What do observers in relative motion agree on?

Question

What are the measurements on which two observer in relative motion will agree? Other than the speed of light.

Answer 1

They would agree about what the laws of physics are.

The principle of relativity:

"The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion. OR: The laws of physics are the same in all inertial frames of reference."

https://en.m.wikipedia.org/wiki/Postulates_of_special_relativity

Peter mentions spacetime interval as invariant. I would add that $\vec{E}\cdot \vec{B}$ and $E^2 - c^2 B^2$ are invariant, where $\vec{E}$ and $\vec{B}$ are the electric and magnetic fields seen in any uniformly moving frame of reference.

Answer 2

They all agree on the spacetime interval.

Answer 3

We can categorise objects depending on how they transform under relativistic transformations. For example velocity (strictly speaking four-velocity) transforms as a vector. For objects to be the same in all frames they need to transform as Lorentz scalars.

Peter mentions that the spacetime interval is the same for all observers, and this is an example of a scalar. In fact it's the length of a displacement vector given by:

$$ \Delta s^2 = \eta_{\alpha\beta} x^\alpha x^\beta $$

where $\mathbf{x}$ is the vector and $\eta$ is the Minkowski metric tensor. In general anything given by an equation of this type will be a scalar and the same for all observers. The magnitude of the four velocity is another example, though this is a rather boring example since it's always equal to the speed of light. In fact the speed of light is an invariant in SR precisely because it's the magnitude of a four velocity.

Actually I don't think there are many exciting scalars in special relativity. Moving to general relativity adds some more important scalars such as the scalar curvature and the Kretschmann scalar.

Answer 4

Not exactly sure this is the kind of answer you were looking for but I'll take a shot.

"Relativity" is actually a poor name for Einstein's theory but in historical context, he was arguing against the idea of an unmoving aether which provided an absolute or foundational frame of reference which would always produce the same same measurements for all observers. This got stuck in people's heads that "all things (events) are relative and different for each observer so no one can agree on any event so all events are subjective," which is not true.

All observers and in all frames of references can convert all other measurements in all other frames of references to the local frame as long as they can observer or communicate with other observers in other frames.

One could argue that since all observations/measurement in all frames of references can be converted to meaningful observations/measurements in all other frames of reference, that all observers do observe and measure the same event in same way, once the information about or from other frames of reference, are combined with the local observations/measurements.

For example, lets consider a variant of the famous three observers observing a ball being bounced up and down on a moving train, except in this case, each external observer has fire a weapon such that the weapon/weapon-effect strike the ball simultaneously regardless of frame of reference from which the strike is observed.

Consider a science fiction scenario in three different Weapon Platform Targeting Systems (WPTS), each with a different frame of reference for a moving target. The target is rotating around its axis of travel which causes a sensory or weapons cluster on one exterior side to rotate around the circumference of the ship. (Analogous to the ball being bounced up and down on the moving train.)

To disable the cluster, all three WPTS must strike the cluster with a simultaneous attack (could be kinetic or directed energy doesn't really matter. To drag in time dilation, lets assume all the powered spacecraft are moving at 10% speed of light.

(This is a variant of the bouncing ball on a train experiment except the observers have to communicate and coordinate.)

The experiment occurs out in interstellar space so that the only orbits are those around the galactic core which we'll ignore.

WTPS1 is on a space station, which is not under acceleration. The target is moving in a trajectory that is a displaced tangent to the station. The velocity of the target appears as X.

WTPS2 is on a spacecraft that is mirroring the motion of the target such that the target appears to remain motionless in respect to it. The velocity of the target appears as 0.

WTPS3 is exactly behind the target but accelerating faster such that the apparent velocity of the target is Y towards WTP3.

WTP1 would observe the target cluster to move in a broken sinusoidal wave, broken up when the cluster is on the other side of the target and unobservable. WTP2 would observe the target cluster moving along a straight line, appearing at the "top" and moving down to the "bottom" before disappearing and then at a specific time appearing at the top again. WPTS3 would observe the target cluster moving in a spiral pattern with the spiral growing outward as WPTS3 closed on the target.

As well, each WPTS would observe the other 2 WPTS moving at different velocities relative to the observing WPTS.

To hit the target cluster, each WPTS will fire at a different relative time and at different vector. Each WPTS will have to calculate when the other two will fire in the context of its own time frame.

This is pretty easy to do, assuming each WPTS knows the velocity and capability of the weapons being fired.

Simple doppler radar would give each WPTS all the information it needs to convert its own measurements to the other's frames of reference. Alternative, by prior arraignment, each could flash a beacon at predetermined internal within their own time frame e.g. every five seconds. As the other two WPTS would measure the beacons flashing at intervals other than five seconds, it could use that difference to calculate the observations of the target made from the two WPTS's frames of reference.

Each WPTS could also calculate at what local time the target cluster's rotation around the circumference of the target craft, it would be exposed to a strike from the other two WPTSs.

From all four frames of reference, the Target and each of the three WPTS, each WPTS would appear to fire at different local times but all would observe the strikes hit the target at the point in local time.

If multiple observers in multiple frames of reference can nevertheless convert their local observations into all other frames they observe or communicate with to the extent they coordinate their individual actions upon a common observed event, then arguably they actually agree on all observations.