Twin paradox: who decided who is the younger one

[bobc2]

As a matter of fact, I presented a well-known alternative objective picture in an earlier thread* and again referred to it in my post #19 here; it seemed to directly answer the OP's question. So, what did I miss (or what did you miss)?

*https://www.physicsforums.com/showthread.php?t=539826&page=2 ; see also p.3

Thanks for jumping, harrylin. Your contribution is always good. I've read the Langevin paper again. It seems to me to be his summary of the special relativity theory with no new (for the time) interpretation. His summary seems to me to be consistent with Einstein's presentation of the theory. He refers to Minkowski's "world" and "world line" without any new embellishments.

I agree that his references to the 4-dimensional continuum, implied with his references to space and time, would be more consistent with ghwellsjr's emphasis of the 4-dimensional continuum as being three spatial dimensions and one time dimension. So, beyond that I don't see this as refuting a 4-dimensional space concept.

Logically, once you have a 4-dimensional continuum where you refer to the 4th dimension as time (provided the continuum is consistent with special relativity), you are automatically implying a 4th spatial dimension. This is manifestly so when looking at any space-time diagram representing a 4-dimensional continuum. And X4 is equivalent to ct. Different cross-section cuts through the 4-dimensional continuum (whether you use X4 or ct) do not separate out time from spatial dimensions.

This is obvious when you imagine extruding a 3-D space along the "time dimension." That is, an observer, when advancing along his world line, always observes a continuous sequence of 3-D spaces. Thus, it is seen that the world line at each successive point populates the sequence of 3-D spaces. But that means that as you move along the world line you move through space.

Einstein often spoke of the 4-dimensional continuum in the context of 3 spatial and 1 time (again, consistent with hgwellsjr's comments). He spoke of the problem of "Now" in special relativity and commented that physicists make no distinction between past, present and future. He seemed clear in his understanding of an external objective physical reality that embodies a 4-dimensional continuum. And he used

ds^2 = dX1^2 + dX2^2 + dX3^2 - dX4^2

as a 4-dimensional line element.

 

[yoron]

You seem both to define it similarly to me?

So how about defining where you differ in your definitions? If I get you right Bob? You want SpaceTime to be a 'absolute' 4-dimensional, not differing 'time' from the other 'dimensions', as a unified 'jello' of sort? And that this 'jello' represent all observers?

Or do I get you wrong there?

And the other definition seems to be one in where 'time' has a unique flavour, although being a 'dimension' too? But both agree on that in 'reality' all 'dimensions' must be included for a time dilation, that is if I read you right?

 

[harrylin]

Thanks for jumping, harrylin. Your contribution is always good. I've read the Langevin paper again. It seems to me to be his summary of the special relativity theory with no new (for the time) interpretation. His summary seems to me to be consistent with Einstein's presentation of the theory. He refers to Minkowski's "world" and "world line" without any new embellishments.

I agree that his references to the 4-dimensional continuum, implied with his references to space and time, would be more consistent with ghwellsjr's emphasis of the 4-dimensional continuum as being three spatial dimensions and one time dimension. So, beyond that I don't see this as refuting a 4-dimensional space concept.

I think that the 4-dimensional space concept (in the way of a rather weird 4D physical space, not just mathematical space) is incompatible with Langevin's concept of a physical space: a stationary ether, of which we can detect the existence by a change of motion relative to it.

Logically, once you have a 4-dimensional continuum where you refer to the 4th dimension as time (provided the continuum is consistent with special relativity), you are automatically implying a 4th spatial dimension. [..]

Only if you interpret that 4th dimension as a spatial dimension; and clearly Langevin can not have meant that, for the reason that I just mentioned.
For example also temperature is a dimension, see http://en.wikipedia.org/wiki/Dimensional_analysis.

Einstein often spoke of the 4-dimensional continuum in the context of 3 spatial and 1 time (again, consistent with hgwellsjr's comments). He spoke of the problem of "Now" in special relativity and commented that physicists make no distinction between past, present and future. He seemed clear in his understanding of an external objective physical reality that embodies a 4-dimensional continuum.

Even Einstein stressed that (according to him, perhaps not Minkowski) space is a three-dimensional continuum, and that in contrast the "four dimensional space" of Minkowski is the "world" of events (brackets his).
- http://www.bartleby.com/173/17.html (that fits rather nicely with Langevin's speech; I wonder if he was influenced by it?).

Events take place in the physical world; the "4D continuum" of events is not the physical world itself. Another way to put it: "the map is not the territory".

As this is of course mostly* a matter of metaphysics, the point here is that the same mathematics has been interpreted or explained in very different ways, based on very different views of the physical world.

Harald

*only mostly: I don't think that you can walk in the negative time direction

PS I hope that the above also answers some of yoron's questions: there is no need to believe in a unified 'jello' of sorts!

 

[bobc2]

You seem both to define it similarly to me?

So how about defining where you differ in your definitions? If I get you right Bob? You want SpaceTime to be a 'absolute' 4-dimensional, not differing 'time' from the other 'dimensions', as a unified 'jello' of sort? And that this 'jello' represent all observers?

Or do I get you wrong there?

And the other definition seems to be one in where 'time' has a unique flavour, although being a 'dimension' too? But both agree on that in 'reality' all 'dimensions' must be included for a time dilation, that is if I read you right?

Yoron, I think you are pretty much on target with your summary here. And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority.

I recall one physicist writing on this subject fairly recently commenting that most physicists now embrace the "block universe" concept. He provided no basis for making that statement, and I'm doubtful it is true. The UK physicist, Cox, who does a lot of popular physics presentations with books and videos, championed the block universe concept on BBC. It was available on YouTube for a while until BBC had it blocked. In Brian Greene's video he illustrates the concept of observers having different cross-section views of the 4-dimensional continuum by slicing a loaf of bread in different directions. There of course is nothing original with me about this subject, and you can find many physicists accept the concept if you just do some googling on "block universe" and "block time". I don't believe it is an "out-of-the-mainstream" subject.
.
I hope the airing of these views has been fruitful for visitors to the thread. I'll have to mull over harrylin's post. He always does a good job with his posts. Maybe there is more to say--and maybe we've all pretty much presented our views. Perhaps others will be motivated to research this question.

 

[harrylin]

Yoron, I think you are pretty much on target with your summary here. And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority.
I recall one physicist writing on this subject fairly recently commenting that most physicists now embrace the "block universe" concept. He provided no basis for making that statement, and I'm doubtful it is true. The UK physicist, Cox, who does a lot of popular physics presentations with books and videos, championed the block universe concept on BBC. It was available on YouTube for a while until BBC had it blocked. In Brian Greene's video he illustrates the concept of observers having different cross-section views of the 4-dimensional continuum by slicing a loaf of bread in different directions. There of course is nothing original with me about this subject, and you can find many physicists accept the concept if you just do some googling on "block universe" and "block time". I don't believe it is an "out-of-the-mainstream" subject.

It is often claimed that the interpretation that you presented here - although you indicated that you do not like it very much - is the majority interpretation. It might be interesting to do a poll here, similar to the one about interpretations of QM; but popularity is not all-important.

My point was that an alternative interpretation has been known from the start, and again other interpretations exist (e.g. "physical relativity" as expressed by Harvey R. Brown). It's not necessary to get hung up on what may be the most popular interpretation, especially for those for who that interpretation doesn't make much sense.

I hope the airing of these views has been fruitful for visitors to the thread. I'll have to mull over harrylin's post. He always does a good job with his posts. Maybe there is more to say--and maybe we've all pretty much presented our views. Perhaps others will be motivated to research this question.

Thanks! Let's hope that it was useful for someone. And perhaps Passiday, who started this topic, has a comment.

 

[ghwellsjr]

Here's another example. The hyperbolic calibration curves were computed in MatLab.

Why in your first example do you call the hyperbola curve "Proper Distance" and in the second one "Proper Time"?

 

[nitsuj]

I stopped reading Brian Greene's book fabric of spacetime because of him laying the block universe idea so heavily as the way to interpret 4D. I wasn't up for the same force feeding of strings.

A second book I got called Relativity a brief introduction by Russell Stannard (a more [STRIKE]scientific approach to explaing[/STRIKE] accurate explanation of SR/GR) also proposes the block universe as a better interpretaion of 4D. Superior to Greene though, Stannard does mention that this is not the interpretation of most physicists, and also lists oddities of both conceptions.

As I slowly understand spacetime diagrams better, I see the perspective of Greene and Stannard more clearly, but think it's taking the idea of spacetime diagrams a little too far (visualizing 4Ds at right angles to each other, and coming up with block universe or whatever).

 

[Passiday]

Guys, I am overwhelmed by your feedback. I asked what I thought a pretty basic question, but got a very elaborate explanations, with accompanying heated discussion. I bow in front of your commitment.

I kind of dislike the oversimplification in science, you know, when talking about not-that-trivial things. I think, most of people when read about the twin paradox, take it at face value and then they think they've got the relativity. But I've got confused about the complex trajectories of both twins, in the field of gravity of the Earth and the Sun. The most important point I take from this discussion is that the frame of reference is irrelevant (the aging difference holds disregarding the FoR), and what serves here as the definition of the speed is "that what happens to an object that just felt acceleration". However, the fact that there would be an acceleration to be felt even in totally empty universe, that escapes my deep understanding, feels like mystery. But I am happy to learn that Einstein and Mach were not that clear on that subject either :)

 

[nitsuj]

However, the fact that there would be an acceleration to be felt even in totally empty universe, that escapes my deep understanding, feels like mystery.

Why would feeling acceleration in "totally empty universe" be mysterious?

 

[Dale]

I asked what I thought a pretty basic question, but got a very elaborate explanations, with accompanying heated discussion.

By the way, I very much liked your thread title. For me it calls to mind this mental image of Einstein and his contemporaries in a secret meeting behind closed doors in a room filled with cigar smoke taking a vote to decide which twin is younger.

 

[tom.stoer]

To define the 'lengths' of the curves you said "Here the "length" and therefore the proper time is calculated according to the strange 4-dim. relativistic Pythagoras t² - x²." The variables "t" and "x" in that expression are not arbitrary, they must be inertial coordinates, ...

Of course not!

The definiton of the line integral is completely independent from any specific reference frame. If you want to calculate something you can use a coordinate system i.e. you introduce t and x. But the geometric property 'the length of a timelike curve C in a pseudo-Riemannian manifold' does not depend on a coordinate system. Neither does the geometric property 'this timelike curve is longer than that timelike curve' depend on a coordinate system.

 

[bobc2]

Why in your first example do you call the hyperbola curve "Proper Distance" and in the second one "Proper Time"?

The upper and lower cones (inside the cones) are normally referred to as "timelike" and the outside of the cone is referred to as "spacelike." Points on the cone sufrace are "lightlike."

The term "Proper Time" normally refers to measurements along world lines inside the light cone. I usually stick to that convention unless I am trying to emphasize the X4 distance along the 4th dimension in accordance with X4 = ct. Actually, some authors use units of spatial distance for Proper Time (usually with the greek tau symbol).

 

[bobc2]

Then why did you label your axes X₁ and X₄ instead of x and ct like everyone else does? Can you show me an example of a Minkowski space-time diagram not promoting Block Universe Theory that is labeled like you did yours?

Here is a copy of a space-time diagram from the textbook "The Geometry of Spacetime" by Gregory L. Naber. He uses X1, X2, X3, and X4 notation throughout the book (along with ct and tau at times).

The author never mentions block time, nor can you find the term in the index. He does not engage in any interpretations of spacetime--just deals with the math. He takes a fairly formal approach to the subject.

 

[ghwellsjr]

We had a misunderstanding. I was referring to the following space-time diagram in the wiki link:This shows the planes of simultanaeity for the travel twin, which are exactly the diagonal lines I mentioned in earlier posts (yes, they represent 3-D cross-sections of a 4-dimensional space, x2 and x3 being supressed).

I can see lines of simultaneity in the drawing and you say they represent 3-D cross-sections of a 4-D space, so why do you call them "planes of simultaneity" instead of "volumes of simultaneity"?

 

[ghwellsjr]

...look closely and you will see that I included that hyberbolic calibration curves so that we could make comparisons between the two coordinate systems. I used the 5 year calibration curves for each of the two inertial start events for the traveling twin.

I'm not understanding what these calibration curves are for or even how you derived them. On the bottom one, you show it going through the 5 year Proper Time for the traveling twin and through the 5 year Proper Time of the stationary twin.

Later you said:

My slanted lines were not meant to represent some standard space-time configuration. And, again, they were certainly not intended to be the planes of simultaneity shown in the sketch above (we had already discussed those more than once in this thread). Again, I just added them to the space-time diagram so you could follow the synchronous proper times of the twins.

Are you saying that you have a calibration scheme that allows you to determine that the twins have synchronized clocks or that it provides some means by which to "calibrate" them? It seems rather trivial to me if all you want to do is draw a line from a particular Proper Time for one twin to the same Proper Time for the other twin but then I don't understand why you would need a calibration curve.

 

[bobc2]

I'm not understanding what these calibration curves are for or even how you derived them. On the bottom one, you show it going through the 5 year Proper Time for the traveling twin and through the 5 year Proper Time of the stationary twin.

Later you said:

Are you saying that you have a calibration scheme that allows you to determine that the twins have synchronized clocks or that it provides some means by which to "calibrate" them? It seems rather trivial to me if all you want to do is draw a line from a particular Proper Time for one twin to the same Proper Time for the other twin but then I don't understand why you would need a calibration curve.

I'll add some more commentary a little later. For now, let me provide these sketches with a little bit of commentary.

Equation 1) is derived from the upper left sketch. We have sketched the space-time diagram for a red guy moving to the left and a blue guy moving to the right. Both red and blue are moving at the same speed with respect to the rest black system. This is necessary in order that line lengths on the screen for red and blue will have the same scaling (one inch along a red coordinate has the same physical value as one inch on the corresponding blue coordinate). If you don’t use symmetric coordinate systems in this way, you must use hyperbolic calibration curves to compare physical distances among coordinate systems. This is what we wish to make clear with the hyperbolic curve derivation accompanying the sketches.

These slanted coordinate systems arise from special relativity theory. These unusual looking coordinates are selected as the only coordinates that always yield the same speed for light: c. That’s because the world line of a photon of light always bisects the angle between the X4 coordinate and the X1 axis.

Note that the X4 axis for a moving observer is rotated with respect to the rest system X4 axis (the slope is proportional to the speed). Then, the moving observer’s X1 axis is rotated so as to always maintain symmetric rotation with respect to a photon world line (which is always rotated to a 45-degree angle in the rest system).

Now, we see that the blue X1 axis is perpendicular to the red X4 axis. You will find this is the situation for any pair of symmetric coordinate systems. Further you can always find a rest system for which observers moving relative to each other will move in opposite directions at the same speed. So, contrary to some objections, this derivation is not a special case—it has completely general application. This allows us to write the Pythagorean Theorem equation involving the red and blue coordinates. The time dilation Lorentz transformation equation can be derived directly from the Pythagorean Theorem.

Here, we just want to derive the Proper Time hyperbola equation, i.e., equation 2) above. This equation may be modified for time scaling, using X4 = ct (we use units of years for time and use the compatible units of light-years distance along the X1 axis as shown when plotting the graphs for equation 3).

Equation 4) is shown for a constant value of 10 for the Proper Time. See the corresponding plot. This plot shows the points along a hyperbolic curve in the black rest system that correspond to a fixed Proper Time value of 10 years. The red slanted lines terminating on the hyperbola represent example world lines (time axes) associated with possible observers moving at various speeds.

So, even though the line lengths on the computer screen are different in the rest black rectangular coordinate system, the Proper Times are all the same.

 

[ghwellsjr]

Bob, I appreciate the time you are spending on this. I can see now what your curve is.

But you can get the same curve using the Lorentz Transform by plugging in t=10 and x=0 and then sweeping β from +infinity to -infinity and plotting the locus of [t',x'] points. You are showing how a single event can transform into all possible frames.

But why? What has that got to do with anything?

Did you discover this on your own or can you point me to an on-line reference that explains this calibration curve and what its purpose is?

 

[bobc2]

Did you discover this on your own or can you point me to an on-line reference that explains this calibration curve and what its purpose is?

ghwellsjr, let me get back to you later this afternoon with more complete response to your questions. I did a quick google search and did not come across discussions of proper time that included the space-time diagram with hyperbolic curves. I'll look some more. In the meantime here is a figure from the Naber special relativity textbook. But, no--this stuff is definitely not original with me by any means. My first encounter with the proper time calibration curves was in an udergraduate course on Modern Physics. They were also used by my special relativity prof in grad school.

 

[ghwellsjr]

But can't you give me a quick idea of why you do it? What are you calibrating? How do you use the curves once they are drawn?

 

[yoron]

Bob, you wrote "And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority."

Well, I think of it this way too, but when I think of 'time' I see it as a very 'local definition', radiation and gravity then becoming what gives us the 'whole unified experience' of SpaceTime. Doing the later one might assume that 'locality' solves it all, but if it is so that 'time', or better expressed, your local 'clock' defines all other frames of reference then there still will exist all those other 'frames of reference' defining you relative their 'clocks'. So even if 'times arrow' can be defined locally SpaceTime is very much like a jello to me.

Which makes it very understandable that some want 'time' to be anything than what it is :)

Eh, the last one was a slight joke relative entropy.

 

[bobc2]

But can't you give me a quick idea of why you do it? What are you calibrating? How do you use the curves once they are drawn?

Hi, ghwellsjr. Here is the short story. Example a) is a spacetime diagram with black rest frame and blue frame moving relative to rest frame. But you cannot compare times between the black frame and the blue frame. Example b) uses the hyperbolic calibration curves which allow you to compare times between t and t'. And you can see how much time dilation there is for the blue guy looking at a clock along black's world line (t axis). When blue's calendar says 30 years, he "sees" (correcting for light travel time, etc.) black's calendar showing about 26 years. You can measure the slope of blue's time axis to see how fast he is moving with respect to the black rest system.

By the way, notice that the X1 axis of blue is tangent to the hyperbolic curve at the time point of interest.

 

[bobc2]

Bob, you wrote "And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority."

Well, I think of it this way too, but when I think of 'time' I see it as a very 'local definition', radiation and gravity then becoming what gives us the 'whole unified experience' of SpaceTime. Doing the later one might assume that 'locality' solves it all, but if it is so that 'time', or better expressed, your local 'clock' defines all other frames of reference then there still will exist all those other 'frames of reference' defining you relative their 'clocks'. So even if 'times arrow' can be defined locally SpaceTime is very much like a jello to me.

Which makes it very understandable that some want 'time' to be anything than what it is :)

Eh, the last one was a slight joke relative entropy.

Thanks for the comments and insight, yoron. You've given me something I'll have to reflect on for a while. You have hit on a point that has to be considered. I think of special relativity locally, but at the same time can envision a continuous sequence of light cones along world lines curving through curved space-time--the cones tipping more and more as they approach massive objects. In that sense I favor a more global application of special relativity.

 

[yoron]

Well, I think you can see it both ways, you start from a whole 'perspective', I start from a local. But as long as we both agree on that SpaceTime existing for all observers we should meet at some, eh :) 'point'. To me it feels simpler to define 'times arrow' from locality but the 'Jello' won't go away because of that. It just makes me look at 'frames of reference' and 'time' from another angle.

As I see it this was the way Einstein defined SpaceTime too, as a 'whole', using 'c' as the constant defining it, together with Gravity/acceleration relative motion. Maybe a little simplistic, but?

 

[harrylin]

Why would feeling acceleration in "totally empty universe" be mysterious?

Easy: if one accelerates relative to nothing, there would also be nothing to cause an effect from it.

 

[harrylin]

[..] I think, most of people when read about the twin paradox, take it at face value and then they think they've got the relativity. But I've got confused about the complex trajectories of both twins, in the field of gravity of the Earth and the Sun.

In the usual (SR) discussion the time dilation due to gravity fields are neglected, and indeed they just add unnecessary complexity for the understanding of SR time dilation.

The most important point I take from this discussion is that the frame of reference is irrelevant (the aging difference holds disregarding the FoR), and what serves here as the definition of the speed is "that what happens to an object that just felt acceleration".

Sorry but that is wrong: it has nothing to do with "feeling". Please read again the discussion by Langevin: he uses the orbit around the far away planet for the turn-around, so that the acceleration is not felt. What matters is the change of velocity.

However, the fact that there would be an acceleration to be felt even in totally empty universe, that escapes my deep understanding, feels like mystery. But I am happy to learn that Einstein and Mach were not that clear on that subject either :)

Although he was always a bit foggy about such topics, Einstein admitted (at least around 1918-1924) that "empty space" can't be truly empty. Indeed, such a view is inconsistent with field theory. See for example: http://en.wikisource.org/wiki/Ether_and_the_Theory_of_Relativity

Harald

 

[tom.stoer]

Easy: if one accelerates relative to nothing, there would also be nothing to cause an effect from it.

In that sense an empty universe has still the geometrical property to define geodesics. So you feel acceleration w.r.t. these geodesics (I guess this is not what Mach had in mind).

 

[ghwellsjr]

Hi, ghwellsjr. Here is the short story. Example a) is a spacetime diagram with black rest frame and blue frame moving relative to rest frame. But you cannot compare times between the black frame and the blue frame. Example b) uses the hyperbolic calibration curves which allow you to compare times between t and t'. And you can see how much time dilation there is for the blue guy looking at a clock along black's world line (t axis). When blue's calendar says 30 years, he "sees" (correcting for light travel time, etc.) black's calendar showing about 26 years. You can measure the slope of blue's time axis to see how fast he is moving with respect to the black rest system.

By the way, notice that the X1 axis of blue is tangent to the hyperbolic curve at the time point of interest.

Thanks again, Bob, for putting your time into making these graphics. I now understand what the calibration curve is for and how it is used.

I get the impression that back in the "old" days, before computers or even calculators, there must have been preprinted Minkowski diagrams available with the calibration curves already in place so that the user could label the black axes, draw in his sloping blue axis for whatever β he was interested in, and then he could easily use the calibration curves to label his blue axis--all without doing any calculation except determining the slope of the blue axis, which he could get from a lookup table (along with γ and its reciprocal).

But we have computers now which I'm sure you used to calculate and plot the calibration curves which is more work than simply plotting the blue axis with its appropriate labels.

You point out that the tangent of the calibration curve at the blue axis allows you to easily see the time dilation on the black axis but it is even easier to see if you start at the 30-year point on the black axis and just look at the horizontal (tangent) line going over to the blue axis and see the time dilation there 26 years. And once you know that, you also know that at 30 years for blue, he will "see" 26 years for black.

If the whole purpose of this is to graphically show on a Minkowski the reciprocal nature of time dilation, then why didn't you point this out?

However, all the same things can be shown using just the Lorentz Transform (which is the source of the information that gets drawn on a Minkowski diagram) so why not just stick with the exact numbers that you get from the Lorentz Transform, now that we all have computers and calculators? They work even when the values of β are close to zero or close to one where the Minkowski diagram becomes very difficult to evaluate.

 

[bobc2]

Thanks again, Bob, for putting your time into making these graphics. I now understand what the calibration curve is for and how it is used.

And thanks for your ideas on this subject.

I get the impression that back in the "old" days, before computers or even calculators, there must have been preprinted Minkowski diagrams available with the calibration curves already in place so that the user could label the black axes, draw in his sloping blue axis for whatever β he was interested in, and then he could easily use the calibration curves to label his blue axis--all without doing any calculation except determining the slope of the blue axis, which he could get from a lookup table (along with γ and its reciprocal).

But we have computers now which I'm sure you used to calculate and plot the calibration curves which is more work than simply plotting the blue axis with its appropriate labels.

I'll bet you are right about that. And yes, I used MatLab to do the math, then copied and pasted into Microsoft Paint to add a couple of things.

You point out that the tangent of the calibration curve at the blue axis allows you to easily see the time dilation on the black axis but it is even easier to see if you start at the 30-year point on the black axis and just look at the horizontal (tangent) line going over to the blue axis and see the time dilation there 26 years. And once you know that, you also know that at 30 years for blue, he will "see" 26 years for black.

A really good point. Thanks for pointing that out.

If the whole purpose of this is to graphically show on a Minkowski the reciprocal nature of time dilation, then why didn't you point this out?

I don't know. I guess I was originally more focused on using the diagram to emphasize it's representation of the 4-dimensional continuum and possibility of viewing objects as 4-dimensional as well.

However, all the same things can be shown using just the Lorentz Transform (which is the source of the information that gets drawn on a Minkowski diagram) so why not just stick with the exact numbers that you get from the Lorentz Transform, now that we all have computers and calculators? They work even when the values of β are close to zero or close to one where the Minkowski diagram becomes very difficult to evaluate.

You have a good point there. If space-time diagrams don't do anything for folks, then just stick to the calculations as you say.