter. We can't have this distance indicated in terms of time because the observers disagree about elapsed time.

The next thing to consider is that this thought experiment shows that the two observers disagree about the elapsed time between two specific events (light switched on, light reaching mirror). Turns out that this implies that the two observers will disagree about the elapsed time between *any* two events, such as time time between beats of the travelers heart. Otherwise an observer would be able to detect if they were the one in motion by comparing the results of the "light clock" with their heartbeat. This is the other postulate of special relativity, that either observer can declare themself to be the one at rest.

I suggest taking a look at Feynman's very well written treatment of this thought experiment. Thanks to Caltech, it is freely available on the Web:
http://www.feynmanlectures.caltech.edu/I_15.html section 15.4
He describes why the two observers agree on vertical distance, and why the time dilation computed for the "light clock" applies to any other "clock".

Cheers

 

[Grimble]

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Alice%20and%20Bobs%20light.png?attachauth=ANoY7coZsmLJQ0JAxFV72ZNxkbJBRzT6MCrPSCK-3J6r6q7Izraagl5byes4jkgMWSCX6HuegfgHK2T9WMjDRqFf4hiwqHDURP6iadByaTSGqqOQYVo6YMF5qgO27Bsr8136cganLCdkHQqxiFovfcGOXsUYCROHx5EmthiNTOzJ4cGRCeFtWBCgKdvX2gbuqm0ZWxrLfoiZJDt1VDrKloK8g7rMD7fa4nq766aEDc9T6Ygg0PEWufVX2U79EzbTBU7jnRGqgrycgQN2nox_xutMu9kDyjdx4HgjrbFyc0S49x0eXgZKP38%3D&attredirects=0

This is not an Minkowski Spacetime diagram. If anything it is a simple Newtonian diagram.
A simple diagram of the Frame of Reference of a resting observer, Alice; Bob is traveling along the x axis. He has a light clock with the light direction along his y axis. As he travels the path of the light in his clock is rotated by his motion. The speed of the light along the rotated path is 'c'. (The rotated path is only observed by Alice, Bob only sees his light travel at 'c' along his y axis).
Bob is traveling at 0.6c relative to Alice. After Bob has traveled 0.6 units, his light will have traveled 1 unit. along the rotated path, as measured by Alice.
Alice also has a light that emits a flash along her y-axis at Bob's departure.
Both lights travel in the y direction at the same rate, yet Bob's light also travels at 0.6c along the x-axis and has therefore traveled 1 unit along the rotated path, as measured by Alice, when Bob has traveled 0.6 units x-wards.
The increased duration of Bob's light journey (1 unit) is due to the relative speed between Alice and Bob and is only measured by Alice who measures Bob's light to have traveled 1 unit when her light has traveled 0.8 units; as Bob's light will be measured by Bob. Both lights are moving at 'c'; therefore the time passing for Bob, as measured by Alice, and only as measured by Alice, is measured to be 1 unit rather than 0.8 units.

That is purely a change in the time measured by Alice. Bob's clock has only run slow as measured by Alice, Not as measured by Bob. Indeed its rate is unchanged except as viewed by Alice due to her relative speed.

Now, before anyone complains that I am trying to rewrite anything, it seems to me to be crucial to this scenario, to relativity the experiments and all, to aver that Bob's unchanged passage of time is no more real than Alice's changed perception of time passing for Bob.

In Alice's Frame Bob's time does run slow and is measured to run slow. The Muon's time run's slow, because it is measured in the Earth's Frame of Reference in which it does indeed run slow.

Surely that is what is so important and fundamental to the whole concept and theory of relativity - it is all relative! No one observer's view is more correct than any other. Is that not what relativity is all about?

And if one swapped Alice and Bob in that thought experiment, would Bob not find that Alice's time runs slow? - And that both view are correct at the same time, whoever is considered to be the one moving? After all, all movement is and can only be relative...

 

[PeterDonis]

This is not an Minkowski Spacetime diagram. If anything it is a simple Newtonian diagram.

Yet you appear to be trying to demonstrate something about SR from it. I don't understand what you are doing or why you think it is valid. You can't use a "Newtonian diagram" to demonstrate something about SR. The two theories are inconsistent.

In fact, I'm not even sure your diagram correctly represents Newtonian physics. You appear to be assuming that in Newtonian physics, Alice's light travels 0.8 units in the same time that Bob's light travels 1 unit. I don't see anything in Newtonian physics that would lead to that result.

It seems to me that you are expending a lot of effort trying to invent new conceptual tools for something that you don't yet understand. That's not very likely to be a good strategy; so far it certainly hasn't appeared to work for you in this thread. I think you would be better served by cracking open a basic SR textbook, like Taylor & Wheeler, and trying to learn to use the conceptual tools that have already been invented by people who thoroughly understand the subject matter. Or you could try Einstein's own book for the layman, linked to in post #84.

 

[PeterDonis]

When we draw this out in detail in the drawing on the right, we have the
Spacetime Interval measured, t = 0.8seconds
The time measured on the moving clock, t' = 1 second
The time it would take Zach and clock B to travel distance x = 1/cvt' = 0.6 seconds

No, this is wrong--both your diagram (the one on the right) and your calculation as quoted just now. The equation for the interval that you gave, which is correct, is

$$
(\Delta S)^2 = (\Delta t)^2 - (\frac{1}{c} \Delta x)^2
$$

However, you are interpreting the terms wrong. The spacetime interval $\Delta S$ is the same as the time measured on the moving clock. The time interval $\Delta t$ is the coordinate time, i.e., the time according to the clock that stays at rest. So the correct calculation is: $\Delta t = 1$ (1 second elapsed on the clock that stays at rest); $\Delta x = v \Delta t = 0.6$ (the moving clock travels 0.6 light seconds in 1 second, both distance and time being measured according to the frame of the clock at rest); so $\Delta S = 0.8$ (0.8 seconds elapsed on the moving clock).

If you draw a proper spacetime diagram of the above, note that the triangle you draw will not obey the ordinary Pythagorean theorem of Euclidean geometry--the side of the triangle $\Delta S$ will be what looks like the "hypotenuse" of the triangle, and the side $\Delta t$ will be the vertical leg (the side $\Delta x$ will be the horizontal leg--your diagram did get that one thing right), even though $\Delta t$ is longer than $\Delta S$. That is because the geometry of spacetime is not Euclidean; it's Minkowskian. There is no way to draw an undistorted diagram of Minkowski spacetime on a Euclidean piece of paper; you have to accept that some things in the diagram will not work the way ordinary diagrams in Euclidean geometry work.

 

[Grimble]

Yet you appear to be trying to demonstrate something about SR from it. I don't understand what you are doing or why you think it is valid. You can't use a "Newtonian diagram" to demonstrate something about SR. The two theories are inconsistent.

In fact, I'm not even sure your diagram correctly represents Newtonian physics. You appear to be assuming that in Newtonian physics, Alice's light travels 0.8 units in the same time that Bob's light travels 1 unit. I don't see anything in Newtonian physics that would lead to that result.

It seems to me that you are expending a lot of effort trying to invent new conceptual tools for something that you don't yet understand. That's not very likely to be a good strategy; so far it certainly hasn't appeared to work for you in this thread. I think you would be better served by cracking open a basic SR textbook, like Taylor & Wheeler, and trying to learn to use the conceptual tools that have already been invented by people who thoroughly understand the subject matter. Or you could try Einstein's own book for the layman, linked to in post #84.

Thank you Peter, I understand that this is not a Newtonian, nor a Minkowski diagram. It was intended to be neither but to take Newtonian Mechanics with the stipulation that v cannot exceed 'c'; examining how that one simple change drawn using Newtonian Mechanics, affected the outcome.

You say:

Alice's light travels 0.8 units in the same time that Bob's light travels 1 unit. I don't see anything in Newtonian physics that would lead to that result.

but that is the whole point; that by making that one simple stipulation, that we know the the speed of Bob's light (as measured by Alice), leads to the inevitable conclusion that the time in Bob's Frame is dilated by the Lorentz factor with respect to Alice's time, but only Alice perceives this effect.

 

[vanhees71]

Thank you Peter, I understand that this is not a Newtonian, nor a Minkowski diagram. It was intended to be neither but to take Newtonian Mechanics with the stipulation that v cannot exceed 'c'; examining how that one simple change drawn using Newtonian Mechanics, affected the outcome.

Obviously it is rather a pretty undefined drawing rather than a diagram helping to understand anything about relativity! You should update your knowledge about space-time diagrams if you like to understand special relativity properly. It's not such a complicated topic by the way!

 

[Dale]

that by making that one simple stipulation, that we know the the speed of Bob's light (as measured by Alice), leads to the inevitable conclusion that the time in Bob's Frame is dilated by the Lorentz factor with respect to Alice's time, but only Alice perceives this effect

Yes, this is correct. I think that the diagram confusion is a matter of presentation and communication, but it seems that you have understood conceptually.

 

[PeterDonis]

by making that one simple stipulation, that we know the the speed of Bob's light (as measured by Alice), leads to the inevitable conclusion that the time in Bob's Frame is dilated by the Lorentz factor with respect to Alice's time, but only Alice perceives this effect.

Not from the diagram you gave. You apparently missed part of my point (which, to be fair, I expanded on more in post #105, subsequent to the post you responded to): in SR, Bob's light does not travel 1 unit in the same time that Alice's light travels 0.8 units. In every frame, every light ray travels 1 unit in 1 unit of time. So in Alice's frame, Bob's light travels 1 unit in the same amount of time that Alice's light travels 1 unit. What makes Bob's frame look time dilated, with respect to Alice's frame, is that in Alice's frame, Bob's light has to travel farther than 1 unit to reach Bob's mirror, and so takes longer to reach Bob's mirror than Alice's light takes to reach Alice's mirror--whereas in Bob's frame, Bob's light only has to travel 1 unit to reach Bob's mirror (and in Bob's frame, Alice's light has to travel farther than 1 unit to reach Alice's mirror, and therefore takes longer in this frame to reach Alice's mirror than Bob's light takes to reach Bob's mirror). And since Bob's light reaching Bob's mirror is what counts as one "tick" of Bob's time, Bob's clock is running slow relative to Alice's frame (and Alice's clock is running slow relative to Bob's).

So just making "one simple stipulation", that Bob's light and Alice's light both travel at the same speed, is not enough; you also have to fully account for the implications of that, which are that it is impossible for Bob's light to travel 1 unit in the same time that Alice's light travels 0.8 units.

it seems that you have understood conceptually.

I'm not sure I agree. See above and my post #105.

 

[Grimble]

I am not sure what the problem is here, Peter, because reading your post #109, you are describing exactly what I am saying.​

What makes Bob's frame look time dilated, with respect to Alice's frame, is that in Alice's frame, Bob's light has to travel farther than 1 unit to reach Bob's mirror, and so takes longer to reach Bob's mirror than Alice's light takes to reach Alice's mirror--whereas in Bob's frame, Bob's light only has to travel 1 unit to reach Bob's mirror [...]. And since Bob's light reaching Bob's mirror is what counts as one "tick" of Bob's time, Bob's clock is running slow relative to Alice's frame (my bold)

"
Yes: "Bob's clock is running slow relative to Alice's frame" - Bob's light, measured by Alice, takes 1 unit of time to travel i unit of distance in the same time that Alice measures her own light to travel 0.8 units of distance in 0.8 units of time. It takes longer in Bob's frame, measured by Alice because "Bob's clock (Bob's time indeed) is running slow relative to Alice's frame"
So it is Alice's measure of time in her frame that differs from her measure of Bob's time between two events.
She measures light to travel at the same rate 'c', in both her frame and Bob's frame; they cannot travel at different rates therefore the passage of time has to differ, Bob's clock runs slow.

And yet in your very next paragraph you state:

So just making "one simple stipulation", that Bob's light and Alice's light both travel at the same speed, is not enough; you also have to fully account for the implications of that, which are that it is impossible for Bob's light to travel 1 unit in the same time that Alice's light travels 0.8 units.

I think here you seem to be indefinite in using the phrase "in the same time" because that is precisely what it isn't! they are different times because Bob's clock is measured by Alice to run slow. So it is not "in the same time" so much as between two events.

The spacetime interval for Alice's light to travel 0.8 units in Alice's frame is 0.8, the proper time, as it must be as Alice is at rest.
(ΔS)² = (Δt)² - (¹/_cΔr)² becomes
(ΔS)² = (Δt)²
and the Spacetime interval for Bob's light to travel 0.8 units in Bob's frame is 0.8, the proper time, as it must be as Bob is at rest.

While for the Spacetime interval for Bob's light to travel 1 unit in Alice's frame (ΔS)² = (1)² - (0.6)²
(ΔS)² = 1 - 0.36 = 0.64
ΔS = 0.8
The invariant Spacetime interval.

All of which brings me back to your post #105.

However, you are interpreting the terms wrong. The spacetime interval ΔSΔS\Delta S is the same as the time measured on the moving clock.

Now here is the problem: a time-like Spacetime interval as defined in Wiki is

The measure of a time-like spacetime interval is described by the proper time interval, [PLAIN]https://upload.wikimedia.org/math/3/5/5/355deec8daaddf14b3d7c610cb90e75a.png:

(proper time interval).
The proper time interval would be measured by an observer with a clock traveling between the two events in an inertial reference frame, when the observer's path intersects each event as that event occurs.

Now, as I see it that describes Bob, not Alice.So if the STI is proper time, it has to be measured in the clock's own frame? Measuring that from Alice's frame makes it the coordinate time.
But you then say:

The time interval ΔtΔt\Delta t is the coordinate time, i.e., the time according to the clock that stays at rest.

Ah but yes it is "the time according to the clock that stays at rest." but it is the time read on the moving clock, according to the clock that stays at rest.

So the correct calculation is: Δt=1Δt=1\Delta t = 1 (1 second elapsed on the clock that stays at rest)

1 second elapsed, measured from the clock that stays at rest.

; Δx=vΔt=0.6Δx=vΔt=0.6\Delta x = v \Delta t = 0.6 (the moving clock travels 0.6 light seconds in 1 second, both distance and time being measured according to the frame of the clock at rest)

Agreed.

so ΔS=0.8ΔS=0.8\Delta S = 0.8 (0.8 seconds elapsed on the moving clock).

Proper time, measured on the moving clock, by the observer moving with the clock.

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Alice%20and%20Bobs%20light%20clocks%20x%2Cy.png?attachauth=ANoY7cqB8EpgaGCeY6Ya0-Xwd6YR0yOUN0MXzotENjkerckDXmcSW-TAPlUyfTPy5VUs4dlVKSmD8uomd19vBesz6aMSYnJ43hdYE1CIjVT1WhdeRvN5tlGSRd3uEO2LtV6tJfDTh4wQUjpqw4wrhKv_t1DP75afXSluQu_kdifYn-EDcj9GAE0ro0SHjZiWTc7Pp7iPECBuSMGUaghramMJKU72AQyHvt7GBK83TPiRNg_cxKBdVQjDfa3W4HroCtRgwCmyC1j8WplLTJLkDhrdWkFDaPoeIFhs227-WNiqZqAmsN8PO4E%3D&attredirects=0
The first Diagram depicts Alice's and Bob's light clocks separating at 0.6c. Drawn from the perspective of an independent observer permanently situated midway between the two clocks. Each clock is moving at 0.3c relative to the independent observer.
When the clocks are 0.6 units apart the light in each clock will have traveled 1 unit to the mirror in that clock.
The following two diagrams are drawn for the individual Frames of Reference for Alice and Bob. Note, not times are depicted only the relative positions of the clocks and the lights in those clocks.
When the lights in the clock have traveled 1 unit, measured within each clock be the owner of that clock, both lights will have arrived at their respective mirrors. However the path seen by each observer of the other's light will be longer, due to the additional lateral displacement of the other's clock relative to each observer. Each light in a moving clock will be measured to have traveled 1.25 units in the stationary observer's Frame of Reference.
The light in each stationary clock will have traveled 1 unit, while the light in each moving clock will have traveled 1.25 units.
The lights have to be traveling at 'c'.
Therefore, as each observer will measure that, when the moving light has traveled 1 unit, it has only moved 0.8 units along the Y axis, that while 1 unit of time is measured to have passed in the moving frame only 0.8 units of time have passed in the stationary frame, because both lights are moving at 'c'.

This is simple mechanics using no more than simple geometry. It is a necessary consequence of the two postulates, that all movement is relative, not based on some fixed, privileged reference body; and the invariance of the speed of light. That is how it seems to me.​

 

[PeterDonis]

Bob's light, measured by Alice, takes 1 unit of time to travel i unit of distance in the same time that Alice measures her own light to travel 0.8 units of distance in 0.8 units of time

Read what you just wrote here. It contradicts itself. You say Alice measures Bob's light to take 1 unit of time, in the same time that Alice measures her own light to take 0.8 units of time. 1 unit of time is not the same as 0.8 units of time; Alice can't make two measurements "in the same time" if one of them takes 1 unit of her time and the other only takes 0.8 units of her time. That doesn't make sense.

So it is Alice's measure of time in her frame that differs from her measure of Bob's time between two events.

This doesn't make sense either. Alice can't directly measure "Bob's time". She can only measure the time it takes Bob, or Bob's light, to travel with reference to her own time.

I think here you seem to be indefinite in using the phrase "in the same time" because that is precisely what it isn't!

"Time" here means time relative to Alice's frame. You said that yourself in what I quoted at the top of this post. I'm using "time" in the same sense you are using it there.

The spacetime interval for Alice's light to travel 0.8 units in Alice's frame is 0.8, the proper time, as it must be as Alice is at rest.

No, it isn't. The spacetime interval for Alice to travel 0.8 units of time in Alice's frame is 0.8, the proper time. But the spacetime interval for light to travel 0.8 units of time in Alice's frame is zero--light travels 0.8 units of distance in 0.8 units of time, for an interval of $0.8^2 - 0.8^2 = 0$. Light always has a spacetime interval of zero; that's another way of stating the postulate that light always travels at the same speed in all inertial frames.

the Spacetime interval for Bob's light to travel 1 unit in Alice's frame

Is also zero. But the spacetime interval for Bob himself to travel some distance in Alice's frame is not. The spacetime interval for Bob to travel for 1 unit of time in Alice's frame is, as you say, 0.8, since Bob travels 0.6 units of distance in that time. That is Bob's elapsed proper time between those two events. But those two events on Bob's worldline are not the same as the two events, in spacetime, that Bob's light travels between--nor are they the same as the two events that Alice travels between in the same proper time of 0.8 units. If you had taken my advice earlier to use the standard conceptual tools of SR, such as spacetime diagrams, instead of trying to invent your own, this would be obvious. But your idiosyncratic conceptual tools do not show correct relationships between spacetime events, so you can't use them to make correct conclusions about spacetime intervals.

as I see it that describes Bob, not Alice.

It describes both of them. Take the numbers given above. For Alice, $\Delta t$ is 0.8 and $\Delta r$ is 0, so $\Delta \tau$ is 0.8. For Bob, $\Delta t$ is 1 and $\Delta r$ is 0.6, so $\Delta \tau$ is 0.8. So it is true that $\Delta \tau$ is the same for both Alice and Bob for the particular pairs of events chosen.

But $\tau$ is not a coordinate; it does not label unique events, and $\Delta \tau$ does not label unique spacetime intervals. The two intervals above, for Alice and Bob, are different spacetime intervals--different line segments between different pairs of events in spacetime--that just happen to have the same arc length $\Delta \tau$ of 0.8 units. Once again, if you were using spacetime diagrams, this would be obvious.

it is "the time according to the clock that stays at rest." but it is the time read on the moving clock, according to the clock that stays at rest.

No, it isn't. The time the moving clock actually reads, as it travels along a particular worldline between two events, is the $\Delta \tau$ along that worldline between those two events. It is not the same as $\Delta t$, the coordinate time between those two events.

Let's pick another pair of events for Bob to make this clear. Let's ask how much time elapses on Bob's clock when he travels for 0.8 units of time in Alice's frame, i.e., $\Delta t$ is 0.8--the same as it was for Alice in the numbers given above. Bob travels 0.48 units of distance in that time, so his $\Delta \tau$ is $\sqrt{0.8^2 - 0.48^2}$, or 0.64. So Bob's $\Delta \tau$ for this pair of events is 0.64 units--which, of course, is the coordinate time $\Delta t$ of 0.8 times Bob's time dilation factor of 0.8 relative to Alice.

1 second elapsed, measured from the clock that stays at rest.

I'm not sure what the difference is between this and what I said.

The first Diagram depicts Alice's and Bob's light clocks separating at 0.6c. Drawn from the perspective of an independent observer permanently situated midway between the two clocks. Each clock is moving at 0.3c relative to the independent observer.

Wrong. You are not using the relativistic velocity addition formula. You need to use that formula to find a $v$ such that $2v / (1 + v^2) = 0.6$. The correct solution to that equation is not $v = 0.3$. Try it. And then you will need to rework all the rest of your numbers.

This is simple mechanics using no more than simple geometry.

As far as I can tell, it is the same incorrect reasoning based on a confusion about the meaning of $\Delta \tau$ that I pointed out above. I really think it would be a good idea for you to stop using your ad hoc idiosyncratic conceptual tools and try using the standard tools of SR instead. Your personal conceptual tools are confusing you, not helping you.

 

[Grimble]

I am sorry Peter, but we seem to be talking at cross purposes. I am very grateful for your comments yet language and semantics continues to cause problems in how we each understand the other.

This doesn't make sense either. Alice can't directly measure "Bob's time". She can only measure the time it takes Bob, or Bob's light, to travel with reference to her own time.

Alice measures Bob's light travel between point (0,0) and point (0.6,0.8). A distance of 1 unit. Light traveling 1 unit must take 1 time unit. That is a given. The second postulate. So Bob's light measured by the distance it travels in Alice's Frame travels 1 unit of distance in 1 unit of time.
In Alice's Frame those two events - at point (0,0) and (06.08) - are 0.8 time units apart.
So in a time measured at 0.8 time units, Bob's light travels 1 distance unit, in what can only be 1 time unit.
Alice measures Bob's light to travel 1 unit in 0.8 time units. So Bob's clock, that is the time passing in Bob's frame, as measured by Alice is greater than the time passing in her frame. t' = γt. Time dilation.
Which is all about what Einstein was saying when he wrote

We were led to that conflict by the considerations of Section VI, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment in each second of time. But, according to the foregoing considerations, the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body). Hence it cannot be contended that the man in walking travels the distance w relative to the railway line in a time which is equal to one second as judged from the embankment.

So I really do not understand why you have such a problem as you expressed here:

Read what you just wrote here. It contradicts itself. You say Alice measures Bob's light to take 1 unit of time, in the same time that Alice measures her own light to take 0.8 units of time. 1 unit of time is not the same as 0.8 units of time; Alice can't make two measurements "in the same time" if one of them takes 1 unit of her time and the other only takes 0.8 units of her time. That doesn't make sense.

 

[Grimble]

"Time" here means time relative to Alice's frame. You said that yourself in what I quoted at the top of this post. I'm using "time" in the same sense you are using it there.

No, you are insisting that there is only one 'time' that Alice is measuring.
The time Alice measure in her frame is Proper time. The time she 'measures' (or calculates using the Lorentz transformation equations) for Bob's frame is coordinate time.
This really is very basic. You really should try and read what I write, and try and understand what I write, rather than trying to make it read what you want it read, it seems that you are determined to misrepresent everything I write because you are so convinced that it is wrong.
It may be wrong.

Sometimes I do use the wrong terms, phrases or constructions usually because trying to speak 'physics' is like using a foreign language in which I am unsure of the vocabulary, syntax and correct phraseology. I am trying to learn more and improve yet sometimes it seems that some (not you particularly) take delight in pulling apart every sentence without any attempt to read what I mean.

I think here you seem to be indefinite in using the phrase "in the same time" because that is precisely what it isn't! they are different times because Bob's clock is measured by Alice to run slow. So it is not "in the same time" so much as between two events.

you replied:

"Time" here means time relative to Alice's frame. You said that yourself in what I quoted at the top of this post. I'm using "time" in the same sense you are using it there.

What I mean here is that Alice is measuring time on her clock; proper time as you have averred and time for the light in Bobs clock to travel between two events.
The first event is at (0,0) in Alice's frame and would be (0,0) in Bob's frame.
The second event is at (0.6,0.8) in Alice's frame and would be at (0,08) in Bob's frame (because the clock is stationary in his frame).
The proper time between those events in Alice's frame is τ = √1 - 0.36 = √0.64 = 0.8.
The proper time between those events in Bob's frame is τ = 0.8.
But the time Alice is 'measuring' (according to the distance Bob's light travels, in Alice's frame, between those two events) has to be coordinate time; the time measured on the moving clock, the time dilated moving clock, would be the time it takes for the light to travel the distance of 1 unit between (0,0) and (0.6,0.8) in Alice's frame, which must be 1 unit of time at 'c'.

Why are you denying the facts of Time Dilation? Because that is all that I am describing.

 

[Grimble]

The spacetime interval for Alice's light to travel 0.8 units in Alice's frame is 0.8, the proper time, as it must be as Alice is at rest.

No, it isn't. The spacetime interval for Alice to travel 0.8 units of time in Alice's frame is 0.8, the proper time. But the spacetime interval for light to travel 0.8 units of time in Alice's frame is zero--light travels 0.8 units of distance in 0.8 units of time, for an interval of 0.82−0.82=00.82−0.82=00.8^2 - 0.8^2 = 0. Light always has a spacetime interval of zero; that's another way of stating the postulate that light always travels at the same speed in all inertial frames.

You are quite right I was getting my terminology mixed up there:

the Spacetime interval for Bob's light to travel 0.8 units in Bob's frame is 0.8, the proper time, as it must be as Bob is at rest.

Is also zero. But the spacetime interval for Bob himself to travel some distance in Alice's frame is not. The spacetime interval for Bob to travel for 1 unit of time in Alice's frame is, as you say, 0.8, since Bob travels 0.6 units of distance in that time. That is Bob's elapsed proper time between those two events. But those two events on Bob's worldline are not the same as the two events, in spacetime, that Bob's light travels between--nor are they the same as the two events that Alice travels between in the same proper time of 0.8 units. If you had taken my advice earlier to use the standard conceptual tools of SR, such as spacetime diagrams, instead of trying to invent your own, this would be obvious. But your idiosyncratic conceptual tools do not show correct relationships between spacetime events, so you can't use them to make correct conclusions about spacetime intervals.

And again there, the Spacetime Interval is for Alice, not her light which as you say would be zero.
I understood though, that the person who, carrying his clock along his worldline, passes through the two events; the point where the clock he is carrying emits the light pulse and the point where the light pulse, in the clock he is carrying is reflected in its mirror, i.e. one tick of his clock is Bob.
Pedantically viewed, one could say that in spacetime there are two events, the emission of the light pulse and its reflection, which would be 1 light unit from Bob; so Bob's worldline would not actually pass through both events. Yet those two events the emission and reflection of the light pulse in Bob's clock are exactly the same two events, which have the coordinates (0,0) and (0,08) in Bob's frame and (0,0) and (0.6,0.8) in Alice's frame. Events are unique but have different coordinates in different frames, but they are still the same events... or am I getting this mixed up as well?

 

[Grimble]

No, it isn't. The time the moving clock actually reads, as it travels along a particular worldline between two events, is the ΔτΔτ\Delta \tau along that worldline between those two events. It is not the same as ΔtΔt\Delta t, the coordinate time between those two events.

That is just being pedantic! That is contradicting every time anyone has said the moving clock runs slow!
EINSTEIN: RELATIVITY: THE SPECIAL AND GENERAL THEORY

As judged from K, the clock is moving with the velocity v; as judged from this reference-body, the time which elapses between two strokes of the clock is not one second, but

seconds, i.e. a somewhat larger time.

How many times do we read about observers reading a different time on a moving clock? Which we all know is an impossible feat! It is just a shorthand phrase that everyone understands is not to be taken literally. A clock will only show one time.

So the correct calculation is: Δt=1Δt=1\Delta t = 1 (1 second elapsed on the clock that stays at rest)

1 second elapsed, measured from the clock that stays at rest.

I'm not sure what the difference is between this and what I said.

The difference is Δt (coordinate time) is 1 second measured on the moving clock, by the observer with the clock at rest (using their proper time)
τ = t/γ = t√(1 - v²/c²)
τ² = t² - (¹/_cvt)²
τ² = t² -(¹/_cx)²
S² = τ² = t² -(¹/_cx)²

 

[Grimble]

Wrong. You are not using the relativistic velocity addition formula. You need to use that formula to find a vvv such that 2v/(1+v2)=0.62v/(1+v2)=0.62v / (1 + v^2) = 0.6. The correct solution to that equation is not v=0.3v=0.3v = 0.3. Try it. And then you will need to rework all the rest of your numbers.

Yes, point taken, it would perhaps have been better to have left the central observer out, but I have been told so many times that one cannot have a 'god's view'... . But anyway, those are not intended to be Spacetime diagrams. If anything they are Newtonian. I left time out deliberately and using no more than simple euclidean geometry(?) and Newtons laws of motion we can see and deduce time dilation. The speed of separation of the two clocks is 0.6c, less than the speed of light. That none of the speeds exceeds the speed of light is a stipulation that one can make.

 

[PeterDonis]

Alice measures Bob's light travel between point (0,0) and point (0.6,0.8).

Sure, if you pick that particular event--which, by the way, has coordinates (1, 0.6, 0.8) in Alice's frame if you are using proper spacetime coordinates (the "1" is the $t$ coordinate). But Alice also measures Bob's light travel from event (0, 0, 0) to event (0, 0.3, 0.4), or event (0, 0.9, 1.2), etc., etc. Bob's light travels on a continuous worldline. So if you are going to pick out event (1, 0.6, 0.8) for special consideration, what picks that event out? If your answer is, because that's where the light is after 1 unit of time in Alice's frame, then you need to be picking the right events to compare it to, and you are not. See below.

In Alice's Frame those two events - at point (0,0) and (06.08) - are 0.8 time units apart.

No, they aren't. The spatial points (0, 0) and (0.6, 0.8) aren't 1 time unit apart; that makes no sense, spatial points don't have a "time distance" between them. The events (0, 0, 0) and (1, 0.6, 0.8) on the worldline of Bob's light ray are 1 time unit apart in Alice's frame, as I just said above--and as can easily be confirmed by calculating the spacetime interval between (0, 0, 0) and (1, 0.6, 0.8) and confirming that it is zero, as it must be for a light ray.

The events you now appear to be referring to are events on Alice's worldline: those events are (0, 0, 0) and (0.8, 0, 0)--in other words, after 0.8 time units in Alice's frame, Alice is located at spatial coordinates (x, y) = (0, 0)--because she's always at those spatial coordinates in her own rest frame. Those two events are 0.8 time units apart, yes--but what does that have to do with the two events you picked out for Bob's light above? Answer: nothing whatsoever. If you want to see where Alice is, in her frame, at the same time as Bob's light is at spatial coordinates (0.6, 0.8), then you need to look at where Alice is after 1 unit of time in her frame. At that time she is at (t, x, y) = (1, 0, 0). So 1 time unit has elapsed for her--which should be so obvious as to not even need mention, but you have managed to confuse yourself into not believing it somehow.

Also, as you can see, Alice never occupies spatial coordinates (0.6, 0.8) in her frame. So those spatial coordinates have nothing to do with the time elapsed on Alice's clock. In fact, Bob never occupies those spatial coordinates either. After 1 unit of time in Alice's frame, Bob is at coordinates (t, x, y) = (1, 0.6, 0)--he has moved 0.6 units along the x axis, which is perfectly consistent with his speed of 0.6 relative to Alice. Bob never moves at all along the y axis, so his y coordinate is always zero; it's never 0.8.

This is what I mean about refusing to use standard tools. A standard tool in SR is an inertial frame--an assignment of a unique set of four numbers, (t, x, y, z), to each event. Here we always have z = 0, so we can ignore that coordinate; but instead of writing down, correctly, the (t, x, y) coordinates of all events of interest and then looking at their relationships, you are writing down (x, y) coordinates only--and not always the right ones, at that, as the above shows--and trying to reason about them without including the t coordinate. That doesn't work, and your posts are just illustrating that fact.

Another standard tool is a spacetime diagram. Try imagining a diagram (or drawing a projection of it on a sheet of paper) where Alice's worldline goes from (0, 0, 0) to (1, 0, 0); Bob's worldline goes from (0, 0, 0) to (1, 0.6, 0); and Bob's light goes from (0, 0, 0) to (1, 0.6, 0.8). That is a correct diagram that shows the correct relationships between those three objects and how they travel in 1 unit of time in Alice's frame. None of your diagrams show that kind of relationship, and in fact they are confusing you into thinking the relationship is something different and incorrect.

You have bombarded me with several more posts, but at this point I'm not even going to respond to them. You need to look at what I wrote above and take a big step back and start from scratch. Write down the proper coordinates for all events of interest in Alice's frame. Then look for relationships between them. If you keep on trying to use your personally invented tools, or trying to convince me that your analysis is correct without starting from scratch and using the standard tools, you are just going to confuse yourself further, and there will be no point in continuing this thread, and it will be shut down.

 

[PeterDonis]

That is just being pedantic!

I won't comment on the rest of your posts, but I have to comment on this. I am not being pedantic. I am telling you, repeatedly now, that you are getting the physics wrong. You need to get the physics right. Taking the suggestions I made in my previous post would be a good start.

 

[the_emi_guy]

Yet you appear to be trying to demonstrate something about SR from it. I don't understand what you are doing or why you think it is valid...

Obviously it is rather a pretty undefined drawing rather than a diagram helping to understand anything about relativity! You should update your knowledge about space-time diagrams if you like to understand special relativity properly. It's not such a complicated topic by the way!

The OP is not inventing anything new here. He is simply working through the very common "light clock" exercise that appears is virtually every introductory text on SR. This exercise does not require, and is introduced prior to, Minkowski space-time diagrams and the concept of proper time.
I already mentioned the Feynman text, but you can also look in Wikipedia under Time dilation to find examples of Grimble's diagrams (attached below)

There may be some improper terminology involved (Grimble is high school level after all) but let's not throw the baby out with the bath water.

 

[PeterDonis]

The OP is not inventing anything new here.

I'm aware of that. But he still needs to do the analysis properly, even if it's an analysis that has been done many times before. Also, he does appear to be trying to invent new tools to do the analysis with, and that doesn't seem to be working well.

This exercise does not require, and is introduced prior to, Minkowski space-time diagrams and the concept of proper time.

That might be true in some texts, yes. It's perfectly possible to correctly analyze the light clock using just inertial coordinates and the Lorentz transformations. But you still need to do it correctly; I pointed out several ways in which the OP was not.

you can also look in Wikipedia under Time dilation to find examples of Grimble's diagrams (attached below)

The diagrams you show are not the same as the diagrams Grimble has been posting (and which are confusing him, not helping him). I agree that your diagrams are fine and can be used to help with a correct analysis of the light clock. The key is to be clear about which frame one is using, and about which time coordinates in that frame go with the various events being illustrated.