Misleading Spacetime Diagrams

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There are some traps for the unwary in spacetime diagrams. I'll note a few; does anyone have more?

Let me use my favorite actors, Bob and Alice. Bob is the stationary observer; Alice is the moving one. Both are in inertial frames (or however one prefers to phrase it). We look at Alice's direction of movement and create a coordinate system for her such that her motion is completely in the positive X direction. We then set up Bob's axes to be parallel to each of Alice's.

There is some point at which Bob's and Alice's x coordinates are the same—as determined by Bob. For Bob, all the x coordinates on his worldline are 0. We set his clock time (t) to 0 at this point. For Alice, the situation is the same: all the x' coordinates on her worldline are 0 and we set her clock time (t') to 0.

We can now use the spacetime diagram to solve problems related to Bob and Alice, so this is very useful. It's like a geometric version of the Lorentz transform.

One pitfall is that when Bob and Alice have the same x coordinate, it is very tempting to think that they are at the same spot (at least, I keep falling into that trap). Some problems are actually stated that way and then the spacetime diagram is drawn, but the spacetime diagram looks no different if Bob and Alice are separated by wide distances along the y and z axes. While I can say that both Bob and Alice have a clock time of 0 at this magic location, I cannot say that they can observe each other's clocks to read 0 without adding that their y and z separation is 0—this information is not included in the diagram.

Another pitfall that I just ran into is in determining if two events have spacelike, timelike, or lightlike separation. I was thinking that I could just look at the angles made by a line connecting the two events on a spacetime diagram. Thinking about it, I decided I couldn't because we cannot ignore the y and z coordinates in making the determination (at least, I don't think so).

Neither of these are the diagram's fault but they are easy traps to fall into. Are there other things which spacetime diagrams mislead us about?
 

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  • #2
robphy
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Of course, a civil engineer designing a building needs more than a [single] planar diagram since the building is three-dimensional.

The usual (1+1)-dimensional position-vs-time graph (aka spacetime diagram)
is useful for problems involving one spatial dimension... in Galilean physics and Special Relativity.

If you have more than one spatial dimension to deal with,
then you need a higher-dimensional diagram.

I don't think that "diagrams mislead"... it's more that one has to learn how to read them correctly.
Failure to do so is "misreading" by the reader (not "misleading").
We have learned to read position-vs-time diagrams in PHY 101 because we have learned
to not treat the diagram with Euclidean geometry (since the x-vs-t graph is nonEuclidean).
 
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  • #3
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I don't think that "diagrams mislead"... it's more that one has to learn how to read them correctly.
It might be better to say that there is an unstated assumption about the y and z axes that must be recognized to read them correctly. When someone is drawing a spacetime diagram they expect that their audience will be aware of this; the diagram will mislead if this expectation is not met.
 
  • #4
robphy
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It might be better to say that there is an unstated assumption about the y and z axes that must be recognized to read them correctly. When someone is drawing a spacetime diagram they expect that their audience will be aware of this; the diagram will mislead if this expectation is not met.
That's fair.

My main point is that this issue
is an issue concerning ANY DIAGRAM that really needs more dimensions for a complete description.
It is not an issue unique to a spacetime diagram.

So, the term "spacetime" should be dropped from the title of this thread
... otherwise, it is "misleading" to pin the problem on spacetime diagrams.
 
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  • #5
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My main point is that this issue
is an issue concerning ANY DIAGRAM that really needs more dimensions for a complete description.
Yes, any diagram can be misread, but my questions was specifically about spacetime diagrams.

The answer depends on the knowledge of the person viewing the diagram, of course. People who know nothing about them and people who understand them fully are unlikely to be misled (for different reasons).

Typically, I've encountered these as a word problem turned into a diagram. The words add information to the diagram outside of the basic constraints: 2 observers, both moving at constant velocity. The trap comes if one starts from a diagram and then starts assuming additional constraints.

Let's revisit Bob and Alice. Let's say I state that an event occurs at (5, 7) relative to Bob. When does Alice see the event? The right answer is "who knows?" The wrong answer is to draw 45° lines from (5,7) until one of them intersects Alice's worldline and then read t or t'. Maybe you've never been there, but I have.

I suspect if I asked some physics teachers the question, they might have a list. I also suspect all common errors are variations of forgetting that one can only ignore the y and z values for some calculations, not all of them.

When someone is drawing a spacetime diagram they expect that their audience will be aware of this; the diagram will mislead if this expectation is not met.
It's not just the audience. Someone (like me) can draw a diagram to explore a problem and mislead oneself.
 
  • #6
PeterDonis
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Let's say I state that an event occurs at (5, 7) relative to Bob. When does Alice see the event? The right answer is "who knows?" The wrong answer is to draw 45° lines from (5,7) until one of them intersects Alice's worldline and then read t or t'.
I don't understand; on a correctly drawn spacetime diagram for 1 spatial dimension, light does travel on 45 degree worldlines. So what you are calling the "wrong" procedure is in fact the right one. If there are additional spatial dimensions involved, one has to instead look at the full light cone, but that's still perfectly well defined; it's not a matter of "who knows".
 
  • #7
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I don't understand; on a correctly drawn spacetime diagram for 1 spatial dimension, light does travel on 45 degree worldlines. So what you are calling the "wrong" procedure is in fact the right one. If there are additional spatial dimensions involved, one has to instead look at the full light cone, but that's still perfectly well defined; it's not a matter of "who knows".
Hi, Peter!

On a correctly drawn spacetime diagram that shows only 1 spatial dimension, light will appear to travel only on 45° worldlines. That doesn't mean that the situation being diagrammed has only one spatial dimension. Spacetime diagrams can be used for two observers in 3-dimensional space.

We can use these diagrams to answer certain questions, such as what Alice's (x', t') coordinates are for an event that Bob places at (5, 7). But we cannot use them to answer questions that require knowing the missing dimensions.

In the following diagram, the light worldline intersects Alice's worldline at Alice's (0, 9 point something) coordinates, but we can't say that 9 point something is when Alice sees the event, because we only know how far Alice is from the event in the x dimension. Without the missing y and z values for Alice and the event, the answer is "who knows?"

Clipboard01.png

For another example, we can look at the origin of both sets of axes. We can say that both Bob's and Alice's coordinates are (0, 0), but we can't say what time Bob would see if he looks at Alice's clock. At time 0, they could be separated by many light years.

This was my point about diagrams being misleading. We sometimes silently add assumptions to the diagram that aren't there. If we assume that the y and z values are the same for all observers and events, then there are a lot more questions we can answer.

Spacetime diagrams can have more restrictions, but as far as I know, they only require that we have two observers moving inertially and that we set up axes for each that are oriented in a specific way.
 
  • #8
Ibix
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Spacetime diagrams can have more restrictions, but as far as I know, they only require that we have two observers moving inertially and that we set up axes for each that are oriented in a specific way.
They don't even require that. It's usually of use to have the frames of some perpetually inertial observers, but it's not required. You can just pick an arbitrary frame and draw on that one, whether or not it's anyone's rest frame.

More generally, I don't really understand your objections. Sure spacetime diagrams suppress some information which can be relevant, but a well drawn one would either make sure that the information isn't relevant (##x=y=0## for all objects, for example) or note the shortcomings on the description.
 
  • #9
PeterDonis
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Spacetime diagrams can be used for two observers in 3-dimensional space.
Two-dimensional spacetime diagrams shouldn't be used for situations where more than two dimensions are relevant. But spacetime diagrams are not restricted to representing only two dimensions, although they are harder to read if they are representing three (you will find examples of these in various textbooks).

There is also the option of using more than one diagram for a given scenario, in order to capture more information than can be shown in a single diagram.

Without the missing y and z values for Alice and the event, the answer is "who knows?"
Then that is an issue with the problem specification, not the diagram. The problem specification should give this information. Perhaps it can't be represented in a diagram, but that doesn't mean there is no way to convey the information at all.
 
  • #10
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They don't even require that. It's usually of use to have the frames of some perpetually inertial observers, but it's not required. You can just pick an arbitrary frame and draw on that one, whether or not it's anyone's rest frame.
Sure, but then you're limited to either one observer or of having to add the restriction that all the observers that are moving relative to the chosen rest frame need to be moving in parallel directions.

No that there's anything wrong with that. :-) It's just that I love the fact that one can pick any two observers moving inertially and set up a system that allows one to visualize the interesting stuff using only a 2D graph.

If I add the restriction that all motion is in a parallel direction, I can have multiple observers and observers in non-inertial frames, which can be neat for certain problems.
More generally, I don't really understand your objections.
Well, the other day I was looking at someone's diagram, which was dotted with events, and I was thinking that I could determine if the event pairs were timelike, spacelike, or lightlike by just looking at the angle formed by a line connecting the events. The person who made the diagram didn't specify that the events all had the same y and z coordinates, so this was an example of me falling into a seductive trap (I figured out my mistake pretty quickly).

At an earlier point, I thought I could determine the clock times that one observer sees on another's clock. Again, this is true only if I add some restrictions or learn something about the missing y and z coordinates—it is not universal to all spacetime diagrams. On the other hand, I can calculate the clock time for the other observer at any point—no additional assumptions required.

I'm not critiquing spacetime diagrams—at all. I love 'em.
 
  • #11
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Two-dimensional spacetime diagrams shouldn't be used for situations where more than two dimensions are relevant.
The diagram I posted is just a diagram. There is absolutely nothing wrong with it.

A conclusion that the intersection of Alice's worldline and the event's light's worldline tells you anything about when Alice sees the event is not supported by the diagram. I wasn't presenting the diagram to show how to calculate when Alice sees the event; I was presenting to show that you can't use this diagram to answer the question.
Then that is an issue with the problem specification, not the diagram. The problem specification should give this information.
The problem specification was that this was a basic, standard spacetime diagram. There's nothing wrong with the problem specification or with the answer that one can't determine when Alice sees the event from it, not even by drawing the light worldline.

If I gave this diagram to someone and asked them when Alice saw the event, the correct answer would be "who knows?"

You might prefer a different problem, but that wouldn't be the problem I'm illustrating. My point is that it is easy to fall into the trap of thinking that the solution is 9 point something. I realize no one in this group would make that mistake, but I wonder how many wrong answers I'd get if I asked this question on an undergrad physics test.
 
  • #12
PeterDonis
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The diagram I posted is just a diagram. There is absolutely nothing wrong with it.
Sure there is: you say so yourself:

I was presenting to show that you can't use this diagram to answer the question.
Which means it's the wrong diagram to answer that question.

The problem specification was that this was a basic, standard spacetime diagram.
By "problem specification" I don't mean a general rule about diagrams, or anything else. I mean the information that is provided with a specific problem: for example, if we have a scenario involving Alice and Bob and a light signal between them, the problem specification should include information about the worldlines of Alice and Bob and the emission of the light signal. Specifying the worldlines of Alice and Bob and the event on Bob's worldline at which the light signal is emitted is sufficient information to determine at what event on Alice's worldline she receives the light signal. If that information can't be conveyed in a single diagram, then it needs to be conveyed some other way; and if a single diagram is the wrong tool to compute the answer, then some other tool that is the right tool should be used.
 
  • #13
PeterDonis
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If I gave this diagram to someone and asked them when Alice saw the event, the correct answer would be "who knows?"
That's because you did not provide them sufficient information about the scenario to compute the answer. That's an issue with your problem specification.

I wonder how many wrong answers I'd get if I asked this question on an undergrad physics test.
If you gave an undergraduate physics test and expected people to compute correct answers with insufficient information, you would not get very far as a physics teacher.

If you framed the question specifically as "do you have enough information to find the answer?", that would be different, but of course that framing would notify people that you might be giving them insufficient information, and that would change how they respond.
 
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  • #14
robphy
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A spacetime diagram to describe a (1+3) Minkowski-spacetime is 4-dimensional.
Similarly,
A position-vs-time plot to describe a particle moving in space is 4-dimensional.

Out of convenience, if the situation can be represented fully on a two-dimensional plane
(by choosing axes appropriately, if possible), then we can draw a two dimensional diagram and remind the viewer that two dimensions are suppressed.

If one is studying a situation from two observers, there's no guarantee that a 2-dimensional diagram
will unambiguously capture the situation.
If the observers meet at an event (so their worldlines are not skewed), and all interesting motion and events of interest happen in the plane spanned by their 4-velocities and have their x-axes chosen to be on this plane, then
a two-dimensional diagram suffices.

Otherwise, a higher-dimensional "diagram" (plot, figure) is needed.


Engineering graphic example:

Given only one orthographic view below (say the top plane),
can the full situation be unambiguously constructed?

Similarly, if a situation in spacetime
is not adequately captured by a single planar spacetime diagram,
then use a higher dimensional spacetime diagram (or multiple planar diagrams).


05fig01.jpg

(source: https://www.oreilly.com/library/view/engineering-graphics-with/9780134271019/ch05.html )
 
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  • #15
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That's because you did not provide them sufficient information about the scenario to compute the answer. That's an issue with your problem specification.
I think we're talking at cross-purposes. The topic you are pursuing with typical PeterDonis tenacity (and I like your tenacity on some things) is not a topic I'm particularly interested in.

The question that I was interested in can probably be dropped. I suspect all the answers involve forgetting about the y/z coordinates.

At this point in the thread, I have the same short list as I started with:
  • You can't calculate clock times seen without additional y/z information.
  • You can't determine whether two events are timelike, spacelike, or lightlike without additional y/z information.
  • If you want to include more than 2 observers or have a non-inertial observer, you need the additional stipulation that everyone's axes are parallel and that all motion is along the X axis.
There may be other bullets I could add to this list. If you have some to contribute, I'd still like to hear them.

When I talk about additional information or stipulations, I mean in addition to the basics of any 2D spacetime diagram. I've described this several times--2 inertial observers, axes parallel, the moving observer's motion only along the X axis (and an arbitrary rest frame still counts as one of the two observers--see the third bullet).

I will admit that the title of this thread was poorly chosen. Diagrams don't mislead; people misunderstand diagrams.
 
  • #16
PeterDonis
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The question that I was interested in can probably be dropped. I suspect all the answers involve forgetting about the y/z coordinates.
Of course. If information about the y and z coordinates is relevant, and you don't include it, obviously you can't expect to get the right answer.

I will admit that the title of this thread was poorly chosen. Diagrams don't mislead; people misunderstand diagrams.
I agree that people can misunderstand diagrams. I would also say that diagrams (or indeed any information) can be presented in a way that invites misunderstanding. For example, if you have a problem where you know more than one spatial dimension is relevant, and you never mention that fact while presenting a 2-D diagram, I would say you have invited misunderstanding.

When I talk about additional information or stipulations, I mean in addition to the basics of any 2D spacetime diagram.
I would say the general rule here is that only one spatial dimension should be relevant. If more than one is relevant, you should not be using a single 2D diagram. You might use a 3D diagram (or rather a 2D drawing that is a projection of a 3D diagram), or multiple 2D diagrams, or something else.
 
  • #17
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If one is studying a situation from two observers, there's no guarantee that a 2-dimensional diagram
will unambiguously capture the situation.
Yes, of course, but...

The standard spacetime diagram can take any four-dimensional problem involving two inertial observers and translate it to a two-dimensional graph that can unambiguously answer certain questions. That's the beauty of it, isn't it? So it isn't like I can just examine the objects in question and determine if a 2D representation is unambiguous—it will be unambiguous for certain kinds of questions and ambiguous for others.
 
  • #18
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I agree that people can misunderstand diagrams. I would also say that diagrams (or indeed any information) can be presented in a way that invites misunderstanding. For example, if you have a problem where you know more than one spatial dimension is relevant, and you never mention that fact while presenting a 2-D diagram, I would say you have invited misunderstanding.
Hmm...when I asked the original question, I had in mind that the person creating the diagram and the person interpreting it were one and the same.

The person in question was me. The misunderstandings I listed were traps I've fallen into (and dug my way out of). It's unlikely I'm the only one who has made some of these mistakes, so I was hoping for a list that might include ones I hadn't thought of.

Yes, this forum group is probably the wrong place to ask. A physics professor experienced at teaching undergrads may have accumulated a list of common ones. Or not. I don't know how much time one dwells on spacetime diagrams in physics classes.
 
  • #19
robphy
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a two-dimensional graph that can unambiguously answer certain questions. That's the beauty of it, isn't it?
Unambiguous….
restricting to “certain” questions
… um, ok. I guess that’s true since you have a lot of latitude to choose the certain questions….
 
  • #20
PeterDonis
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when I asked the original question, I had in mind that the person creating the diagram and the person interpreting it were one and the same.
In other words, you think up a scenario for yourself, give yourself insufficient information, and then try to get the answer?
 
  • #21
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At this point in the thread, I have the same short list as I started with:
  • You can't calculate clock times seen without additional y/z information.
  • You can't determine whether two events are timelike, spacelike, or lightlike without additional y/z information.
  • If you want to include more than 2 observers or have a non-inertial observer, you need the additional stipulation that everyone's axes are parallel and that all motion is along the X axis.

1. Yes you can, from the same diagram. Assuming there is no motion on the y-z surface. If there is a complicated motion (example - rotation on y-z) then this simple two dimensional diagram with t and x only is not suitable as a model for this problem.

2. Yes you can, if you have a point in Minkowski's diagram you can precisely assign to vectors (not events) if they are timelike or spacelike. Timelike - if one event is in "light cone" of another etc.

3. Yes but this is not a problem with Minkowski diagram, which originally has only t and x axes, you cannot from the very idea of x axis describe on it movements along other axes.


For another example, we can look at the origin of both sets of axes. [1] We can say that both Bob's and Alice's coordinates are (0, 0), [2] but we can't say what time Bob would see if he looks at Alice's clock. [3] At time 0, they could be separated by many light years.
[1]. This sentence: "both Bob's and Alice's coordinates are (0, 0)" means that they are both at the same point in spacetime described by Minkowski diagram. If you assume that they are separated along y or z axis this dwo-dimensial diagram is unsuitable for your problem. But this is not a problem with the diagram but with your mathematical model - you are trying to use a wrong tool here.

[2] From above: at (0.0) Bob sees 0 on Alice's watch. This is the very idea of notation (0.0), both A and B are at the same point in spacetime. When you write that they are (0.0) it precisely means that they are in (0.0) (time for both on their watches = 0, they are both at the point 0 on axis x), not one in (0, 0, 1, 7) and the other (4, 0, 2, 3).

[3] No, they cant, because the second cordinate in (0.0) equals zero, which means that they are not separated at all - their x-distance is precisely zero, they are at the same point. It is a direct consequence of [1]

I think I understand your problem now, I suppose you think that using only two axes in the diagram is something wrong because it doesn't describe positions or movements on other axes. But this is exactly the point - to present some problems in simplified form, without bothering with more complicated moves, that can be calculated directly if you understand the main idea of spacetime on t-x model.

The second part of the problem is that you don't understand fully the notation, this convention (a,b) means that event has coordinates (time = a, space = b) in some reference frame (coordinates). And if A and B have the same coordinates - they both are in the same point in space time.

The silent assumption here (understood well by many generations of physicists) is that in you write (t, x) it means that these two coordinates describe the state of your object (s) in full and other dimensions (y and z for example) are irrelevant for the presented problem.
 
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  • #22
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In space-time diagrams all you have to know is there is no relative motion in the Y and Z Direction. They could be in completely different points of space but the diagrams are still valid
 
  • #23
robphy
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In space-time diagrams all you have to know is there is no relative motion in the Y and Z Direction. They could be in completely different points of space but the diagrams are still valid
However, events (T,X,Y,Z) in the Y=2 Z=2 subspace
might be spacelike-related to those events (T,X,Y,Z) in the Y=0 Z=0 subspace.
Since their projections onto the Y=0 Z=0 subspace would discard this information,
the projected-events may not be spacelike related.

Someone unaware could be fooled into thinking that these original events were not spacelike related
since their projected-events were not spacelike related.

So, projections into subspaces that discard important information is bad.
Use a higher-dimensional diagram.
 
  • #24
vanhees71
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Or rather don't use spacetime diagrams at all but formulae. They tell more with less confusion ;-)).
 
  • #25
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Or rather don't use spacetime diagrams at all but formulae. They tell more with less confusion ;-)).
Nah, space-time diagrams are like the Rosetta Stone for relativity.
 

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