How do different observers decide if they are looking at the same thing?

Question

Note: I added some more descriptions, so that anyone who reads can benefit possibly more. I suspect this question become somewhat popular, because there may be many people confused with SR, but how deep is it I am not sure.

Motivation Here SR means Special Relativity, LT means Lorentz transformations.

SR is easy to describe, after you learn about how to back and forth between observers using Lorentz transformations, imo. I know of the physical motivations that leads to LT, but even if you accept them as axioms, I think it suffices for most problems. Note that although you can describe SR as GR with $g=\eta$, in the sense that LT are those linear maps that preserve the "norm" induced by $\eta(v,v)$, I never really needed to think of $\eta$ because I can look at formulas for LT and this is usually enough to solve a problem in SR. I think in the early days when SR existed but GR wasn't, people didn't really treat $\eta$ fundamental, but LT was the main contribution of SR.

I am taking a GR course, and there we are less concerned about what observers see (I mean there are less problems on that) and we are most concerned about solving for $g_{\mu\nu}$ and trajectories of particles etc.

Because of that, I wanted to discover the equations that relate $(t,x,y,z)$ and $(t',x',y',z')$ different observers measure in GR, but I run into the problem of what do I mean by "coordinates of some specific event", as in how does two different observers know they look at the same "event" in GR. In other words, spacetime is some abstract manifold, and an event is some $p\in M$ and $p$ doesn't have to look like a $4-$tuple, nor if it does should the first component be the time, or latter components space. I am asking if someone measures $p$, and how does another person know (if they are moving) they are looking at the same $p$ (which is independent of the reference frame). Below is the simple case of this question, for the SR case.

Question itself:

Let's restrict ourselves to SR. I know that if there are two observers and one of them moves with a constant velocity, say along the $x$ axis, I can relate the coordinates the latter measures to the one measured by the former using LT.

My question is, if this transformation is denoted by $x'=Lx$, and $x$ is interpreted as some event observed by one of the observers, what is $x'$?

What I mean is, I usually think that $x$' is the coordinates that the moving (relative to the other) observer measures for the event x, but how does that observer know they were trying to measure $x'$?

For example, imagine there is a ball $l$ meters away in the x direction. Then I think that the moving observer can measure what is the location of that ball in their reference frame, but how do they know they are supposed to look at that ball?

Like, if space was filled with balls of same color, after boosting, how would the moving observer know they are computing the coordinates of that "same" ball?

For another example, imagine 2 people are in empty space, and one of the moves along $x$ axis with velocity $v$. If $A$ looks at some point, and calls it $(t,x,y,z)$ how does $B$ know which of the four tuples $(t',x',y',z')$ should they tell to person $A$, so that they realize they are related by LT? I mean, it is all black, and you can't really distinguish things.

Otherwise, $x'$ is not a quantity that exists on its own but defined through the Lorentz transformation, which makes it tautological.

Note: For answers like leaf falling from a tree and two people comparing what did they measure for that event, I didn't accept it because in a universe where nothing happens but a leaf fell, two people I think could say these are the same "events", but what if from that same place periodically leafes fall. How do you know which leaf are you looking to compare with other person. I ask this to limit any anthropomorphism involved.

EDIT: My goal was to define "same event" so that I can relate what different observers see in GR (SR is given by LT which is well known). I wasn't fully interested in putting a cartesian system and synchronized clocks per say, since this procedure I think doesn't generalize to GR. Anyway, this question does indeed ask what I wanted to ask.

Answer 1

Many presentations of relativity start with this business of reference frames and observers. Those concepts are useful, but before invoking them, first get thoroughly fixed in your mind that there is ONE spacetime and ONE set of events in spacetime. To help your understanding, get a blank sheet of paper and say to yourself, "this paper will be a diagram showing all the events in some region of spacetime". Mark some dots and lines on your paper. Say to yourself, "Here are some events and sets of events." I won't post an entire introduction here, but the next step is to think about the idea of a clock with a regularly repeating mechanism, and the idea of a fixed and non-infinite speed for signals. One thus builds up the idea of how to specify time and distance by counting clock ticks and using the timing of light signals. Eventually one has a system of assigning spatial and temporal coordinates across the whole of spacetime. You can imagine the system as making a grid on your paper, but, note, the grid does not have to be made of squares or rectangles. It can be made of parallelograms. And there can be many such grids, with different sloping sides and sizes of parallelograms. The Lorentz transformation relates these grids to one another when the units of time and distance have been agreed in some standard way.

Answer 2

You are over-thinking it. An event is a position in spacetime, so in any given frame it has three spatial coordinates and a time coordinate. Consider the event of me typing the letter Q in this sentence. That event happened at a particular time and place. In my frame of reference I would label it with a particular set of coordinates, X- if you are moving relative to me, you would label it with a different set of coordinates in your frame, X'. The Lorenz transformation tells you how those two sets of coordinates for the same event are related, so if you know one set you can calculate the other.

Answer 3

What you ask is not so much a special relativity thing as it is a metrology thing. How do you set up an experiment such that you can measure what you seek to measure.

If you are trying to confirm that the Lorentz transform correctly describes physics, then you would not conduct it in a maximally noisy environment where it is difficult to identify the event being measured in the first place. There are many metrological tricks that can be leveraged. A popular one is to modulate a signal such that both observers can positively identify the specific one.

Once you trust that the Lorentz transform correctly describes physics, then you turn it around and use the Lorentz equations to do something. For example, given noisy observations from those two observers, I can combine them (such as by taking the average of them) to get a higher accuracy observation than either one of them. However, to do that, I must use the Lorentz equations properly to make sure I am averaging apples with apples. For instance, I may choose to use the transformation to convert both of them into a measurement of $x^\prime$. (One already is a measurement of $x^\prime$, but the other would need a conversion to get there)

Answer 4

I see a leaf fall from a tree. You see a leaf fall from a tree. We want to know whether we both saw the same leaf fall from the same tree.

Here's how we can tell: I use my meter sticks and clocks to determine the location $x$ and the time $t$ at which the leaf fell in my frame. I use the Lorentz transformation to compute the location $x'$ and the time $t'$ where that same leaf fell in your frame.

Then I ask you: "That leaf you saw fall----where and when was that?". If your answer matches the $x'$ and $t'$ that I computed, then we observed the same event. If not, not.

Answer 5

Coordinates, like $x$ or $x’$ are not physical quantities. They are convenient mathematical labels that we define.

Specifically they are labels that label real physical events with quadruples of numbers that possess some nice features. Primarily that if a set of events is continuous in the real world, the set of coordinates of those events is also continuous.

A coordinate chart is a smooth invertible map between events in spacetime and points in $\mathbb R^4$. We can make such a mapping by arbitrarily assigning some fiducial events to specific coordinates and arbitrarily assigning a procedure to determine the coordinates of other events. For example, we could have an observer with a clock and radar, and use their radar measurements to define a set of coordinates.

Once we have a coordinate chart, we can define new coordinate charts by making smooth invertible maps from the coordinates in $\mathbb R^4$ to new coordinates in $\mathbb R^4$. These maps are called coordinate transforms. We then know that if a particular coordinate point refers to a given physical event, then the transform of that coordinate refers to the same physical event. We know that because we defined both mappings and definitions are tautologically true. So in this case we know that the red ball at $x$ is the same red ball at $x’$ by definition the definition of the transform.

We could also define a new coordinate chart by choosing different fiducial events and/or a different procedure. In that case we do not know the transformation between the two coordinate systems. So if we knew the coordinates of the same red ball in both coordinate charts, then we could use that to begin building the transformation between the coordinates.

Until now I have not talked about the Lorentz transform. Assuming that spacetime is flat, then if a coordinate chart is made by the radar procedure by an inertial observer, then the Lorentz transform allows us to find new coordinate charts. These new coordinate charts can be obtained either using the radar procedure for a different inertial observer or by using the transformation. If there is any discrepancy between those two procedures then we know that spacetime is not flat, as we had assumed.

Answer 6

I think a way to visualize this is if you imagine that the ball suddenly popped into existence at time $t$ and position $x$ for observer $A$. Then observer $A'$ would see the ball pop into existence at time $t'$ and position $x'$, with $(t', x')$ related to $(t, x)$ by a Lorentz transformation. In this case the event is unambiguous (both observers should agree that something popped into existence at some point). Physically this is a bit unrealistic, but you can imagine other events such as the ball's color changing, or the ball colliding with a wall. Such events are unambiguous.

Alternately, you can suppose that observer $A$ has set up a network of synchronized clocks and detectors that are all at rest in his own frame, and similarly observer $A'$ has his own system of clocks and detectors at rest in his own frame. The trajectory of the ball is observed by $A$ and $A'$ via the collection of all detection events and clock readings (i.e. the set of all tuples $x^\mu = (t, x)$ collected by the detectors over the duration of the experiment). Thus what the observers really observe is a coordinate representation of the ball's worldline in their respective frames, i.e. they measure the path $x^\mu(\tau)$, but only up to unit-length reparametrizations of proper time $\tau$ (i.e. paths $x^\mu(\tau)$ and $x^\mu(\tau + \tau_0)$ are really the "same" worldline and physically indistinguishable). The parameter $\tau$ is simply a label for an arbitrary element of the set of tuples $(t, x)$ collected by the detectors. The ambiguity in choosing the parametrization of $\tau$ is related to the ambiguity you asked about whether $A$ and $A'$ are looking at the same event.

If the particle is moving in a straight line with uniform velocity for ever and ever after with no external interactions, then there is no way to agree on a specific origin of proper time unambiguously. But in practice, there will be a natural way to choose the origin of proper time unambiguously, such as defining $\tau = 0$ to represent the point where the particle collided with a wall, or changed its color, or reached a maximal/minimal speed. Then both observers can use $\tau = 0$ to represent this event, and $x^\mu(\tau)$, ${x^\mu}'(\tau)$ will represent the same events for all other $\tau$ and thus be related by Lorentz transformations.

(Actually, they also need to agree on the direction of time, due to $x^\mu(\tau)$ and $x^\mu(-\tau)$ representing the same worldline too. So there needs to be another unambiguous event that allows defining the direction of time, such as a second collision, or some assumption of entropy increase, or assumption that the ball eventually slows down due to friction forces, etc).

Another option is for the observers to agree on another event independent of the ball, such as a light switch turning on to signal the start of the experiment, and turning off to signal the end of the experiment. Then they can use $\tau = 0$ to label the ball's position when the light switch turned on (assuming the detectors record the state of the light switch as well as the ball), and require that the light turns off for some $\tau \gt 0$ to set the direction of time. This does not require using a distinctive feature in the particle's trajectory.