Special Relativity - Which reference frame experiences which time?

[AronYstad]

Homework Statement

Lisa travels to a dwarf planet. During the travel, she maintains the velocity v = 0.7c.
Kalle, who is observing from Earth, sees the travel take 43 minutes. How long does the travel take according to Lisa?

Relevant Equations

$$\Delta T = \gamma \Delta T_0$$
$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

This was a practice question, so it had the answer with it, which is 31 minutes. However, I'm confused as to why Lisa experiences T₀. It isn't exactly an event happening in Lisa's rocket, but rather her just moving through space. From her perspective, it should look like Earth is moving at the same speed in the opposite direction, right? So why do they experience different times and how do you know which one is T₀?

My LaTeX code wasn't working in preview btw, so idk if it will work when I post.

 

[PeroK]

The data is given in the rest frame of the Earth and the dwarf planet.

It would be a good exercise to describe the same events in the rest frame of the rocket. That should highlight the asymmetry.

And, of course, you should get the same answer to the question!

 

[AronYstad]

Is the rest frame the one that experiences T₀? If so, why does the answer in the key say that Lisa experiences T₀? Also, what exactly do you mean by the asymmetry?

Sorry for being slow at understanding. My native language isn't English, and I started learning special relativity quite recently, so I'm not that used to using these terms.

 

[PeroK]

Is the rest frame the one that experiences T₀? If so, why does the answer in the key say that Lisa experiences T₀? Also, what exactly do you mean by the asymmetry?

Sorry for being slow at understanding. My native language isn't English, and I started learning special relativity quite recently, so I'm not that used to using these terms.

Where and how are you learning SR?

I guess that $T_0$ is the time measured by a moving clock. And $T$ is the coordinate time in the relevant reference frame?

Does that make sense?

 

[AronYstad]

Where and how are you learning SR?

I guess that $T_0$ is the time measured by a moving clock. And $T$ is the coordinate time in the relevant reference frame?

Does that make sense?

I'm learning SR in Sweden, in what I think is equivalent to high school. I'm currently studying Physics 1a.

I was just taught that $T_0$ is the time measured in the same reference frame as the specified event, so I guess the movement to the dwarf planet is the event here. If I understood it correctly, that explanation does make sense, and it makes it quite a lot easier to tackle these types of questions.

 

[PeroK]

I'm learning SR in Sweden, in what I think is equivalent to high school. I'm currently studying Physics 1a.

I was just taught that $T_0$ is the time measured in the same reference frame as the specified event, so I guess the movement to the dwarf planet is the event here. If I understood it correctly, that explanation does make sense, and it makes it quite a lot easier to tackle these types of questions.

Unfortunately, that doesn't make much sense. An event is a physical thing that takes place in all reference frames. Each reference frame assigns time and space coordinates to each event.

Note that this is true in physics generally and is not just in SR.

You need to understand reference frames before can understand SR.

Have you studied the Galilean transformation?

 

[PeroK]

In this question we have only two events: rocket leaves Earth; and, rocket arrives at dwarf planet.

 

[AronYstad]

Unfortunately, that doesn't make much sense. An event is a physical thing that takes place in all reference frames. Each reference frame assigns time and space coordinates to each event.

Note that this is true in physics generally and is not just in SR.

You need to understand reference frames before can understand SR.

Have you studied the Galilean transformation?

Sorry, I might have worded that poorly. What I meant was that $T_0$ is the time measured in the reference frame that's stationary compared to the event.

I have not studied the Galilean transformation.

 

[PeroK]

Sorry, I might have worded that poorly. What I meant was that $T_0$ is the time measured in the reference frame that's stationary compared to the event.

A reference frame cannot be stationary relative to an event.

I have not studied the Galilean transformation.

Look it up on line.

 

[AronYstad]

A reference frame cannot be stationary relative to an event.

Then I have been taught wrong I guess.

 

[PeroK]

Then I have been taught wrong I guess.

Or misunderstood what your teacher was saying.

 

[AronYstad]

Or misunderstood what your teacher was saying.

More likely that, yeah. I'll ask my classmates and teacher after Easter break to see if they can help me understand it better.

 

[PeroK]

More likely that, yeah. I'll ask my classmates and teacher after Easter break to see if they can help me understand it better.

Try this if you don't already have a decent text book:

https://scholar.harvard.edu/david-morin/special-relativity

 

[vela]

Sorry, I might have worded that poorly. What I meant was that $T_0$ is the time measured in the reference frame that's stationary compared to the event.

The proper time $T_0$ is the time measured in the reference frame where the two events (leaving the Earth and arriving at the dwarf planet) happen at the same spatial coordinate. In this problem, that's clearly not in the Earth's rest frame.

In Lisa's frame, both events occur at $x'=0$, so the time she measures is $T_0$. From her perspective, she's at rest and doesn't move from the Earth to the dwarf planet. Instead, it's the planets that are moving. The Earth moves away from her while the dwarf planet comes to her.

 

[malawi_glenn]

I am a physics teacher in sweden also teaching that class of yours. If you want you can PM me

 

[PeroK]

The proper time is the time measured in the reference frame where the two events (leaving the Earth and arriving at the dwarf planet) happen at the same spatial coordinate. In this problem, that's clearly not in the Earth's rest frame.

In Lisa's frame, both events occur at $x'=0$. From her perspective, she's at rest and doesn't move from the Earth to the dwarf planet. Instead, it's the planets that are moving. The Earth moves away from her while the dwarf planet comes to her.

That's fine but proper time is probably another new concept for the OP.

 

[malawi_glenn]

Usually, this special relativity module takes place during 2-3 one hour long lessons. There is no mentioning of Lorentz transformations nor Gallilean transformation. The time dilation formula is "derived" using einstein's light clock thought experiment. That's it.

 

[PeroK]

Usually, this special relativity module takes place during 2-3 one hour long lessons. There is no mentioning of Lorentz transformations nor Gallilean transformation. The time dilation formula is "derived" using einstein's light clock thought experiment. That's it.

What can you learn in three hours? Tragic!

 

[AronYstad]

The proper time $T_0$ is the time measured in the reference frame where the two events (leaving the Earth and arriving at the dwarf planet) happen at the same spatial coordinate. In this problem, that's clearly not in the Earth's rest frame.

In Lisa's frame, both events occur at $x'=0$, so the time she measures is $T_0$. From her perspective, she's at rest and doesn't move from the Earth to the dwarf planet. Instead, it's the planets that are moving. The Earth moves away from her while the dwarf planet comes to her.

Thank you, that makes sense. I can see how I might have misunderstood that.

Regarding the second paragraph, I haven't learnt that notation of $x'$, but I guess it's the spatial coordinate of the events from her perspective. I get now that Lisa measures $T_0$ because of which events are specified in the question, but if it was worded from her perspective and said that the Earth moved away from her, would someone on Earth measure $T_0$ instead, and in that case, would Lisa's time be longer than Kalle's?

 

[malawi_glenn]

What can you learn in three hours? Tragic!

When I teach this module it is usually
1h time dilation
1h lenght contraction
1h relativistic energy

The relativity module is very superficial, but the entire Physics 1a course is a mess. It is like a tapas dinner.
The swedish school agency will change this starting from year 2025 (where they will move this module up to Physics 2 course)

I teach Physics 3 and there we spend about 15 lectures on special relativity :)

 

[PeroK]

Thank you, that makes sense. I can see how I might have misunderstood that.

Regarding the second paragraph, I haven't learnt that notation of $x'$, but I guess it's the spatial coordinate of the events from her perspective. I get now that Lisa measures $T_0$ because of which events are specified in the question, but if it was worded from her perspective and said that the Earth moved away from her, would someone on Earth measure $T_0$ instead, and in that case, would Lisa's time be longer than Kalle's?

Understanding this is an important first step. That's why I suggest analysing the problem from both reference frames.

It's not enough to make a vague argument about the scenario being symmetrical

 

[PeroK]

PS this is where you need mathematical discipline, in terms of clear notation for how you refer to coordinates and clock readings. Usually coordinates in one frame are primed: $x', t'$.

But, if you are not studying coordinate transformations (Galilean or Lorentzian), then it's difficult to know what to suggest!

 

[malawi_glenn]

@AronYstad I will PM you now a page from my book, it is written in swedish so I can't post it here. It covers the time dilation as a special case of the Lorentz transformation so you will not understand the derivation of the forumla. But, there is a nice diagram at the bottom of the page where you can see what is going on which each frames clock.

Note. The problem is will worded. That a person on Earth sees the travel taking 43 min is not pure time dilation. Thing is that we see things by receiving light signals. So, when the rocket has completed the journey, light needs to travel from that location to the stationary observer in the Earth frame. If the distance between the observer and the landing spot for the rocket is L according to the earths frame of reference, you need to add the time L/c.

Time dilation is what is depcited in the diagram above: two stationary clocks C & D in the $\tilde S$ frame (I use tilde instead of prime not to confuse with prime for derivatives) at the origin of that frame, two stationary clocks A & B in frame $S$, one is located at the origin and one at the coordinate $x_1$. The frame $\tilde S$ is moving w.r.t. $S$ with a constant velocity $v$. When the origins of the two frames coincide, clock A and C are stopped. When the origin of $\tilde S$ coincide with $x_1$ clocks B and D are stopped. The ratio of the time differences $t_B - t_A = \Delta t$ and $\tilde t_D - \tilde t_C = \Delta \tilde t$ are equal to $\Delta t / \Delta \tilde t = \gamma$. This is where the "asymmetry" lies, you have two clocks at the same location according to one frame ($\tilde S$), but two clocks at different locations according to the other frame ($S$).

 

[AronYstad]

The problem is will worded.

No, I just didn't translate it correctly. I think a better translation would be "Kalle observes the journey and interprets it as taking 43 minutes." or something like that.

 

[malawi_glenn]

No, I just didn't translate it correctly. I think a better translation would be "Kalle observes the journey and interprets it as taking 43 minutes." or something like that.

Gottcha!

I usually give this kind of problem for my Physics 3 class exams on relativity - how long time it takes for the journey according one stationary observer at Earth, and I get very sad when students just use the time dilation :D

Kalle

 

[vela]

If it was worded from her perspective and said that the Earth moved away from her, would someone on Earth measure $T_0$ instead, and in that case, would Lisa's time be longer than Kalle's?

I'm assuming you're thinking about the same problem just described from Lisa's point of view. If so, then no. The two events happen in the same place only in Lisa's frame regardless of whose perspective is used to describe them. So Lisa always measures $T_0$, and all inertial observers agree on that.

 

[AronYstad]

I'm assuming you're thinking about the same problem just described from Lisa's point of view. If so, then no. The two events happen in the same place only in Lisa's frame regardless of whose perspective is used to describe them. So Lisa always measures $T_0$, and all inertial observers agree on that.

What I meant was that if it's described from Lisa's perspective, can't the events be described as the Earth moving away from her (disregarding everything else in the universe for simplicity's sake)? From Lisa's perspective, the Earth leaves her at one point in space and arrives at some destination at another point in space. However, on the Earth, Kalle would see the Earth in the same position at both points in time.

I get that in an actual situation, it's ridiculous to describe it like everything else moving relative to Lisa, but from her perspective, that's what she observes, right? So I'm just wondering why this wouldn't make $T_0$ be measured on the Earth.

 

[PeroK]

What I meant was that if it's described from Lisa's perspective, can't the events be described as the Earth moving away from her (disregarding everything else in the universe for simplicity's sake)? From Lisa's perspective, the Earth leaves her at one point in space and arrives at some destination at another point in space. However, on the Earth, Kalle would see the Earth in the same position at both points in time.

I get that in an actual situation, it's ridiculous to describe it like everything else moving relative to Lisa, but from her perspective, that's what she observes, right? So I'm just wondering why this wouldn't make $T_0$ be measured on the Earth.

It's not ridiculous. All inertial reference frames are equally valid. That's the first postulate of both Newtonian physics and SR.

But, you can't do physics by waving your hands. You need to do the hard calculations to see why it's not exactly the same from the rocket frame.

In summary, you have one rocket and two planets. That's one difference for a start!

 

[vela]

What I meant was that if it's described from Lisa's perspective, can't the events be described as the Earth moving away from her (disregarding everything else in the universe for simplicity's sake)?

What is "it"?

From Lisa's perspective, the Earth leaves her at one point in space and arrives at some destination at another point in space. However, on the Earth, Kalle would see the Earth in the same position at both points in time.

If the second event is "Earth arrives at some destination," then yes, the time elapsed between the events "Earth leaves Lisa" and "Earth arrives at some destination" according to Kalle is the proper time.

That pair of events is different than the two events "Earth leaves Lisa" and "Dwarf gets to Lisa" that were specified in the original problem.

 

[AronYstad]

What is "it"?

The question/situation.

That pair of events is different than the two events "Earth leaves Lisa" and "Dwarf gets to Lisa" that were specified in the original problem.

I meant that the Earth arrives to some location when the dwarf planet gets to Lisa, since I would assume the Earth and dwarf planet are in the same reference frame according the question. So I can rephrase it to:
The Earth and dwarf planet leave their original locations together (when Lisa leaves Earth), and the Earth and dwarf planet arrive to some destination (when Lisa arrives).

Am I wrong in assuming that the Earth and dwarf planet have the same velocity as each other in all the reference frames here and that Kalle would measure both the Earth and dwarf planet to be stationary, and therefore both events are in the same position relative to Kalle?

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?

Sorry if my explanations are getting more and more confusing. I'm quite tired right now, so my brain isn't the best at writing down my thoughts right now.

 

[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.