World Lines & (x, y, z, t)

Dinosaur

Rational Skeptic
Valued Senior Member
Einstein (I think) in both SR & GR developed the concept of an event specified as occurring at space-time coordinates (x, y, z, t). This leads to the concept of World Lines: A set of (x, y, z, t) coordinates which describe the path of a particle. It might have been Minkowski rather than Einstein who developed the concept of World Lines & the notion of viewing Einstein's theories as Metric Geometery in a 4D Space-Time continuum. The 4D Space-time continuum is often referred to as Minkowski Space, indicating that the concept is probably his & adopted as useful by Einstein.

The concept provides answers to some philosophical arguments which lasted for centuries.

One such argument relates to the concept of the continued existence of an object, either animate or inanimate. What does it mean to claim that an object viewed today is the same object it was yesterday? Note that it is obvious that any object gains or loses atoms/molecules from day to day. It is obvious that the atoms/molecules comprising an object change their configuration from day to day. If an object is defined as a set of related World Lines, the philosophical argument is resolved.

An atom/molecule which moves some distance from the object is no longer a member of the set. Its World Lines are no longer related to the other World Lines in the set.

An atom/molecule which moves from some distance & becomes part of the object becomes a member of the set: Its World Line is now a member of the set.

Atoms/molecules which change their configuation are still part of the set.​

The above model describes a 4D Space-Time continuum which is static: There is no motion.

Whether a person likes or dislikes the notion of a static universe, the model is very useful.

Those who are comfortable with the notion of a motionless 4D universe can accept the model as is.

Those uncomfortable with the notion can make use of the model & reject the notion of a motionless universe. They take the reasonable POV that a model is only a model, it is not reality.

The concept provides answers to some philosophical arguments which lasted for centuries.

One such argument relates to the concept of the continued existence of an object, either animate or inanimate. What does it mean to claim that an object viewed today is the same object it was yesterday? Note that it is obvious that any object gains or loses atoms/molecules from day to day. It is obvious that the atoms/molecules comprising an object change their configuration from day to day. If an object is defined as a set of related World Lines, the philosophical argument is resolved.

An atom/molecule which moves some distance from the object is no longer a member of the set. Its World Lines are no longer related to the other World Lines in the set.

An atom/molecule which moves from some distance & becomes part of the object becomes a member of the set: Its World Line is now a member of the set.

Atoms/molecules which change their configuation are still part of the set.​

The above model describes a 4D Space-Time continuum which is static: There is no motion.
A nice idea to resolve the issue of "is past you the same you as the present, and the future?"
However how small will the spacetime interval between the worldlines have to be for it to be related?

And after defining what constitute a set of worldlines, does it means that if I somehow mange to get close to you such that our spacetime interval is smaller than the specificied value for worldines to be related, does that means at that moment in time w.r.t. you and me, we are part of each other?

Whether a person likes or dislikes the notion of a static universe, the model is very useful.
Those who are comfortable with the notion of a motionless 4D universe can accept the model as is.
Those uncomfortable with the notion can make use of the model & reject the notion of a motionless universe. They take the reasonable POV that a model is only a model, it is not reality.
The only thing I don't get for spacetime models is how to recover the common sense notion that time is flowing

I rmb I asked my professor in my general relativity course, and he explained that the proper time for each observer can be interpret as measuring the "ticking" that we experience in daily life, which give us the notion that time is flowing (w.r.t. each observer itself)
However for the direction, he and Schutz (A First Course in General Relativity) said that actually spacetime diagrams and light cones only tell there's a past and future, but the equations are time reversible thus made no distinction on which half of the light cone is the future and which is the past. He then said you need to put in the observed fact that the universe seemed to have a lower entropy state (less deg of freedom to arrange matter) and the 2nd law of thermodynamics to recover the arrow of time

But would that means, if we reach heat death, then we cannot tell what is past or present. In that case what will spacetime diragrams be like and which direction will the worldlines be extending towards?

...The above model describes a 4D Space-Time continuum which is static: There is no motion.
Well said that man. Many physicists or amateur physicists don't appreciate this. They talk of things like "moving through spacetime" and just don't understand that there can be no motion in an "all-times-at-once" depiction.

Whether a person likes or dislikes the notion of a static universe, the model is very useful. Those who are comfortable with the notion of a motionless 4D universe can accept the model as is. Those uncomfortable with the notion can make use of the model & reject the notion of a motionless universe. They take the reasonable POV that a model is only a model, it is not reality.
It's a useful model. But the map is not the territory.

Secret said:
The only thing I don't get for spacetime models is how to recover the common sense notion that time is flowing.
There's no actual evidence or observation that "supports this common sense notion". If you open up a clock you don't see time flowing through it. You see cogs and sprogs whirring round. We live in a world of space and motion. The flow of time or the passage of time is just a figure of speech.

Farsight's theory of time is worthless. It's contrary to relativity. Just as rotations mix different space directions, Lorentz boosts mix space and time directions.

Let's consider a general possible one:
t' = g0*(t - w*x)
x' = g1*(x - v*t)
y' = y, z' = z
where g0, g1, and w are functions of velocity v.
Now rotate both (t,x,y,z) and (t',x',y',z') 180 degrees around the z-axis. One gets
t' = g0*(t + w*x)
x' = g1*(x + v*t)
y' = y, z' = z
Like the original expression, but with the velocity reversed. w is also reversed, but g0 and g1 are not. So I'll treat it as the inverse of the first one. Combining the two gives
g0 = g1 = g
g[sup]2[/sup]*(1 - v*w) = 1

Now combine two Lorentz boosts, for velocities v1 and v2. It ought to be another Lorentz boost, and we find w(v1)/v1 = w(v2)/v2, so w(v) = K*v and g(v) = (1 - K*v[sup]2[/sup])[sup]-1/2[/sup]

The velocity addition law becomes v12 = (v1 + v2)/(1 + K*v1*v2)

If something that travels at c is always observed to travel at c, then (c - v)/(1 - K*c*v) = c, or K = 1/c[sup]2[/sup]. Thus getting the familiar Lorentz-boost formula.

Let's change the velocity variable: v = c*tanh(u) This gives us
t' = t*cosh(u) - x/c*sinh(u)
x' = x*cosh(u) + c*t*sinh(u)

That looks much like the rotation formula, but with hyperbolic functions. Since rotations preserve distances, let's see what distances are preserved here. It's easy to show that
(c*t')[sup]2[/sup] - x'[sup]2[/sup] - y'[sup]2[/sup] - z'[sup]2[/sup] = (c*t)[sup]2[/sup] - x[sup]2[/sup] - y[sup]2[/sup] - z[sup]2[/sup]

So we get an invariant distance, one that's zero for particles traveling at c.

This, time is much like another space dimension.

This remarkable result was first discovered by Hermann Minkowski in 1908. From the Wikipedia article on him,
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

...Farsight's theory of time is worthless. It's contrary to relativity...
It isn't. See A World without Time: The Forgotten Legacy of Godel and Einstein. But note that it's "time does not exist as we ordinarily understand it". It exists like heat exists. It isn't fundamental, but a hundred years will kill you just as surely as a hundred degrees centigrade.

lpetrich said:
...So we get an invariant distance, one that's zero for particles traveling at c...
You don't understand this at all Loren. It's an invariant interval. The distance isn't zero, it's the light-path length. See Simple inference of time dilation due to relative velocity. Whether the parallel-mirror light clock stays at home or goes on an out-and-back trip, the light-path length is the same. Time is nothing like another space dimension. I can hop backwards a metre but you can't hop backwards a second. That's why time travel is science fiction.

lpetrich said:
...Farsight's theory of time is worthless. It's contrary to relativity...
It isn't. See A World without Time: The Forgotten Legacy of Godel and Einstein.
Yawn. What makes that book revealed truth?
But note that it's "time does not exist as we ordinarily understand it". It exists like heat exists. It isn't fundamental, but a hundred years will kill you just as surely as a hundred degrees centigrade.
Splitting hairs. Time being closely related to space is an important part of relativity, and if you deny that, you deny Minkowski and Einstein. If they are prophets of revealed truth, then that's a VERY wrong thing to do.
lpetrich said:
...So we get an invariant distance, one that's zero for particles traveling at c...
You don't understand this at all Loren. It's an invariant interval. The distance isn't zero, it's the light-path length. See Simple inference of time dilation due to relative velocity. Whether the parallel-mirror light clock stays at home or goes on an out-and-back trip, the light-path length is the same. Time is nothing like another space dimension. I can hop backwards a metre but you can't hop backwards a second. That's why time travel is science fiction.
Actually, the purely-spatial distance traveled is *not* an invariant. It's coordinate-dependent. I can work out the math, and if you can't understand it, it's your problem and your problem only. It's the math that Einstein and Minkowski had used, so if they are prophets of revealed truth, then it would be wrong to deny their math.

In flat spacetime, there are five types of directed interval between two space-time points.
• Forward timelike
• Backward timelike
• Forward null (lightlike)
• Backward null (lightlike)
• Spacelike
In curved spacetime, it's more complicated in general, but this discussion holds true of vectors at each point.

An object doing forward timelike motion through space-time won't be able to reverse direction. No jumping backward in time.

I can hop backwards a metre but you can't hop backwards a second.

I can hop backwards for a duration of time of 1 second, and if you hopped backwards a meter then you hopped for a duration of time too. Guaranteed! You can't hop backwards unless a duration of time occurs. Simple as that.

I'm gonna hop for 1 second and find out how much distance I hopped in that 1 second. You're gonna hop a meter and find out how much time it took you to hop a meter. Do you not see the error of your ways?

lpetrich said:
...An object doing forward timelike motion through space-time...
Duh! You really really don't understand this. Read the thread. There is no motion in spacetime. It's a static "block universe" mathematical model. You can draw world lines in it, but there's nothing moving in it because it depicts all times.

Motor Daddy: tell you what, I'll hop back a hundred metres, you hop back to the middle of last week.

Motor Daddy: tell you what, I'll hop back a hundred metres, you hop back to the middle of last week.

Where is the middle of last week? How many hops does it take to get to the middle of last week?

Duh! You really really don't understand this. Read the thread. There is no motion in spacetime. It's a static "block universe" mathematical model. You can draw world lines in it, but there's nothing moving in it because it depicts all times.

This is like saying there can be no motion or change or gradients in space because space is an "all positions view".

It doesn't make any sense. Motion occurs in spacetime and it means exactly the same thing it has always meant: change in position with respect to time. Lumping space and time together and giving them a common name ("spacetime") doesn't change that one bit.

Considering space and time together isn't even new or specific to relativity. Kids in highschool learn to draw Minkowski diagrams and worldlines all the time. They just don't call them that. They call them distance-time graphs and they like to draw the axes the opposite way around, but otherwise they're exactly the same thing.

Przyk: The following indicates that you do not understand the concept of World Lines:
Motion occurs in space-time and it means exactly the same thing it has always meant: change in position with respect to time. Lumping space and time together and giving them a common name ("space-time") doesn't change that one bit.
Note & think carefully about the following from my Post #1:
The above model describes a 4D Space-Time continuum which is static: There is no motion.

Whether a person likes or dislikes the notion of a static universe, the model is very useful.

Those who are comfortable with the notion of a motionless 4D universe can accept the model as is.

Those uncomfortable with the notion can make use of the model & reject the notion of a motionless universe. They take the reasonable POV that a model is only a model, it is not reality.
The (x, y, z, t) World Line POV is a static model.

Perhaps it's safest to say that motion is a useful definition of a physical correspondence but not a physical fundamental. If it were equally useful we could have a "slope-o-meter" in our vehicles which displayed the ratio of (distance being traveled in Y direction) / (distance being traveled in X direction)

This is like saying there can be no motion or change or gradients in space because space is an "all positions view".
It absolutely isn't, przyk. You can literally move through space. That motion is empirical. But you can't move through time. Surely you read time travel is science fiction? I made it crystal clear. (Hmmn, I can't see you in the first few pages, perhaps you didn't). Think of the stasis box as a glorified freezer wherein you "travel forward through time" by not moving whilst everything else does. Think of a clock. It doesn't literally measure the flow of time, it clocks up some kind of regular cyclical motion and shows a cumulative display called "the time". Time is a cumulative measure of motion, and you can't move through a measure of motion just as you can't literally climb to a higher temperature. In similar vein you can't move through spacetime. Don't confuse space with spacetime. The latter is static, like Dinosaur said.

It doesn't make any sense. Motion occurs in spacetime and it means exactly the same thing it has always meant: change in position with respect to time. Lumping space and time together and giving them a common name ("spacetime") doesn't change that one bit.
It does. It's a popscience myth that you can move through spacetime. You can move through space, but that isn't what spacetime is. There are no worldlines in space. You can't look up and point out a light cone.

Considering space and time together isn't even new or specific to relativity. Kids in highschool learn to draw Minkowski diagrams and worldlines all the time. They just don't call them that. They call them distance-time graphs and they like to draw the axes the opposite way around, but otherwise they're exactly the same thing.
No problem there. But you represent motion with a sloped line. The graph itself is static. And like you said, otherwise they're the same thing. The map is not the territory.

Considering space and time together isn't even new or specific to relativity. Kids in highschool learn to draw Minkowski diagrams and worldlines all the time. They just don't call them that. They call them distance-time graphs and they like to draw the axes the opposite way around, but otherwise they're exactly the same thing.
*Anything* can be graphed as a function of time. That does not mean any special relationship with time, the way that space has.

I think that the success of Lorentz boosts makes absolute hash out of Farsight's theory of a fundamental difference between space and time. If anything, Farsight's theory is a throwback to late 19th cy. ether theories.

His theory is that space and motion are fundamental but time is not. It reminds me of this from Martin Gardner's Fads and Fallacies in the Name of Science (1952):
He has strong compulsions to focus his attacks on the greatest scientists and the best-established theories. When Newton was the outstanding name in physics, eccentric works in that science were violently anti-Newton. Today, with Einstein the father-symbol of authority, a crank theory of physics is likely to attack Einstein in the name of Newton. This same defiance can be seen in a tendency to assert the diametrical opposite of well-established beliefs. Mathematicians prove the angle cannot be trisected. So the crank trisects it. A perpetual motion machine cannot be built. He builds one. There are many eccentric theories in which the “pull” of gravity is replaced by a “push.” Germs do not cause disease, some modern cranks insist. Disease produces the germs. Glasses do not help the eyes, said Dr. Bates. They make them worse. In our next chapter we shall learn how Cyrus Teed literally turned the entire cosmos inside-out, compressing it within the confines of a hollow earth, inhabited only on the inside.
In his chapter "Down with Einstein!", MG discussed physics crackpottery. He mentioned two phases of it, and I think that we are starting to see a third phase:
1. Anti-Newton
2. Anti-Einstein, often claiming to restore Newton
3. Anti-present-day-physics, often claiming to restore Newton and Einstein

It absolutely isn't, przyk. You can literally move through space.

Where did I say otherwise?

But you can't move through time.

That depends on how you define "move" as applied to time. I can neither accept nor reject an idea that you haven't even defined.

If you're talking about something like jumping back to the 17th century, then obviously I can't do that, and when did I ever say I could? When did Minkowski ever say you could?

Don't confuse space with spacetime

Why in the world would anyone confuse space with spacetime?

It does. It's a popscience myth that you can move through spacetime.

Again, that depends on how you define "move". The way I see it, there are at least two ways of answering that, both of which are just playing with definitions and neither of which will give any real insight into any physics.

The first answer you could give is that "move" means displacement in space and space only, measured with respect to time, and so motion through time is by definition meaningless.

The second answer you could give involves adopting a more abstract definition of motion, in which you measure change in both position and time with respect to some other measurable parameter (e.g. proper time). I don't think it's controversial to say that the universe evolves and ages around you as you do, and that the ratio (e.g. the relativistic time dilation factor) is measurable. If some physicists feel like calling that a type of "motion", with it understood that it is this more abstract meaning that is being referred to, then why shouldn't they?

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I'll now derive the five kinds of space-time interval in special relativity. To keep the equations simple, I'll collapse the three space dimensions into one, x, and set c = 1. The time is, of course, t. Thus, the space-time interval vector is {t,x}.

The interval is T = t[sup]2[/sup] - x[sup]2[/sup], a Lorentz-invariant quantity. The three possibilities:
• If T > 0, then |t| > |x| -- timelike
• If T == 0, then |t| == |x| -- null
• If T < 0, then |t| < |x| -- spacelike
In all these cases, positive x and negative x are connected by a 180-d rotation. However, positive t and negative t are a different story.

For a timelike interval, t >= sqrt(T) or t <= sqrt(T). Since Lorentz boosts make t vary continuously, t > 0 and t < 0 are separate cases.

Likewise, for a null interval, t > 0 or t < 0. t cannot equal 0. Thus, as before, t > 0 and t < 0 are separate cases.

But for a spacelike interval, t > 0 is continuous with t < 0, going through t = 0, without a split by sign of time. Thus, |x| >= sqrt(-T).

Thus, the five cases that I'd posted on earlier.

lpetrich said:
...If anything, Farsight's theory is a throwback to late 19th cy. ether theories...
It isn't my theory, it's Einstein's theory. And you're the crank dismissing Einstein. Where do you get your physics from? The Discovery Channel? New Scientist? Go and look at Aether theories on Wikipedia and note the quote by Robert B Laughlin:

"It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed [..] The word 'ether' has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum..."

Go and look at Einstein's 1920 Leyden Address where he said space is the aether of general relativity, and go and look at arxiV. Then shut up.

View attachment 7079

Perhaps it's safest to say that motion is a useful definition of a physical correspondence but not a physical fundamental. If it were equally useful we could have a "slope-o-meter" in our vehicles which displayed the ratio of (distance being traveled in Y direction) / (distance being traveled in X direction)

IMO, Airplane instrumentation (relationship to the horizon) comes close

That depends on how you define "move" as applied to time. I can neither accept nor reject an idea that you haven't even defined.
No it doesn't. I can hop back a metre but you can't hop back a second. Or forward. Humpty-Dumpty physics doesn't change that. You know what I mean by that? See Wikipedia. Lewis Carroll was having a dig at slippery evasion.

If you're talking about something like jumping back to the 17th century, then obviously I can't do that, and when did I ever say I could? When did Minkowski ever say you could?
You didn't and he didn't, but there's plenty of so-called physicists out there who believe in the possibility of time travel. They believe time is on a par with space, and they do not appreciate that it is a dimension of measure rather than a dimension that offers freedom of movement.

przyk said:
Why in the world would anyone confuse space with spacetime?
Beats me, przyk. But they do. Google on space is curved. And remember the Wheeler quote: "Matter tells space how to curve. Space tells matter how to move". Also note that Kip Thorne is an advocate of time travel, and that Wheeler promoted the notion that a positron was an electron travelling back in time. It grieves me that Feynman was associated with that.

przyk said:
Again, that depends on how you define "move". The way I see it, there are at least two ways of answering that, both of which are just playing with definitions and neither of which will give any real insight into any physics.

The first answer you could give is that "move" means displacement in space and space only, measured with respect to time, and so motion through time is by definition meaningless.
And how do you "measure it with respect to time"? Using a clock. Let's open up that clock. What do we see? Time flowing through it, like it's some kind of cosmic gas meter? No. We see cogs moving. We are measuring that motion with respect to some other motion.

przyk said:
The second answer you could give involves adopting a more abstract definition of motion, in which you measure change in both position and time with respect to some other measurable parameter (e.g. proper time).
Ah you mean proper time as measured on your local clock? Let's open up that clock. What do we see? Time flowing through it, like it's some kind of cosmic gas meter? No. We see cogs moving.

przyk said:
I don't think it's controversial to say that the universe evolves and ages around you as you do, and that the ratio (e.g. the relativistic time dilation factor) is measurable.
No problem with that ratio.

przyk said:
If some physicists feel like calling that a type of "motion", with it understood that it is this more abstract meaning that is being referred to, then why shouldn't they?
Because it isn't a type of motion. It's a ratio between one motion and another.

Ipetrich,
This, time is much like another space dimension.
IMO, time does not even pre-exist as a dimension. I believe time is an emergent result and property of the function of change in space.

This remarkable result was first discovered by Hermann Minkowski in 1908. From the Wikipedia article on him,
The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

IMO, there is no union of two plenums (space and time). Time is a "child of change" in the physical world and can only move forward because it is always a result of a space event now and in the past. There is no future time, the future is a permissive condition and comes into being in measurable durations along with reality itself.

Static space would be non-causal and in the absence of change time is a meaningless term.
Dynamic space IS casual and the "duration" of the change is measurable which results in a chronological history of "duration" in reality.

IMO, the concept of Time is similar to the concept of Thought, they are non-causal abstractions.