Confused by Lorentz transformation equation

[FeynmanFtw]

TL;DR Summary: Solving a problem regarding a train going past a station using length contraction and the Lorentzian transformation.

I'll dive straight in. I encountered a problem where there is a train travelling at 0.6c going past a station, length 500 m when measured by an observer at rest relative to the station. The question asks what is the perceived length of the station relative to an observer on the train.

I simply used the length contraction formula:

$$L = \frac{L_0}{\gamma} = \frac{500}{1.25} = 400 \textnormal{ m}$$

In the answer, it uses the equation:

$$\Delta x' = \gamma(\Delta x - v\Delta t) = \frac{5}{4}\cdot 500 = 625 \textnormal{ m}$$

I'm pretty sure the first answer is correct, but I must admit I'm really confounded by the answer from the question. I haven't fully covered the Lorentzian transformation equations so I don't feel confident in tackling that part yet, so I would appreciate some help on this!

Could it be a mistake in the mark scheme (they do seldom happen, unfortunately).

 

[Orodruin]

It feels like you have not presented the actual problem statement verbatim. If you have changed anything in the exposition and/or given your own recollection of the problem, this may affect the problem interpretation, in particular when it comes to relativity problems. As presented here, yes you could apply the length contraction formula, but I do not want to give this as a definitive answer before I have seen the exact problem statement.

 

[Ibix]

Yes, you have the correct answer to the problem you've stated. Theirs is the correct answer to a related question, so I'd be interested to see the exact wording of the question, not your paraphrase.

(Beaten to it by Orodruin, I see.)

 

[Sagittarius A-Star]

$$L = \frac{L_0}{\gamma} = \frac{500}{1.25} = 300 \textnormal{ m}$$

$$ \frac{500}{1.25} = 400$$Source:
https://www.wolframalpha.com/input?i2d=true&i=Divide[500,1.25]

 

[FeynmanFtw]

$$ \frac{500}{1.25} = 400$$Source:
https://www.wolframalpha.com/input?i2d=true&i=Divide[500,1.25]

Ah, that was a typo, I meant to write 400, not 300.

 

[FeynmanFtw]

Yes, you have the correct answer to the problem you've stated. Theirs is the correct answer to a related question, so I'd be interested to see the exact wording of the question, not your paraphrase.

(Beaten to it by Orodruin, I see.)

"Calculate, for the reference frame of the train, the distance between the light beacons at each end of the station".

 

[Sagittarius A-Star]

Ah, that was a typo, I meant to write 400, not 300.

As others mentioned, could there be also a typo in the problem text (mix up train/station)?

 

[FeynmanFtw]

It feels like you have not presented the actual problem statement verbatim. If you have changed anything in the exposition and/or given your own recollection of the problem, this may affect the problem interpretation, in particular when it comes to relativity problems. As presented here, yes you could apply the length contraction formula, but I do not want to give this as a definitive answer before I have seen the exact problem statement.

The exact question asks about a train going past a station with two light beacons at either end, that are said to give a pulse at the same time as observed by someone at the station, and just as the train starts to pass the station platform.

The question first asks to calculate the Lorentz factor, and then asks about the distance between light beacons. I imagine the confusion may lie here, because I'm interpreting this as asking about the distance between the beacons as the length of the station.

The only other thing I can assume is the perceived distance travelled by the train before observing the second light (not the beacon but the actual light signal), but SURELY this must be less than the length of the platform regardless?

 

[Orodruin]

Beaten to it by Orodruin, I see.

It would be worse if our answers were not coherent ...

"Calculate, for the reference frame of the train, the distance between the light beacons at each end of the station".

This still sounds like an excerpt of s as question, not the full formulation. There is a reason that the homework rules require you to reproduce the problem statement exactly as given.

 

[Ibix]

"Calculate, for the reference frame of the train, the distance between the light beacons at each end of the station".

Hm. Presumably the lights are attached to the platform and flash simultaneously in the platform rest frame. If so, their answer is the distance between the locations of the lights when they flash (non-simultaneously in the train frame). Yours is the distance between the lamps.

 

[FeynmanFtw]

Hm. Presumably the lights are attached to the platform and flash simultaneously in the platform rest frame. If so, their answer is the distance between the locations of the lights when they flash (non-simultaneously in the train frame). Yours is the distance between the lamps.

This may be the solution, it seems, but then the question statement may be unintentionally ambiguous. However, in the answer scheme it says that I can use either equation (length contraction or the Lorentz transformation) to get the right answer. But the only way I can get the other length using the Length contraction equation is by swapping the values for L, which to my mind doesn't make sense at all (surely L_0 is the proper length?!)

Also, how can the distance between the locations of the lights when they flash be greater than the length of the station?

 

[jambaugh]

I wish to add a few details. To formulate a relativity problem correctly, the best practice is to identify the two space-time events that define the differences which answer the questions of "how long" or "what duration" express the solution.

In your example, an "observer on the train" perceives time in the frame where the train itself is "stationary" and the stations are passing by at 0.6c. The peculiarity of your question that I see is that the length of the train is given as 500m as seen by an observer relative to the station. This means that there are two events, that are, in the station frame, simultaneous ([itex]\Delta t=0[/itex]) as seen by the station observer, and with spatial separation in that frame of [itex]\Delta x=500m[/itex]. The train's frame is boosted by the factor [itex] v=c\cdot tanh(B)[/itex] where [itex]B=tanh^{-1}(0.6)\approx 0.6931472[/itex].
Under this frame transformation:
[tex]\Delta x' = cosh(B)\Delta x + c\cdot sinh(B) \Delta t[/tex] [tex]c\Delta t' = sinh(B)\Delta x + c \cdot cosh(B)\Delta x[/tex]
or [tex]\Delta x' = 1.25 \Delta x + c\cdot 0.75\Delta t[/tex] [tex]c\Delta t' = 0.75\Delta x + c \cdot 1.25\Delta x[/tex]
Or [tex]\Delta x' = 1.25 (500m)+ c\cdot 0.75(0)=625m[/tex] [tex]c\Delta t' = 0.75(500m) + c \cdot 1.25(0)=375c =375\text{ light meters}[/tex] This gives the moving observers observation of the length of the train and the time of passage.
In my not so humble opinion, the easiest way to express a relativistic transformation is as a "pseudo"-rotation (i.e. using hyperbolic trig) of a pair of space-time event points using the common units of distance and c*time, e.g. meters and light-meter for example.)

 

[Sagittarius A-Star]

I encountered a problem where there is a train travelling at 0.6c going past a station, length 500 m when measured by an observer at rest relative to the station.

I assume, that "500 m" is meant to be the contracted length of the train.

The exact question asks about a train going past a station with two light beacons at either end, that are said to give a pulse at the same time as observed by someone at the station, and just as the train starts to pass the station platform.

... and ...

The question asks what is the perceived length of ... relative to an observer on the train.

I assume, that they ask for the spatial distance between the light beacon event locations with respect to the train-frame. This would be the un-contracted length of the train.

 

[Ibix]

Also, how can the distance between the locations of the lights when they flash be greater than the length of the station?

They're moving and they don't flash at the same time.

 

[Sagittarius A-Star]

The question first asks to calculate the Lorentz factor, and then asks about the distance between light beacons.

That's indirect speech. As others mentioned, why do you not write the complete problem text verbatim?

 

[Mister T]

I encountered a problem where there is a train travelling at 0.6c going past a station, length 500 m when measured by an observer at rest relative to the station.

Perhaps it's the train that has a length of 500 m as measured in the station's rest frame.

 

[Orodruin]

The exact question asks about a train going past a station with two light beacons at either end, that are said to give a pulse at the same time as observed by someone at the station, and just as the train starts to pass the station platform.

The question first asks to calculate the Lorentz factor, and then asks about the distance between light beacons. I imagine the confusion may lie here, because I'm interpreting this as asking about the distance between the beacons as the length of the station.

The only other thing I can assume is the perceived distance travelled by the train before observing the second light (not the beacon but the actual light signal), but SURELY this must be less than the length of the platform regardless?

You see, this is still not quoting the question verbatim. Even after several people asking you to do so repeatedly. Please follow the guidelines in the future.

Regardless, it is sufficient information to resolve your issue. The problem is most likely (we still don’t know for sure because you refuse to post the actual problem text) asking for the distance between the two flash events in the train system - not the length of the platform as you stated in the OP. These are not the same thing.

The length contraction formula can indeed still be used, but then you need to use it the other way around. The reason it can be used like that is that the events are known to be simultaneous in the platform frame.

 

[FeynmanFtw]

I assume, that "500 m" is meant to be the contracted length of the train.

No, it is the length of the station (platform).

 

[FeynmanFtw]

You see, this is still not quoting the question verbatim. Even after several people asking you to do so repeatedly. Please follow the guidelines in the future.

Regardless, it is sufficient information to resolve your issue. The problem is most likely (we still don’t know for sure because you refuse to post the actual problem text) asking for the distance between the two flash events in the train system - not the length of the platform as you stated in the OP. These are not the same thing.

The length contraction formula can indeed still be used, but then you need to use it the other way around. The reason it can be used like that is that the events are known to be simultaneous in the platform frame.

I'm not refusing to do anything. I just didn't expect there to be such an issue with the wording, but I guess such is the world of SR.

"A train is travelling past a station at a speed of 0.6c. Two light beacons, A & B, are positioned at either end of the station. They turn on at the same time according to observers on the station, as soon as the train starts to pass the station.

The station is 500 m long, as measured by observers on the station.

a) Calculate the Lorentz factor for the train.

b) Calculate, in the reference frame of observers on the station, the length contraction of the train.

c) Calculate, in the reference frame of the train, the distance between the two light beacons.

d) Calculate, in the reference frame of the train, the time between the light beacons turning on."

This is the problem question at hand.

 

[PeroK]

I'm not refusing to do anything. I just didn't expect there to be such an issue with the wording, but I guess such is the world of SR.

"A train is travelling past a station at a speed of 0.6c. Two light beacons, A & B, are positioned at either end of the station. They turn on at the same time according to observers on the station, as soon as the train starts to pass the station.

The station is 500 m long, as measured by observers on the station.

a) Calculate the Lorentz factor for the train.

b) Calculate, in the reference frame of observers on the station, the length contraction of the train.

c) Calculate, in the reference frame of the train, the distance between the two light beacons.

d) Calculate, in the reference frame of the train, the time between the light beacons turning on."

This is the problem question at hand.

I tend to agree with your initial answer. The distance between the beacons is independent of whether the beacons are on, off or flashing. And, that distance is the contracted length of the platform.

A separate question is the distance (in the reference frame of the train) between the two events: first beacon flashes and second beacon flashes.

 

[FeynmanFtw]

I tend to agree with your initial answer. The distance between the beacons is independent of whether beacons are on, off or flashing. And, that distance is the contracted length of the platform.

A separate question is the distance (in the reference frame of the train) between the two events: first beacon flashes and second beacon flashes.

I appreciate the response. That said, I must say I'm confused by how the other answer works. Since the platform is moving at 0.6c relative to the train, plus the 2nd light beacon has emitted a light signal travelling towards the train at c (correct addition of relativistic speeds notwithstanding), how on Earth can the perceived distance between these two lights be greater than the length of the platform? Also, shouldn't the platform itself (as has been established) be contracted, hence 'bringing' the further beacon closer to me?

 

[PeroK]

I appreciate the response. That said, I must say I'm confused by how the other answer works. Since the platform is moving at 0.6c relative to the train, plus the 2nd light beacon has emitted a light signal travelling towards the train at c (correct addition of relativistic speeds notwithstanding), how on Earth can the perceived distance between these two lights be greater than the length of the platform? Also, shouldn't the platform itself (as has been established) be contracted, hence 'bringing' the further beacon closer to me?

Let's forget the simultaneity of the beacons flashing in the platform frame. In fact, let's forget relativity for a moment. Suppose we have a plane moving at a humble ground speed of 250m/s. The captain switches on the fasten seat belt sign. That event takes place at some location relative to the Earth's surface. Each passenger, after a variable delay, fastens their seat belt. Those events could take place practically any distance from the initial "switch on seat belt" event - again relative to the Earth's surface. If a passenger waits 10 seconds, then they fasten their seat belt about 2.5 km ahead of the first event. Etc.

Does that make sense?

 

[Orodruin]

I just didn't expect there to be such an issue with the wording, but I guess such is the world of SR.

Such is often the case in many physics subjects. You would be surprised how many misunderstandings are based on people misinterpreting things about a problem formulation or, as you have, add on additional assumptions of their own that turn out false. That is why the homework guidelines require you to post the problem as stated - verbatim.

I tend to agree with your initial answer. The distance between the beacons is independent of whether the beacons are on, off or flashing. And, that distance is the contracted length of the platform.

A separate question is the distance (in the reference frame of the train) between the two events: first beacon flashes and second beacon flashes.

Based on the provided formulation, I agree. If they wanted the other answer, the formulation is sloppy at best.

I appreciate the response. That said, I must say I'm confused by how the other answer works. Since the platform is moving at 0.6c relative to the train, plus the 2nd light beacon has emitted a light signal travelling towards the train at c (correct addition of relativistic speeds notwithstanding), how on Earth can the perceived distance between these two lights be greater than the length of the platform? Also, shouldn't the platform itself (as has been established) be contracted, hence 'bringing' the further beacon closer to me?

The beacons will not send their flashes simultaneously in the train frame. The one at the end if the platform will flash first. In the time delay between flashes, the other beacon moves away from where the first flash originated. Even if the beacons are closer due to length contraction, the time delay is enough for the other beacon to move so far away that the distance between the flash origins is longer than the station in the station rest frame.

 

[FeynmanFtw]

Let's forget the simultaneity of the beacons flashing in the platform frame. In fact, let's forget relativity for a moment. Suppose we have a plane moving at a humble ground speed of 250m/s. The captain switches on the fasten seat belt sign. That event takes place at some location relative to the Earth's surface. Each passenger, after a variable delay, fastens their seat belt. Those events could take place practically any distance from the initial "switch on seat belt" event - again relative to the Earth's surface. If a passenger waits 10 seconds, then they fasten their seat belt about 2.5 km ahead of the first event. Etc.

Does that make sense?

That makes total sense, and I've never really had issues with this, nor Galilean transformations to any degree (so far). That said, no matter how I try to approach this problem, I keep encountering a bit of a brick wall (which suggests to me that I'm understanding something fundamental about SR incorrectly).

I've tried to also go at the problem differently, factoring in time dilation, because on paper it shows me the answer they want, but intuitively I'm struggling.

I assume that the first beacon is observed by the train as soon as it turns on, as the train should observe this pulse coming towards it at c, and since the question says that the train passes the station as soon as the lights turn on, we can assume that, for all intents and purposes, the train and the station observers can agree that this is t = 0.

In the mean time, light from B (as observed by the station) will travel across the platform using the equation:

$$ t = \frac{500}{c} \approx 1.67 \times 10^{-6} \textnormal{ s}$$

However the train is moving at 0.6c, therefore experiences time dilation:

$$ \Delta t = \gamma \Delta t_0 = \frac{5}{4} \cdot 1.67 \times 10^{-6} \approx 2.1 \times 10^{-6} \textnormal{ s}$$

Finally, multiplying this by c, gives me the answer that's in the mark scheme:

$$ x = 2.1 \times 10^{-6} \cdot c = 625 \textnormal{ m} $$

Here's my issue. Is that the distance that light seems to have travelled as measured by the train? Am I any closer to understanding this properly?

 

[Sagittarius A-Star]

c) Calculate, in the reference frame of the train, the distance between the two light beacons.
d) Calculate, in the reference frame of the train, the time between the light beacons turning on."

• Your answer in the OP is correct related to question c).
• The given answer, cited (unfortunately also not complete nor verbatim) in the OP, is so far correct, related to question d).

 

[PeroK]

That makes total sense, and I've never really had issues with this, nor Galilean transformations to any degree (so far). That said, no matter how I try to approach this problem, I keep encountering a bit of a brick wall (which suggests to me that I'm understanding something fundamental about SR incorrectly).

I've tried to also go at the problem differently, factoring in time dilation, because on paper it shows me the answer they want, but intuitively I'm struggling.

I assume that the first beacon is observed by the train as soon as it turns on, as the train should observe this pulse coming towards it at c, and since the question says that the train passes the station as soon as the lights turn on, we can assume that, for all intents and purposes, the train and the station observers can agree that this is t = 0.

In the mean time, light from B (as observed by the station) will travel across the platform using the equation:

$$ t = \frac{500}{c} \approx 1.67 \times 10^{-6} \textnormal{ s}$$

However the train is moving at 0.6c, therefore experiences time dilation:

$$ \Delta t = \gamma \Delta t_0 = \frac{5}{4} \cdot 1.67 \times 10^{-6} \approx 2.1 \times 10^{-6} \textnormal{ s}$$

Finally, multiplying this by c, gives me the answer that's in the mark scheme:

$$ x = 2.1 \times 10^{-6} \cdot c = 625 \textnormal{ m} $$

Here's my issue. Is that the distance that light seems to have travelled as measured by the train? Am I any closer to understanding this properly?

Unfortunately, I think that calculation only works by accident. You have two misconceptions (which are encouraged by this sort of question, IMO):

1) SR is a theory of spacetime transformations between reference frames. It does not depend on the observations by any particular observer. The location of the train is of no consequence. The only relevant point is that the train and the platform have a relative speed of ##0.6c## in the direction aligned with the platform. In other words, the train could be nowhere near that particular station and the answer would be the same - as long at the train is moving in the given direction.

2) SR is not dependent on light signals from events reaching an observer. SR applies equally in a darkened room with no electromagnetic radiation. The fact that they are light-emitting beacons is totally irrelevant. The only relevant point is that two events at either end of the platform occur simultaneously in the platform frame. It could be two station porters each tossing a coin simultaneously at either end of the platform.

For the problem where we are trying to find the distance between the two flashing events, you could muddle through using length contarction, time dilation and relativity of simultaneity. But, really, this is a job for the Lorentz Transformation. Then the approach is clear:

- Identify the coordinates of the events in the platform frame.
- Transform those coordinates to the train frame.

Nothing to do with light, observations or any of that other extraneous stuff. It's a pure and simple coordinate transformation.

 

[FeynmanFtw]

• Your answer in the OP is correct related to question c).
• The given answer, cited (unfortunately also not complete nor verbatim) in the OP, is so far correct, related to question d).

Yes, yes it is. I literally said that in the first post. "In the answer, it uses the equation, etc".

 

[Sagittarius A-Star]

Yes, yes it is. I literally said that in the first post. "In the answer, it uses the equation, etc".

Does the cited answer also say, that it relates to question d), and does it continue to calculate
##\Delta t'## from ##\Delta x'=625m## and ##\Delta x=500m##?

Is there also an explicit answer to question c) ?

 

[FeynmanFtw]

Unfortunately, I think that calculation only works by accident. You have two misconceptions (which are encouraged by this sort of question, IMO):

1) SR is a theory of spacetime transformations between reference frames. It does not depend on the observations by any particular observer. The location of the train is of no consequence. The only relevant point is that the train and the platform have a relative speed of ##0.6c## in the direction aligned with the platform. In other words, the train could be nowhere near that particular station and the answer would be the same - as long at the train is moving in the given direction.

2) SR is not dependent on light signals from events reaching an observer. SR applies equally in a darkened room with no electromagnetic radiation. The fact that they are light-emitting beacons is totally irrelevant. The only relevant point is that two events at either end of the platform occur simultaneously in the platform frame. It could be two station porters each tossing a coin simultaneously at either end of the platform.

For the problem where we are trying to find the distance between the two flashing events, you could muddle through using length contarction, time dilation and relativity of simultaneity. But, really, this is a job for the Lorentz Transformation. Then the approach is clear:

- Identify the coordinates of the events in the platform frame.
- Transform those coordinates to the train frame.

Nothing to do with light, observations or any of that other extraneous stuff. It's a pure and simple coordinate transformation.

I guess I still have some work to do, but seriously, wrapping my head around this subject has proven very troublesome.

I hope I'm not asking for too much if you could perhaps show me how this coordinate transformation would take place (as per the homework guidelines, the solution is already apparent and I'm asking for an alternative explanation).

As an appropriate starting point, can we state the coordinates of the beacons A and B, relative to A, for example, such that A is at the origin, and B is 500 m away. Then we use the Lorentz transformation to change both coordinates of A and B. Am I on the right track?

 

[FeynmanFtw]

Does the cited answer also say, that it relates to question d) and does it continue to calculate
##\Delta t'## from ##\Delta x'=625m## and ##\Delta x=500m##?

Is there an explicite answer to question c) ?

The answer is not backed up by any explanation. It literally reads as I've written in the first post. Just the delta equation, then replaced with numbers, with the final numerical answer. It also says underneath "One may use ##L\cdot \gamma = L_0##".

That is literally...it.

 

[PeroK]

As an appropriate starting point, can we state the coordinates of the beacons A and B, relative to A, for example, such that A is at the origin, and B is 500 m away. Then we use the Lorentz transformation to change both coordinates of A and B. Am I on the right track?

Yes, the Lorentz Transformation is given in your OP.

$$\Delta x' = \gamma(\Delta x - v\Delta t) = \frac{5}{4}\cdot 500 = 625 \textnormal{ m}$$

In fact, they didn't even specify an origin. The origin is irrelevant to the distance between two events. To expand on that equation:

We have two simultaneous events with locations ##x## and ##x + \Delta x## in the platform frame (##\Delta x = 500m##). Simultaneous means ##\Delta t = 0##.

The above is a legitimate form of the Lorentz Transformation, as:
$$x'_1 = \gamma(x_1 - vt_1), \ x'_2 = \gamma(x_2 - vt_2)$$$$\Rightarrow \Delta x' = x'_2 - x'_1 = \gamma(x_2 - vt_2) - \gamma(x_1 - vt_1) = \gamma([x_2-x_1] - v[t_2-t_1])$$$$\Rightarrow \Delta x'= \gamma(\Delta x - v\Delta t)$$And, you can start to see my point that the location of the train, the origin, any observers and any light signals are irrelevant. It's all about the coordinates!

 

[Sagittarius A-Star]

The answer is not backed up by any explanation. It literally reads as I've written in the first post. Just the delta equation, then replaced with numbers, with the final numerical answer. It also says underneath "One may use ##L\cdot \gamma = L_0##".

That is literally...it.

From that cited partly answer, you can continue to answer question d) with the inverse LT:
##\Delta x = \gamma (\Delta x' + v \Delta t')##
##500m = \gamma (625m + v \Delta t')##
... and then solve for ## \Delta t'##.

 

[Orodruin]

1) SR is a theory of spacetime transformations between reference frames. It does not depend on the observations by any particular observer. The location of the train is of no consequence. The only relevant point is that the train and the platform have a relative speed of 0.6c in the direction aligned with the platform. In other words, the train could be nowhere near that particular station and the answer would be the same - as long at the train is moving in the given direction.

(My emphasis)
I’d rather say SR is a theory of spacetime geometry. The transformations between inertial frames are of course part of that in the same manner that rotations form part of Euclidean geometry, but I would say that the fundamental part is the geometry.

 

[PeroK]

(My emphasis)
I’d rather say SR is a theory of spacetime geometry. The transformations between inertial frames are of course part of that in the same manner that rotations form part of Euclidean geometry, but I would say that the fundamental part is the geometry.

That's the next insight, of course. The initial insight is that the events don't have to be beacons emitting light signals. The LT doesn't care whether the events are luminous beacons or geiger-counter clicks in a sealed box.

 

[Ibix]

I hope I'm not asking for too much if you could perhaps show me how this coordinate transformation would take place (as per the homework guidelines, the solution is already apparent and I'm asking for an alternative explanation).

The fundamental physics hiding under the examples is as follows.

Spacetime is the background on which all physics happens.

A reference frame is just a way of "drawing" a coordinate system on that background. It makes it easier to do physics because you can assign numbers to different locations and express physical quantities as a function of those numbers.

A different reference frame is just a different way of drawing coordinates. Just like you can draw different grids on a piece of paper, you can define different grids on spacetime.

On a piece of paper, if we agree the origin and units for our grid, the coordinates of a point using one syatem are related to the coordinates in another by a rotation. On spacetime, if we agree the origin and units for our grids, the coordinates in two grids are related by the Lorentz transforms.

Any object you see around you is actually 4d - the 3d object you see now and an extent in time. A sphere, for example, is actually a 4d analog of a cylinder - a spherical cross section and a long length in the time direction.

Imagine drawing a very long narrow rectangle on a piece of paper. Draw a grid on it. What is the x-extent of the rectangle according to the grid? Different grids will measure different widths because their x axis makes a different angle to the rectangle. Now remember that frames in relativity are just different grids on spacetime. The extent of an object in the x direction depends on the choice of grid - that's length contraction.

Imagine two points drawn on a piece of paper at the same y coordinate. Using a different grid, will they be at the same y' coordinate? Similarly, two events in spacetime that have the same t coordinate won't have the same t' coordinate. This is the relativity of simultaneity.

Two points with different y coordinates will have some y-separation. Using a different grid they will have a different y'-separation. Similarly, two events in spacetime with some t-separation will have a different t'-separation. This is time dilation.

Note that all effects like time dilation and length contraction are just consequences of different choices of the imaginary grid you chose to use.

Go and look up Minkowski diagrams, which are just maps of spacetime drawn on a piece of paper. You can draw multiple grids on them and begin to build intuition for how measurements made with the grids relate.