Special Relativity - Which reference frame experiences which time?

[PeroK]

Too many words. Time for some mathematics!

 

[vela]

What I'm trying to say is, can't the situation be described as two events which are that the Earth and dwarf planet move together?

No. An event happens at a place and time. The Earth and the dwarf planet aren't at the same point in space at the same time so they can't be described by the same event.

Kalle's perspective:

• Event 1: Lisa leaves Earth ($x=0$) at t=0.
• Event 2: Lisa arrives at the dwarf planet ($x=D$) at $t=T$ (different place and time).

Lisa's perspective:

• Event 1: Earth leaves Lisa ($x'=0$) at t'=0.
• Event 2: Dwarf planet arrives at Lisa ($x'=0$) at $t'=T_0$ (same place, but different time).

 

[PeroK]

No. An event happens at a place and time. The Earth and the dwarf planet aren't at the same point in space at the same time so they can't be described by the same event.

Kalle's perspective:

• Event 1: Lisa leaves Earth ($x=0$) at t=0.
• Event 2: Lisa arrives at the dwarf planet ($x=D$) at $t=T$ (different place and time).

Lisa's perspective:

• Event 1: Earth leaves Lisa ($x'=0$) at t'=0.
• Event 2: Dwarf planet arrives at Lisa ($x'=0$) at $t'=T_0$ (same place, but different time).

It's worth pointing out that this applies to classical physics as well as SR. This is valid whatever the speed of the rocket. In the classical, non-relativistic case (where the rocket has a speed much less than the speed of light), we have $T = T_0$. That's the Galilean time transformation - which is simple as there is assumed to be a single universal time in non-relativistic physics.

SR is more complicated because $T \ne T_0$.

It's also worth emphasising that you must be able to describe a problem in terms of events and their coordinates. It's of fundamental importance that you do this.

 

[AronYstad]

No. An event happens at a place and time. The Earth and the dwarf planet aren't at the same point in space at the same time so they can't be described by the same event.

Kalle's perspective:

• Event 1: Lisa leaves Earth ($x=0$) at t=0.
• Event 2: Lisa arrives at the dwarf planet ($x=D$) at $t=T$ (different place and time).

Lisa's perspective:

• Event 1: Earth leaves Lisa ($x'=0$) at t'=0.
• Event 2: Dwarf planet arrives at Lisa ($x'=0$) at $t'=T_0$ (same place, but different time).

What I meant was that since we describe everything on Earth as one object, Earth, then, since the Earth and the dwarf planet have the same reference frame and velocity, why can't we describe them as one single object.

Although now, once I have thought about it for a bit, I guess that it's because the situation is simplified in such a way that the Earth is one solid object which is at one point in space at a time, and the dwarf planet is also described as one solid object that is also at one point in space at a time. And an event can't be spread out over space, so the situation has to be described like that to get any useful information.

 

[PeroK]

It's simpler than that. An event is a point in spacetime. E.g. "today, 3pm, at Oxford Circus" is an event. "Stockholm and London" is not an event.

 

[vela]

What I meant was that since we describe everything on Earth as one object, Earth, then, since the Earth and the dwarf planet have the same reference frame and velocity, why can't we describe them as one single object.

In this problem, you can treat the Earth and the dwarf planet as points in space because they're very small compared to the relevant length scale. The Earth's radius, for example, is about 6400 km, which is much smaller than the distance Lisa travels, which will be on the order of an astronomical unit, which is approximately 150,000,000 km.

 

[jbriggs444]

What I meant was that since we describe everything on Earth as one object, Earth, then, since the Earth and the dwarf planet have the same reference frame and velocity, why can't we describe them as one single object.

You can treat them as a single body. An body with a non-negligible spatial extent. An "extended body" if you will. As opposed to a "pointlike object".

As has been pointed out, you cannot treat them as a single event.

An extended body (such as a rocket, a meter stick or a pair of distant planets) will be length contracted when considered from the point of view of a frame of reference against which it is seen to be moving. That is, the length of the body as measured by third party observers will be shorter than the length of the body as measured by observers who are moving along with the body.

In the case at hand, length contraction can help reconcile traveler Lisa's account of events with an Earthbound observer's account of events.

A complete understanding of how the same object can be measured to have different lengths is more complicated. It can be approached by carefully defining what the "length of an object" actually means in terms of experimental procedures and then exploring the relativity of simultaneity. The explanation might include terms such as "world tube", "cross section", non-Euclidean interval measures and a Euclidean analogy with the width of a strip of paper.