How Many Reference Frames Can Be Coincident at the Origin?

[Lynch101]

TL;DR Summary

Can an infinite number of reference frames be made coincident at the origin?

In a previous thread, I referenced an argument from the book the Scientist as Philosopher by Friedel Weinert, in which he talks about the reference frames of two relatively moving observers. He made a statement that I had't thought about before, but when I was reading it this time around a question popped into my head. It's pretty trivial question but just one I was wondering about. In the point he is making, he mentions that

we must introduce the coordinate system of observer O...The second observer’s reference frame can be made coincident with that of O, by a convenient choice.138 That means that both coordinate systems can be made to coincide at the origin.

As I say, it's a pretty trivial question but is there a limit on the number of reference frames that can be made coincident at the origin, or it possible to make an infinite number of reference frames coincident at the origin i.e. the reference frame of every observer in the Universe?

 

[Ibix]

it possible to make an infinite number of reference frames coincident at the origin

Reference frames that share an origin are related by a Lorentz transform which is parameterised by $v$, which is a continuous variable. There are, therefore, uncountably infinite numbers of reference frames sharing a common origin.

i.e. the reference frames of every observer in the Universe?

Practically, "every observer in the universe" steps outside the bounds of special relativity, so you cannot define global reference frames. So this bit is not correct, unless you restrict yourself to a mass-free special relativistic universe.

 

[Vanadium 50]

Are you asking are there an infinite number of reference frames that can label a particular event as (0,0,0,t)? Yes. Or are you asking if every reference frame can label a particular event as (0,0,0,t)? No.

every observer in the Universe?

Or are you talking about an expanding universe, where SR does not apply globally?

 

[Lynch101]

Reference frames that share an origin are related by a Lorentz transform which is parameterised by $v$, which is a continuous variable. There are, therefore, uncountably infinite numbers of reference frames sharing a common origin.

Does this mean uncountably infinite numbers of reference frames who share an origin, but not necessarily all sharing the same origin?

Practically, "every observer in the universe" steps outside the bounds of special relativity, so you cannot define global reference frames. So this bit is not correct, unless you restrict yourself to a mass-free special relativistic universe.

Pardon my denseness, but I presume that phrasing doesn't matter, "every observer in the Universe"? The same is true if we talk about all relatively moving reference frames being made coincident at the origin?

 

[Lynch101]

Are you asking are there an infinite number of reference frames that can label a particular event as (0,0,0,t)? Yes. Or are you asking if every reference frame can label a particular event as (0,0,0,t)? No.

The latter sounds more like what I'm wondering. Just in the example above, with two relatively moving observers, both coordinate systems are made to coincide at the origin. Is it possible to make the frame of a third observer similarly coincide at the origin; and then a fourth; a fifth; sixth; and so on...?

Or are you talking about an expanding universe, where SR does not apply globally?

Honestly, I don't know but I don't think so.

 

[Ibix]

Does this mean uncountably infinite numbers of reference frames who share an origin, but not necessarily all sharing the same origin?

...um, what? They only have one origin each.

Pardon my denseness, but I presume that phrasing doesn't matter, "every observer in the Universe"? The same is true if we talk about all relatively moving reference frames being made coincident at the origin?

"Observer" and "reference frame" are not interchangeable terms.

In the flat spacetime of special relativity you can define global reference frames. Thus if I pick an origin and announce it, anyone anywhere or anywhen moving with any velocity can use that same event as the origin of their reference frame (although a different choice may be more convenient).

In curved spacetime, such as the one we actually inhabit, you cannot define global inertial reference frames. Thus two distant observers cannot even define reference frames that cover each others' locations, let alone discuss a shared origin.

 

[PeterDonis]

is there a limit on the number of reference frames that can be made coincident at the origin

You're thinking of it backwards. "The origin" is not predetermined. First, you have to pick which event you want to be the origin. In the case from Friedel Weinert that you describe, the origin is the event where the two observers pass each other.

Once you've picked an event to be the origin, there are an infinite number of reference frames you can have whose origin is that event, since there are a continuous infinity of possible 4-velocity vectors at the origin that can serve as the timelike basis vector for a reference frame (corresponding to the continuous infinity of possible relative velocities that two observers who meet each other at the chosen event can have).

the reference frame of every observer in the Universe?

You could, in principle, pick one single event in 4-d spacetime and have every observer (more precisely, every observer who remains inertial for all time) use a reference frame that had that event as the origin. But for almost all such observers, that origin event would not be on their worldline. Normally, in relativity scenarios, a reference frame is chosen so the origin event is on the worldline of whatever observers are of interest (as in the Friedel Weinert scenario, where the event where the two observers meet each other is chosen as the origin since it's the only event that is on both observers' worldlines). That is not strictly required by the math, but it's often very helpful for simplifying things.

Note, btw, that in all of this, the term "Universe" really means "flat Minkowski spacetime", which does not describe our actual universe.

 

[Dale]

Summary:: Can an infinite number of reference frames be made coincident at the origin?

The set of all inertial frames forms a group called the Poincare group. It is a ten dimensional group. Three dimensions are spatial rotations, three dimensions are boosts, and four are spacetime translations. The rotations and boosts form a six dimensional subgroup that share an origin. So for each event in spacetime there are as many reference frames with that event as the origin as there are points in R6. I.e. infinite.

 

[Lynch101]

You're thinking of it backwards. "The origin" is not predetermined. First, you have to pick which event you want to be the origin. In the case from Friedel Weinert that you describe, the origin is the event where the two observers pass each other.

Once you've picked an event to be the origin, there are an infinite number of reference frames you can have whose origin is that event, since there are a continuous infinity of possible 4-velocity vectors at the origin that can serve as the timelike basis vector for a reference frame (corresponding to the continuous infinity of possible relative velocities that two observers who meet each other at the chosen event can have).

You could, in principle, pick one single event in 4-d spacetime and have every observer (more precisely, every observer who remains inertial for all time) use a reference frame that had that event as the origin. But for almost all such observers, that origin event would not be on their worldline. Normally, in relativity scenarios, a reference frame is chosen so the origin event is on the worldline of whatever observers are of interest (as in the Friedel Weinert scenario, where the event where the two observers meet each other is chosen as the origin since it's the only event that is on both observers' worldlines). That is not strictly required by the math, but it's often very helpful for simplifying things.

Note, btw, that in all of this, the term "Universe" really means "flat Minkowski spacetime", which does not describe our actual universe.

Ah OK, I think I get the point about picking the event first.

In the Weinert example, the event is where the two observers pass each other. Is it possible to have a third observer passing such that that the passing event includes all three and all three then have the "passing event" as their origin? If so, can the same be done for a third, fourth, fifth, etc.?

 

[Lynch101]

The set of all inertial frames forms a group called the Poincare group. It is a ten dimensional group. Three dimensions are spatial rotations, three dimensions are boosts, and four are spacetime translations. The rotations and boosts form a six dimensional subgroup that share an origin. So for each event in spacetime there are as many reference frames with that event as the origin as there are points in R6. I.e. infinite.

This is a little above my pay grade unfortunately, but I'll hopefully get there at some point. I appreciate the reply though, because it gives me information to look up. Thank you Dale.

 

[jbriggs444]

This is a little above my pay grade unfortunately, but I'll hopefully get there at some point.

If you disregard the stuff about being a "group", then what @Dale has written is pretty clear and very close to your pay grade.

If you start with one coordinate system, you can come up with a new coordinate system with a rotation. The rotation can be a roll, a pitch, a yaw or a combination of all three. That is three free parameters. Or the new coordinate system can have a relative x velocity, y velocity, z velocity or a combination of all three. That is another three free parameters.

So without changing the origin, you have six free parameters that you can use to choose a new coordinate system.

Now the part about being a group...

The idea is that you consider the set of coordinate system transformations identified by the six parameters (roll, pitch, yaw, vx, vy, vz). If you take a coordinate system and transform it twice (or three, four, five or more times) each time using a transformation selected from this set, you will find that the final resulting transformation was already a member of the original set.

In the sense of abstract algebra, this is to say that this set of transformations is "closed" under the operation of composition. Along with a few more simple facts, this is enough to ensure that this [sub-]set of transformations forms an algebraic "group".

[I am a not sure about whether or how the handedness and time reversal symmetries are normally integrated into the group. I suppose that one could glue on two additional discrete-valued dimensions -- (right-handed/left-handed, normal/time-reversed)]

 

[Ibix]

In the Weinert example, the event is where the two observers pass each other. Is it possible to have a third observer passing such that that the passing event includes all three and all three then have the "passing event" as their origin?

Why would the third observer need to pass the first two? Why not just use "those two guys pass each other" as the origin? I mean, I can use latitude and longitude without ever having been anywhere near its origin. All I need is someone to tell me how latitude and longitude work and my current location (or the location of some landmark) in that coordinate system. Ditto reference frames in special relativity, except you need to specify a location in space and time.

 

[Lynch101]

Why would the third observer need to pass the first two? Why not just use "those two guys pass each other" as the origin? I mean, I can use latitude and longitude without ever having been anywhere near its origin. All I need is someone to tell me how latitude and longitude work and my current location (or the location of some landmark) in that coordinate system. Ditto reference frames in special relativity, except you need to specify a location in space and time.

I don't think there is any need for the third observer to pass the first two, it was just an idle thought that occurred to me. So it's not so much a question of necessity but one of possibility. I'm just wondering if there is a limit on the number of observers who can pass each other in this way, such that their coordinate systems can be made to coincide at the origin.

 

[Lynch101]

If you disregard the stuff about being a "group", then what @Dale has written is pretty clear and very close to your pay grade.

If you start with one coordinate system, you can come up with a new coordinate system with a rotation. The rotation can be a roll, a pitch, a yaw or a combination of all three. That is three free parameters. Or the new coordinate system can have a relative x velocity, y velocity, z velocity or a combination of all three. That is another three free parameters.

So without changing the origin, you have six free parameters that you can use to choose a new coordinate system.

Now the part about being a group...

The idea is that you consider the set of coordinate system transformations identified by the six parameters (roll, pitch, yaw, vx, vy, vz). If you take a coordinate system and transform it twice (or three, four, five or more times) each time using a transformation selected from this set, you will find that the final resulting transformation was already a member of the original set.

In the sense of abstract algebra, this is to say that this set of transformations is "closed" under the operation of composition. Along with a few more simple facts, this is enough to ensure that this [sub-]set of transformations forms an algebraic "group".

[I am a not sure about whether or how the handedness and time reversal symmetries are normally integrated into the group. I suppose that one could glue on two additional discrete-valued dimensions -- (right-handed/left-handed, normal/time-reversed)]

thank you jbriggs. I read "10 dimensional group" and I think a barrier went up in my mind saying, "I won't be able to understand this". Your post has made it much clearer. Not saying I fully understand it, but I have a better insight into it.

When I hear about strong theory being 11 dimensional, is it in the same sense as this? or is it different again?

 

[Pencilvester]

I'm just wondering if there is a limit on the number of observers who can pass each other in this way, such that their coordinate systems can be made to coincide at the origin.

@Ibix is saying you can have an infinite number of observers’ reference frames have their origins at that originally mentioned reference event, but none other than the first two need to physically be at that event.

That said, you’re free to create a new hypothetical scenario, but the one you seem to be asking about is constrained more by biology than by special relativity. You can only cram so much biomass into any given volume and still call it biomass.

 

[Ibix]

I don't think there is any need for the third observer to pass the first two, it was just an idle thought that occurred to me. So it's not so much a question of necessity but one of possibility. I'm just wondering if there is a limit on the number of observers who can pass each other in this way, such that their coordinate systems can be made to coincide at the origin.

You are confusing two separate concepts - how many people can pass through an event and how to define an origin.

How many people can pass through one event is obviously limited because you can't have many people at the same point at the same time - we don't fit. This does not affect me choosing the event "those two guys meet" as my origin of coordinates, whether I was there or not. I was not present at the birth of Christ, but I still use that as the origin of my time coordinate.

So the answer to "how many people can pass each other" is "a few". The answer to "how many people's rest frames can share an origin" is "everybody".

 

[Lynch101]

@Ibix is saying you can have an infinite number of observers’ reference frames have their origins at that originally mentioned reference frame, but none other than the first two need to physically be at that event.

Yes, thank you Pencilvester, I realized what Ibix was saying eventually.

That said, you’re free to create a new hypothetical scenario, but the one you seem to be asking about is constrained more by biology than by special relativity. You can only cram so much biomass into any given volume and still call it biomass.

hahaha I get you. I was thinking that with two passing observers there would be a certain minimum distance between them, given they can't both occupy the same space. In this way, I was wondering if we could add more observers?

EDIT: I hadn't seen Ibix's post immediately prior to yours, I was going off the previous ones. I think I get the point clearly now.

 

[Lynch101]

You are confusing two separate concepts - how many people can pass through an event and how to define an origin.

How many people can pass through one event is obviously limited because you can't have many people at the same point at the same time - we don't fit. This does not affect me choosing the event "those two guys meet" as my origin of coordinates, whether I was there or not. I was not present at the birth of Christ, but I still use that as the origin of my time coordinate.

So the answer to "how many people can pass each other" is "a few". The answer to "how many people's rest frames can share an origin" is "everybody".

I was thinking that if two people pass each other, they cannot occupy the same space so there must be some physical distance between them. In this way I was wondering if a third person could be passing in the same way.

I think I see the point now though. If we add a third person, then they might be, for arguments sake, directly behind one of the other two. If they use the same origin, then they would not mark themselves as being at the origin of their own reference frame but perhaps one unit of measurement away?

 

[Lynch101]

Is the above roughly how an "Earth centred inertial frame" would work?

 

[jbriggs444]

When I hear about strong theory being 11 dimensional, is it in the same sense as this? or is it different again?

It is the same. The number of "dimensions" in an abstract space is the number of parameters required to identity a position within that space.

You could start here to get the foundations of a vector space. That article talks about the dimensionality of a vector space in terms of the number of basis vectors required to span the space.

There are other types of "spaces", but there is a lot of commonality.

 

[PeterDonis]

Is it possible to have a third observer passing such that that the passing event includes all three and all three then have the "passing event" as their origin? If so, can the same be done for a third, fourth, fifth, etc.?

Mathematically, yes. It's implied by what I said here:

Once you've picked an event to be the origin, there are an infinite number of reference frames you can have whose origin is that event, since there are a continuous infinity of possible 4-velocity vectors at the origin that can serve as the timelike basis vector for a reference frame (corresponding to the continuous infinity of possible relative velocities that two observers who meet each other at the chosen event can have).

In a practical sense, as others have pointed out, you can't fit an infinite number of people into a single event. (Strictly speaking, you can't even fit two, since people have a finite size.) But mathematically in SR "observers" are idealized as points and they are idealized as being able to pass through each other without interference.

 

[PeterDonis]

you can have an infinite number of observers’ reference frames have their origins at that originally mentioned reference event, but none other than the first two need to physically be at that event.

Mathematically speaking, none of the observers need to have their worldlines pass through the event that is chosen as the origin.

In a practical sense, it's almost always more convenient if their worldlines do pass through the origin. But none of them need to, mathematically speaking.

 

[Lynch101]

It is the same. The number of "dimensions" in an abstract space is the number of parameters required to identity a position within that space.

You could start here to get the foundations of a vector space. That article talks about the dimensionality of a vector space in terms of the number of basis vectors required to span the space.

There are other types of "spaces", but there is a lot of commonality.

Thank you jbriggs.

 

[Lynch101]

Mathematically, yes. It's implied by what I said here:
In a practical sense, as others have pointed out, you can't fit an infinite number of people into a single event. (Strictly speaking, you can't even fit two, since people have a finite size.) But mathematically in SR "observers" are idealized as points and they are idealized as being able to pass through each other without interference.

Thank you Peter.

 

[PeterDonis]

I was asking about post #23, not #25. So the correction you were referring to was your post #24 if I’m understanding you correctly?

You are hijacking someone else's thread to no benefit that I can see. Please take any further questions about this to PM conversation.