Simple argument for area conservation under Lorentz transformation?

[bcrowell]

A Lorentz boost along the x-axis conserves area in the x-t plane. Can anyone think of an argument to prove this fact that's simpler than the one below? I don't want to assume the form of the Lorentz transformation, because I want to use the conservation of area as part of a derivation of the Lorentz transformation. I also don't want to assume that c is the same in all frames. The only assumptions are supposed to be:

1 Spacetime is homogeneous and isotropic. No point has special properties that make it distinguishable from other points, nor is one direction distinguishable from another.

2 Inertial frames of reference exist. These are frames in which particles move at constant velocity if not subject to any forces. We can construct such a frame by using a particular particle, which is not subject to any forces, as a reference point.

3 Equivalence of inertial frames: If a frame is in constant-velocity translational motion relative to an inertial frame, then it is also an inertial frame. No experiment can distinguish one preferred inertial frame from all the others.

4 Causality: Observers in different inertial frames agree on the time-ordering of events.

5 No simultaneity: The experimental evidence shows that observers in different inertial frames do not agree on the simultaneity of events.

Start with a rectangular area in the x-t plane, and for simplicity, let this rectangle have unit area. Then the area of the new parallelogram is still 1, by the following argument. Let the new area be A, which is a function of v. By isotropy of spacetime (assumption 1), A(v)=A(-v). Furthermore, the function A(v) must have some universal form for all geometrical figures, not just for a figure that is initially a particular rectangle; this follows because of the definition of affine area in terms of a dissection by a two-dimensional lattice, which we can choose to be a lattice of squares. Applying boosts +v and -v one after another results in a transformation back into our original frame of reference, and since A is universal for all shapes, it doesn't matter that the second transformation starts from a parallelogram rather than a square. Scaling the area once by A(v) and again by A(-v) must therefore give back the original square with its original unit area, A(v)A(-v)=1, and since A(v)=A(-v), A(v)=$\pm1$ for any value of v. Since A(0)=1, we must have A(v)=1 for all v. The argument is independent of the shape of the region, so we conclude that all areas are preserved by Lorentz boosts. The argument is also purely one about affine geometry (it would apply equally well to a Euclidean space), so there is no reason to expect the area A in the (t,x) plane to have any special physical significance in relativity; it is simply a useful mathematical tool in the present discussion.

For anyone who wants to see the whole derivation of the Lorentz transformation from assumptions 1-5, it's here: http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.7 . I'm basically looking to see if I can make the argument simpler and easier to understand for people with less math and physics background.

Thanks in advance!

-Ben

 

[Chronos]

1. Agreed
2. Illogical
3. Agreed
4. Illogical
5. Agreed.

 

[JesseM]

4 Causality: Observers in different inertial frames agree on the time-ordering of events.

This one is false in SR if you're talking about spacelike-separated events (though of course we know they can't have a causal influence on one another unless FTL influences exist)

5 No simultaneity: The experimental evidence shows that observers in different inertial frames do not agree on the simultaneity of events.

Can experimental evidence really show this? Doesn't it depend on what clock synchronization convention you choose to use? No experiment forces us to use Einstein's clock synchronization convention, although if we used a different one the laws of physics wouldn't obey the same equations in different inertial frames.

Scaling the area once by A(v) and again by A(-v) must therefore give back the original square with its original unit area, A(v)A(-v)=1, and since A(v)=A(-v), A(v)=$\pm1$ for any value of v.

Aren't you assuming here that if frame #1 sees frame #2 moving at velocity v, then frame #2 must see frame #1 moving at velocity -v? We can come up with coordinate transformations where this isn't true, so I think you'd need to either show how it follows from assumptions 1-5, or add it as a new assumption.

 

[bcrowell]

This one is false in SR if you're talking about spacelike-separated events (though of course we know they can't have a causal influence on one another unless FTL influences exist)

Thanks for pointing out the problem with that. I should have worded it better. It was intended to refer only to timelike-related events, but at this stage in the derivation the Lorentz transformation hasn't been derived, and there's no notion of light cones. I'll change it to a statement that if one observer says that A causes B, every observer agrees that A causes B, and not the other way around.

Can experimental evidence really show this? Doesn't it depend on what clock synchronization convention you choose to use? No experiment forces us to use Einstein's clock synchronization convention, although if we used a different one the laws of physics wouldn't obey the same equations in different inertial frames.

Another good point, thanks.The proof in the book actually refers back to specific experiments, but without that context, it's probably not that clear what I'm talking about. Basically I'm saying that experiments such as the Hafele-Keating experiment give non-null results.

Aren't you assuming here that if frame #1 sees frame #2 moving at velocity v, then frame #2 must see frame #1 moving at velocity -v? We can come up with coordinate transformations where this isn't true, so I think you'd need to either show how it follows from assumptions 1-5, or add it as a new assumption.

That's a good point. What I've actually given here is the assumptions that go into the derivation, plus one piece out of the middle of the derivation. Before we get to the point where I'm making the area-conservation argument, I've already proved that the transformations are linear. It seems to me that the property you're referring to must be true for linear transformations, whereas for nonlinear transformations there wouldn't be any unambiguous way to associate a single v with a particular transformation. Does that make sense to you, or can you think of an example where the transformation is linear and it doesn't have this property? I'm essentially defining v as the inverse slope of the transformed version of a line that was a constant-x line in the original frame.

Thanks, JesseM, for your comments -- very helpful!

 

[JesseM]

Another good point, thanks.The proof in the book actually refers back to specific experiments, but without that context, it's probably not that clear what I'm talking about. Basically I'm saying that experiments such as the Hafele-Keating experiment give non-null results.

But adopting a different simultaneity convention wouldn't change our predictions about any local physical events, like an airplane clock and a ground clock being compared at a single location.

That's a good point. What I've actually given here is the assumptions that go into the derivation, plus one piece out of the middle of the derivation. Before we get to the point where I'm making the area-conservation argument, I've already proved that the transformations are linear. It seems to me that the property you're referring to must be true for linear transformations, whereas for nonlinear transformations there wouldn't be any unambiguous way to associate a single v with a particular transformation. Does that make sense to you, or can you think of an example where the transformation is linear and it doesn't have this property?

Here's an example of a linear transformation without this property:

x' = 2x - 2vt
t' = t

Suppose you have an object at rest in the x,t frame and pick two events on its worldline, like (x=0,t=0) and (x=0,t=1)...in the x',t' frame they translate to (x'=0,t'=0) and (x'=-2v,t'=1), so the object must have a velocity of -2v in the x',t' frame. The inverse transform would be:

x = (1/2)*x' + vt'
t = t'

So if we pick an object at rest in the x',t' frame which goes through coordinates (x'=0,t'=0) and (x'=0,t'=1), then in the x,t frame the events transform to (x=0,t=0) and (x=v,t=1), meaning the object has a velocity of v in the x,t frame.

 

[cfrogue]

This one is false in SR if you're talking about spacelike-separated events (though of course we know they can't have a causal influence on one another unless FTL influences exist)

May I ask a question?

Causality is normally thought of in terms of the light cone/light sphere.

Is this correct?

 

[JesseM]

May I ask a question?

Causality is normally thought of in terms of the light cone/light sphere.

Is this correct?

Yes, it's normally assumed causal influences can't travel faster than light.

 

[cfrogue]

Yes, it's normally assumed causal influences can't travel faster than light.

Therefore, when considering only one light sphere, how can observers disagree on the ordinality of events associated with that light sphere?

 

[Dale]

They can't disagree about the order of events inside (timelike) or on (null) the light cone, but they can disagree about the order of events outside (spacelike) the light cone.

 

[cfrogue]

They can't disagree about the order of events inside (timelike) or on (null) the light cone, but they can disagree about the order of events outside (spacelike) the light cone.

I did not say inside the light cone.

I said,
"Therefore, when considering only one light sphere, how can observers disagree on the ordinality of events associated with that light sphere"?

This means, when the light cone overtakes an object/event, that becomes part of the absolute past.

Therefore, causality is implemented by the light cone in that the ordinality of events overtaken by the light sphere cannot have disagreement amongst observers.
They may disagree on the timing and position, but not the ordinality.

Is this correct?

If it is correct, any disagreement with statement 4 by the OP is false.

 

[JesseM]

Therefore, when considering only one light sphere, how can observers disagree on the ordinality of events associated with that light sphere?

By "light sphere" you mean a cross-section through the light cone, showing all the points inside the light cone at a single instant in some frame? While all these events can have a causal influence on the future event E that forms the apex of this past light cone, that doesn't mean they can have a causal influence on one another, each event has its own unique past light cone, and nothing outside its own past light cone can have a causal influence on it. So, if different events in the light sphere do not lie in one another's own past light cones, they cannot have a causal influence on one another and different frames can disagree about the order of these events, though all frames agree they lie in the past of the event E at the apex of the light cone that the light sphere was taken from.

 

[cfrogue]

By "light sphere" you mean a cross-section through the light cone, showing all the points inside the light cone at a single instant in some frame? While all these events can have a causal influence on the future event E that forms the apex of this past light cone, that doesn't mean they can have a causal influence on one another, each event has its own unique past light cone, and nothing outside its own past light cone can have a causal influence on it.

I am not deciding a causal influence. That would be silly since the light cone implements the agenda.

I am talking about the ordinality of objects being taken over by the light sphere.

Do you claim two events E1 and E2 can have disagreement amongst observers of ordinality with one light cone?

If so, can you show me the geometric/topological representation?

In other words, do you claim one object is overtaken by the light cone and then the second by one observer, but they are reversed by another observer?

 

[JesseM]

I am not deciding a causal influence. That would be silly since the light cone implements the agenda.

I am talking about the ordinality of objects being taken over by the light sphere.

Yes, and whenever two events cannot causally influence one another (because neither lies in the other event's past light cone), there can be disagreements between inertial frames about their order.

Do you claim two events E1 and E2 can have disagreement amongst observers of ordinality with one light cone?

As long as E1 does not lie in the past light cone of E2 and E2 does not lie in the past light cone of E1, then yes, different frames can disagree about their order.

If so, can you show me the geometric/topological representation?

You mean like a spacetime diagram? I can tell you how to draw one yourself--just draw the x and t axes for frame A, and plot an event E at coordinates x=3 light-years, t=0 years. Then draw worldlines for two objects objects 1 and 2, object 1 being at rest at x=1, and object 2 being at rest at x=6. Then if you plot the future light cone of event E, you'll see that object 1 enters the future light cone of E at x=1, t=2 (event E1) while object 2 enters at x=6, t=3 (event E2), so in this frame E1 happened before E2. But if you then consider a different frame B moving at 0.8c relative to, and you draw a line of simultaneity for frame B which goes through the origin (this line be defined by the function x=0.8*t in units where c=1, so it'll have the inverse slope of the worldline of an observer at rest in frame B, which would be given by the function t=0.8*x), then you'll see that E1 lies above this line of simultaneity while E2 lies below it, so E2 must have happened before E1 in frame B.

In other words, do you claim one object is overtaken by the light cone and then the second by one observer, but they are reversed by another observer?

Yes, that would be true in the example above.

 

[cfrogue]

Yes, and whenever two events cannot causally influence one another (because neither lies in the other event's past light cone), there can be disagreements between inertial frames about their order.

As long as E1 does not lie in the past light cone of E2 and E2 does not lie in the past light cone of E1, then yes, different frames can disagree about their order.

You mean like a spacetime diagram? I can tell you how to draw one yourself--just draw the x and t axes for frame A, and plot an event E at coordinates x=3 light-years, t=0 years. Then draw worldlines for two objects objects 1 and 2, object 1 being at rest at x=1, and object 2 being at rest at x=6. Then if you plot the future light cone of event E, you'll see that object 1 enters the future light cone of E at x=1, t=2 (event E1) while object 2 enters at x=6, t=3 (event E2), so in this frame E1 happened before E2. But if you then consider a different frame B moving at 0.8c relative to, and you draw a line of simultaneity for frame B which goes through the origin (this line be defined by the function x=0.8*t in units where c=1, so it'll have the inverse slope of the worldline of an observer at rest in frame B, which would be given by the function t=0.8*x), then you'll see that E1 lies above this line of simultaneity while E2 lies below it, so E2 must have happened before E1 in frame B.

Yes, that would be true in the example above.

We are not communicating.

When I say overtaken by the light sphere, I am talking about entering the absolute past of the light cone.

There cannot be disagreement about the ordinality of events entering the absolute past.

I do not care about the future light cone and the OP did not in statement 4.

Do you disagree?

 

[JesseM]

We are not communicating.

When I say overtaken by the light sphere, I am talking about entering the absolute past of the light cone.

That doesn't make sense, you can't "enter" the past light cone, you can only exit it. After all, the past light cone can be thought of as a light sphere whose radius is shrinking at the speed of light until it shrinks to a point at the exact position and time of the event E whose past light cone it is, so if you're outside it there's no way to catch up with its boundary without moving FTL.

 

[cfrogue]

That doesn't make sense, you can't "enter" the past light cone, you can only exit it. After all, the past light cone can be thought of as a light sphere whose radius is shrinking at the speed of light until it shrinks to a point at the exact position and time of the event E whose past light cone it is, so if you're outside it there's no way to catch up with its boundary without moving FTL.

No, you do not understand.

The light sphere proceeds at radius ct in all directions.

As it proceeds, it overtakes objects which is then defined as events.

This process of overtaking events and in what order is not in disagreement amoungst observers.


What The Principle of Relativity has done is to make the Light Cone absolute.
It is fair to say that what Einstein did is that he replaced the absoluteness of time and of space with the absoluteness of the speed of light. The Speed of Light is more fundamental than time and space.
http://physics.syr.edu/courses/modules/LIGHTCONE/minkowski.html

Taking as event p a flash of light (light pulse) at time t0, all events that can be reached by this pulse from p form the future light cone of p, while those events that can send a light pulse to p form the past light cone of p.
http://en.wikipedia.org/wiki/Light_cone

Key Concept: Special Relativity in One Sentence.
All speeds are relative, except for the speed of light, which is absolute.
http://www.astronomy.ohio-state.edu/~ryden/ast162_6/notes23.html


An important property of light cones is that they help us identify regions of spacetime which distinguish between relative space and time and absolute future, past, and elsewhere.
http://physics.tamuk.edu/~hewett/Mo...vity/InvariantView/LightCones/LightCones.html

Unlike the velocities of particles, the velocity of light is absolute (frame-independent)
http://philsci-archive.pitt.edu/archive/00003986/01/elsevier2.pdf

Recall that the future Light Cone of an event is the future-history of a light-flash of emitted at that event. It is an absolute surface for the Einstein-Minkowski Spacetime.
http://www.phy.syr.edu/courses/modules/LIGHTCONE/lightconev.html

 

[JesseM]

No, you do not understand.

The light sphere proceeds at radius ct in all directions.

No, only the future light sphere expands at ct. The past light sphere contracts at ct. This is very basic SR, if you don't understand this you really are totally confused! Remember, the past light sphere represents the set of all events at a given moment that could send a signal traveling at the speed of light or less which would be able to reach the future event E that lies at the "top" of the past light cone. 1 year before E, any event within a radius of 1 light-year has time to send a signal to E. But then 1 second before E, only events within a radius of 1 light-second have time to send a signal to E. So, obviously the past light sphere is contracting as you get closer to the time of E.

What The Principle of Relativity has done is to make the Light Cone absolute.
It is fair to say that what Einstein did is that he replaced the absoluteness of time and of space with the absoluteness of the speed of light. The Speed of Light is more fundamental than time and space.
http://physics.syr.edu/courses/modules/LIGHTCONE/minkowski.html

Taking as event p a flash of light (light pulse) at time t0, all events that can be reached by this pulse from p form the future light cone of p, while those events that can send a light pulse to p form the past light cone of p.
http://en.wikipedia.org/wiki/Light_cone

Key Concept: Special Relativity in One Sentence.
All speeds are relative, except for the speed of light, which is absolute.
http://www.astronomy.ohio-state.edu/~ryden/ast162_6/notes23.htmlAn important property of light cones is that they help us identify regions of spacetime which distinguish between relative space and time and absolute future, past, and elsewhere.
http://physics.tamuk.edu/~hewett/Mo...vity/InvariantView/LightCones/LightCones.html

Unlike the velocities of particles, the velocity of light is absolute (frame-independent)
http://philsci-archive.pitt.edu/archive/00003986/01/elsevier2.pdf

Recall that the future Light Cone of an event is the future-history of a light-flash of emitted at that event. It is an absolute surface for the Einstein-Minkowski Spacetime.
http://www.phy.syr.edu/courses/modules/LIGHTCONE/lightconev.html

It's no use trying to get an understanding from out of context quotes if you don't understand the actual concepts and math they're talking about. None of these quotes says it's possible to "enter" a past light cone, and when they talk about light cones establishing an absolute future and past, they only mean that any event E1 or E2 in the past light cone of E will happen earlier than E in all frames, not that there cannot be disagreements about the order of E1 and E2 themselves.

 

[Dale]

I did not say inside the light cone.

I said,
"Therefore, when considering only one light sphere, how can observers disagree on the ordinality of events associated with that light sphere"?

It was not clear (to me) what you meant by that so I just covered all possibilities.

This means, when the light cone overtakes an object/event, that becomes part of the absolute past.

OK, I think you have a little misunderstanding here. Objects are not events. An event is the Minkowski equivalent of a point in Euclidean geometry. Geometrically a classical point object is a Minkowski worldline (with extended objects being 4D hypervolumes). A light cone cannot "overtake" an event, an event is either inside, outside, or on a light cone. An object's worldline can cross a light cone, but worldlines have no temporal ordering. Only events can have a temporal ordering.

Therefore, causality is implemented by the light cone in that the ordinality of events overtaken by the light sphere cannot have disagreement amongst observers.
They may disagree on the timing and position, but not the ordinality.

Is this correct?

No, it is not correct, for the reason indicated above. The phrase "events overtaken by the light sphere" is meaningless.

If it is correct, any disagreement with statement 4 by the OP is false.

JesseM was not so much disagreeing as clarifying, and his clarification was correct and was what was intended by the OP as indicated by the OP's response.

 

[cfrogue]

No, only the future light sphere expands at ct. The past light sphere contracts at ct. This is very basic SR, if you don't understand this you really are totally confused! Remember, the past light sphere represents the set of all events at a given moment that could send a signal traveling at the speed of light or less which would be able to reach the future event E that lies at the "top" of the past light cone. 1 year before E, any event within a radius of 1 light-year has time to send a signal to E. But then 1 second before E, only events within a radius of 1 light-second have time to send a signal to E. So, obviously the past light sphere is contracting as you get closer to the time of E.

Let's see now, the past light cone contracts at ct.

That would imply, since nothing can travel at the speed of light, that the past overtakes all events once they happen.

So, I could see myself being born then.

 

[cfrogue]

No, it is not correct, for the reason indicated above. The phrase "events overtaken by the light sphere" is meaningless.

So, I cannot define an event as to when light stikes an object?

Why?

I am thinking about R of S right now also.

 

[cfrogue]

A Light cone is the path that a flash of light would take through spacetime. As time progresses, the light from the flash spreads out in a circle, and the result is a cone. In reality, there are three space dimensions, so the light actually forms a sphere in space, and the light cone is actually a fourth dimensional shape, but it's much easier to visualize it as a cone.

In special relativity, a light cone (or null cone) is the surface describing the temporal evolution of a flash of light in Minkowski spacetime.


The light cone is constructed as follows. Taking as event p a flash of light (light pulse) at time t0, all events that can be reached by this pulse from p form the future light cone of p, while those events that can send a light pulse to p form the past light cone of p.

Given an event E, the light cone classifies all events in spacetime into 5 distinct categories:

Events on the future light cone of E.
Events on the past light cone of E.
Events inside the future light cone of E are those which are affected by a material particle emitted at E.
Events inside the past light cone of E are those which can emit a material particle and affect what is happening at E.
All other events are in the (absolute) elsewhere of E and are those that can not affect and can not be affected by E.

The above classifications hold true in any frame of reference; that is, an event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference. This is why the concept is so powerful.

As time progresses, each location's past light cone will eventually grow to encompass all locations. The further locations will of course be at far distant times. The Earth's past light cone, at its very edges, includes very distant objects, but only what they looked like long ago, when the universe was young.

http://en.wikipedia.org/wiki/Light_cone

 

[Dale]

So, I cannot define an event as to when light stikes an object?

Why?

Certainly. The intersection of a past light cone and a timelike worldline is at most a single event.

 

[cfrogue]

Certainly. The intersection of a past light cone and a timelike worldline is at most a single event.

So, let's now talk about ordinality of events and the light cone.

 

[JesseM]

Let's see now, the past light cone contracts at ct.

That would imply, since nothing can travel at the speed of light, that the past overtakes all events once they happen.

I don't know what you mean by the past. In relativity we can only talk about the past and future light cones of specific events, different events have different past light cones. What event's past light cone are you talking about? And what do you mean by "overtakes"? Since the past light sphere is shrinking, objects are exiting it as the sphere's radius shrinks, if that's what you mean.

So, I could see myself being born then.

How could you do that? At any given moment, the only events you see are ones that happened right on the surface of your past light cone (ignoring issues like light slowing down in a medium), and there is no point on your worldline that has the event of your birth on the surface of its past light cone.

 

[Dale]

As time progresses, each location's past light cone will eventually grow to encompass all locations.

This sentence is wrong, a location does not have a light cone. An event has a light cone, a location does not.

 

[JesseM]

As time progresses, each location's past light cone will eventually grow to encompass all locations. The further locations will of course be at far distant times. The Earth's past light cone, at its very edges, includes very distant objects, but only what they looked like long ago, when the universe was young.

http://en.wikipedia.org/wiki/Light_cone

This one is confusingly worded, I think they mean "as time progresses backwards". I'll edit that one so it makes more sense.

 

[Dale]

So, let's now talk about ordinality of events and the light cone.

The different events on a single light cone can be timelike, null, or spacelike separated. As I said in post 9 different frames can disagree on the order of the spacelike separated events.

 

[cfrogue]

I don't know what you mean by the past. In relativity we can only talk about the past and future light cones of specific events, different events have different past light cones. What event's past light cone are you talking about? And what do you mean by "overtakes"? Since the past light sphere is shrinking, objects are exiting it as the sphere's radius shrinks, if that's what you mean.

I do not think you see that the light cone implements casuality and that is absolute.
If not, SR falls apart in the sense that there does not exists anything in the universe that can describe the order of events such that the observers cannot disagree.

This would also violate normal human experience which clearly distinguishes events into the absolute past and absolute present.




How could you do that? At any given moment, the only events you see are ones that happened right on the surface of your past light cone (ignoring issues like light slowing down in a medium), and there is no point on your worldline that has the event of your birth on the surface of its past light cone.[/QUOTE]

 

[cfrogue]

The different events on a single light cone can be timelike, null, or spacelike separated. As I said in post 9 different frames can disagree on the order of the spacelike separated events.

You are not listening.

order of the spacelike separated events

This does not exist with a single light sphere.

 

[JesseM]

You are not listening.

order of the spacelike separated events

This does not exist with a single light sphere.

Events within a single light sphere are spacelike separated, if "light sphere" just means the set of all points in the light cone which happened at a specific time t in some inertial frame. Any events which happen simultaneously in a given inertial frame are spacelike separated.

 

[Dale]

I do not think you see that the light cone implements casuality and that is absolute.

Yes, causality is absolute. If event A is the cause of event B then A preceeds B in all reference frames.

 

[Dale]

You are not listening.

order of the spacelike separated events

This does not exist with a single light sphere.

Yes, it does. Consider the intersection of the light cone with any hyperplane of simultaneity for any arbitrary inertial frame. It should be clear that all of those events (a sphere) are spacelike separated.

 

[cfrogue]

This sentence is wrong, a location does not have a light cone. An event has a light cone, a location does not.

You are claiming an event defines the behavior of light?

As time progresses, each location's past light cone will eventually grow to encompass all locations. The further locations will of course be at far distant times. The Earth's past light cone, at its very edges, includes very distant objects, but only what they looked like long ago, when the universe was young.

http://en.wikipedia.org/wiki/Light_cone

 

[cfrogue]

This one is confusingly worded, I think they mean "as time progresses backwards". I'll edit that one so it makes more sense.

Why do you say as time progresses backwards?

 

[JesseM]

I do not think you see that the light cone implements casuality and that is absolute.

Your use of language is so vague! It does "implement causality" in the sense that all observers agree on the order of E and some other event E1 which lies in E's future or past light cone, but that doesn't mean that observers will agree on the order of two events E1 and E2 which both lie in the past light cone of E but don't lie in one another's past or future light cones (i.e. E1 and E2 are spacelike separated)

If not, SR falls apart in the sense that there does not exists anything in the universe that can describe the order of events such that the observers cannot disagree.

For any pair of events where one lies in the other's past light cone, all observers do agree on the order. But two events E1 and E2 can both lie in the past light cone of E (so all observers agree E1 and E2 happened prior to E) yet neither lies in the past light cone of the other one (so different observers disagree about whether E1 or E2 happened first). Is this really so hard to understand? Do you understand that each event has its own individual past light cone?

This would also violate normal human experience which clearly distinguishes events into the absolute past and absolute present.

But normal human experience conflicts with relativity in this sense, since relativity says that when two events don't lie in one another's past or future light cones, there is no absolute truth about which happened earlier and which happened later.