Real life examples of simultaneity

[goodabouthood]

Reference frames change according to motion?

Also what does it mean by -.5c?

I am referring to the diagram I posted above.

 

[bobc2]

Reference frames change according to motion?

Also what does it mean by -.5c?

I am referring to the diagram I posted above.

 

[nitsuj]

No, it's not relative. As I said in post #47:


As far as the gravitational contribution to time dilation goes, yes. During the long orbital times of the two planets, there would be other variations in the observed differences in the clocks but these would average out so that every time the two planets are in the same relative position in the solar system, there will be a steadily increasing time on the Earth clock compared to the Jupiter clock.

Ah okay, thanks ghwellsjr.

I got to think the difference between the two through. Off that bat all I can think of is length contraction being the difference (such as no length contration caused by gravity, just relative motion).

 

[goodabouthood]

Is it correct to say this?

The space and time of an event is different for all reference frames.

Now what about this?

If the train happened to be still and the observer on the ground happened to be still. They both were just at their places with a velocity = 0. Would they both agree that the lightning bolts were simultaneous since they are both at rest? Would they be in the same frame of reference?

 

[ghwellsjr]

Well, what exactly defines a reference frame?

You get to define a reference frame any way you want (as long as you follow Einstein's convention).

Are there infinite number of reference frames?

There are an infinite number of reference frames possible but until we express a definition for one, it doesn't exist. It only exists in our minds not out there is space.

Why do different reference frames have different coordinates?

Part of the answer comes from our choice, the other part of the answer comes from the way nature works.

Also, are these infinite set of clocks in each reference frame always synchronous?

Yes.

Can you give me an example of the coordinates of the reference frame of the train and the coordinates of the reference frame of the ground?

OK, that might help you understand what we're talking about.

I'm going to use the same assumptions and parameters that I used in post #54:

So for simplicity's sake, let's say that the speed of light is 1 foot per nanosecond and let's say the train is traveling at 0.6c and it is 1000 feet long.

I'm also going to assume that the platform is 1000 feet long.

Now the first thing we need to do is define our directions. We could make the direction the train is traveling be along the +X, -X, +Y, -Y, +Z or -Z directions or anything in between but since we later on want to use the standard form to make it easier to use the Lorentz Transform we will make the train travel along the X axis and to conform to popular traditions, we will make the direction of the train be along the +X axis and we will assume that the +Y direction is towards us and the +Z direction is up.

Now we have to select an origin for our Reference Frame. This is where all the coordinates are zero. We could put it at the front end of the platform (the same end as the front of the train) or we could put it at the rear end of the platform or we could put it anywhere else but for simplicity, let's put it at the midpoint of the platform. This means that the front end of the platform has an X-coordinate of 500 feet and the rear end of the platform has an X-coordinate of -500 feet. In this scenario, since there is nothing happening in the Y or Z directions and because those components remain unchanged during the Lorentz Transform we will set those coordinate values to 0 and then ignore them in our expressions and calculations.

Next we have to define the two events of the lightning bolts. Since the problem states them as being simultaneous in the ground frame, we could give their time coordinates any value (in nanoseconds) but for simplicity's sake, we will give them the valuse zero.

Now we are ready to express our two events. Normally, I would use the nomenclature of [t,x,y,z] but since we have agreed to assign zeroes to y an z, I will use the shorthand nomenclature of [t,x]. So here are our two events for the lightning bolts (E1 is in front, E2 is behind:

E1=[0,+500]
E2=[0,-500]

The fact that they both have the same t coordinate means that they are simultaneous.

Now let's define the train Frame of Reference. In order to make things simple, we want to use the standard form so that we can easily use the Lorentz Transform and that means we want to use the same axis directions and units for distance and time and we want their origins to coincide. We will place the origin of the train at its midpoint.

Now we are ready to use the Lorentz Transform. We will use units such that the speed of light equals 1 which means that we are using nanoseconds for time and light nanoseconds (which equal one foot) for distances.

First we have to calculate gamma, γ, from this formula:
γ = 1/√(1-β²)
For β=0.6,
γ=1/√(1-0.6²)
γ=1/√(1-0.36)
γ=1/√(0.64)
γ=1/0.8
γ=1.25

Now the Lorentz Transform has two formulas, one for calculating the new t' coordinate and one for calculating the new x' coordinate from the old t and x coordinates. Here they are:
t'=γ(t-βx)
x'=γ(x-vt)

Since we are only interested in the time coordinate, we will do that calculation for each of our two events here:

t1'=1.25(0-0.6*500)
t1'=1.25(300)
t1'=375

t2'=1.25(0-0.6*-500)
t2'=1.25(-300)
t2'=-375

We can see right away that these two time coordinates are different so the events they go with are not simultaneous. In fact, as a sanity check, we can calculate the difference between them as 750 nanoseconds which is the same value we calculated in post #54 where we used BruceW's shortcut formulat and got 0.75 microseconds.

 

[ghwellsjr]

Reference frames change according to motion?

Also what does it mean by -.5c?

I am referring to the diagram I posted above.

The Lorentz Transform converts the coordinates of any event in one Frame of Reference to the correct coordinates of another Frame of Reference moving with respect to the first Frame of Reference. A value of -.5c simply means the second FoR is moving at .5c along in the -X direction instead of the +X direction.

Is it correct to say this?

The space and time of an event is different for all reference frames.

If you mean the values of the coordinates are different and if the reference frames are in relative motion, then almost always the values of the coordinates will be different, so in general, yes. But it's always the same event, no matter what FoR we use to describe it.

Now what about this?

If the train happened to be still and the observer on the ground happened to be still. They both were just at their places with a velocity = 0. Would they both agree that the lightning bolts were simultaneous since they are both at rest?

If the time coordinates for the two events in a given FoR are equal, then the events are simultaneous. Even if the train stopped somewhere before it got to the platform or somewhere after and so the observer on the train doesn't see the lightning flashes coincidently like the observer at the midpoint of the platform who does see the flashes coincidently, they are still simultaneous because simultaneity has nothing to do with what anybody actually sees. So they both agree and so do all of us watching this scenario that the lightning flashes are simultaneous in the given FoR.

Would they be in the same frame of reference?

Everybody and everything is in all Frames of Reference. If they are both at rest in one ground Frame of Reference, then they are both at rest in all ground Frames of Reference.

I think what may be confusing here is that a lot of people use a shorthand termonology and say the train observer's Frame of Reference or the platform observer's Frame of Reference when what they mean is a Frame of Reference in which the train observer is at rest or a Frame of Reference in which the platform observer is at rest. So if you are using that kind of shorthand termonology and you ask if, when the train is stopped, is the train observer's FoR the same as the platform's FoR?, then this could be true if they have the same origin and the same directions for their axes, etc., but what if this isn't true? Well, the two events will still be simultaneous in all the Frames of Reference that are at rest with each other but the time coordinates in the different frames can be different from each other even though they are the same within each frame.

 

[bobc2]

This thread is about the meaning of simultaneity. The purpose of the train example is to show that in one Frame of Reference (where the ground is at rest), the two lightning bolt events that occur at different locations nevertheless occur at the same time, not because nature demands it, but because the clocks that are located at those two events have been previously synchronized and the problem is stated such that they are simultaneous. That's one point to focus on.

But of course that's exactly why I posted the space-time diagram. The picture shows clearly the two frames of reference and the simultaneous spaces for each. It shows directly the simultaneous spaces that contain each of the key events. And all of those details are in one picture.

The second point to focus on is that the events defined according to one FoR need to be at different locations, otherwise two events at the same location and the same time are not two events but one event (they have the same four coordinates).

That is pretty obvious in the space-time diagram.

The third point to focus on is that in many other FoR's (not all, just some), those same two events, when transformed into a set of new coordinates will have a totally different set of four coordinates and the time components may be different in which case they are not simultaneous in that FoR.

That is also very obvious in the space-time diagram. I know from your past posts that you are one of the well-informed visitors to this forum and are fluent with space-time diagrams and are aware of this. Perhaps you didn't mean to imply that the space-time diagram does not make these things obvious--you probably intended to just add emphasis--and your emphasis is well placed for sure.

But the gal on the train and the guy on the platform are both living in the same world. There's only one world.

I think you should emphasize that there is only one 4-dimensional world. An infinite number of cross-section views are possible. Just like you could cut a length of 2x4 wood in an infinite number of angles and positions along the length of the beam.

The whole point of the different reference frames is that they represent different 3-D spaces. So, you have to be more definitive when you speak of two observers in the same 3-D space. Such a simple statement obscures essential concepts of the 4-dimensional Minkowski space.

The issue is how we describe the 3 spatial coordinates and the 1 time coordinate that define an event--what numbers do we (and they) use? The gal on the train can use a Frame of Reference where the ground is stationary (of which there are an infinite such FoR's) and in which the two events will be simultaneous and the guy on the platform can use a FoR in which the train is stationary (of which there are an infinite such FoR's) and in which the two events will not be simultaneous. But whichever FoR is used, it includes everyone and every thing.

That is true in a sense, but again it is not a careful statement of the situation and obscures a fundamental aspect of special relativity. This description obscures the relationship of a FoR with and actual 3-D space. Of course each FoR includes everyone and everything that is included in that instantaneous cross-section of the 4-D universe. But, the two different FoRs represent two different 3-D spaces. Once that is understood, all of the other SR effects become obvious.

People, other observers and objects don't own their own FoR to the exclusion of the other observers and objects. And it's the FoR that determines if two events are simultaneous not any observers or objects no matter what their states of motion are.

So it's really wrong when discussing the meaning of simultaneity in Special Relativity to link it to particular observers or especially to say that one person lives in a different 3-D world than another one and that's what determines the simultaneity of events, it should only be linked to particular FoR's.

That could give a misimpression of special relativity. The fact that the X1 axis and X4 axis are both rotated symetrically about the 45-degree world line of the photon gives each observer a unique 3-D cross-section for his particular view of that part of the 4-D worl--and his observation of the laws of physics, i.e., speed of light is constant for all observers and laws of physics are the same for all.

 

[goodabouthood]

I appreciate the answers you are giving me. I am still taking some time to digest what you have said.

I think I am having trouble seeing what a frame of reference really is. For example, right now I am sitting still in my chair looking at my computer. What would be my frame of reference?

Another question I have is do different reference frames depend on both motion and position or just motion?

I imagine position changes the frame of reference as well. If my friend was sitting still next to me we would still have different frames of reference even though we are both stationary.

I also know that there really is nothing that is still. It's all relative. Relative to my floor I am still but relative to the Sun I am moving.

I know some of these questions might be a bit obvious to some but I just need to ask them and hope they might help others as well.

Thanks.

 

[goodabouthood]

How about this?

My friend and I are sitting still on the ground and we see a moving train come by at uniform motion. A flash of lightning hits each end of the train. Now me and my friend will still have different frames of reference even though we are both not moving. Right?

I know we would have different spatial differences but would we still agree on the simultaneity of the lightning because we are both still relative to the ground?

What I mean to say is would our time differences change as well even though we are both still?

 

[bobc2]

How about this?

My friend and I are sitting still on the ground and we see a moving train come by at uniform motion. A flash of lightning hits each end of the train. Now me and my friend will still have different frames of reference even though we are both not moving. Right?

I know we would have different spatial differences but would we still agree on the simultaneity of the lightning because we are both still relative to the ground?

What I mean to say is would our time differences change as well even though we are both still?

Here is an example of two guys (black and red) sitting still. But they are in different positions, i.e., displaced from each other along their common X1 direction. So, sitting still, their X1 and X4 coordinates point in the same direction. They are both moving straight up into the 4th dimension along their respective X4 coordinates. If their clocks are synchronized to zero time (t = 0) at the origin of the rest system, then their clocks will both read one second after they have moved 186,000 miles into their future along their 4th coordinate.

Now, we have two events, E1 and E2, that occur simultaneously at the instant one second has lasped. These two events are simultaneous for both the black and red guys, but the events would not be simultaneous for some other guy who was moving at some speed relative to black and red.

Another event, E3, is shown, but that event is in the future of red and black, who have just arrived along their X4 axis at the one second mark. However, that event might have actually occurred for some other observer moving at a different speed relative to red and black.

Clocks are just sign posts along the X4 axis that read out the lapsed time from some start point along X4. You could put time sign posts along the interstate that read lapsed time from your point of departure (but it would assume you were traveling at some fixed speed). Everyone always travels at the speed of light along their 4th dimension, and that makes the clock time sign posts work just fine (you can always compute the distance traveled along X4 by X4 = ct.

 

[ghwellsjr]

I appreciate the answers you are giving me. I am still taking some time to digest what you have said.

I think I am having trouble seeing what a frame of reference really is.

A Frame of Reference is nothing more than a coordinate system with time included but with the remote clocks synchronize according to Einstein's convention.

For example, right now I am sitting still in my chair looking at my computer. What would be my frame of reference?

Well how about that? I'm sitting still in my chair looking at my computer, too. Since we are both sitting still at different places on the surface of the Earth (I presume you're on the Earth and not up in the space station) we can define a common rest frame that includes both of us. There's already a coordinate system for the surface of the Earth that we could use that includes lattitude and longitude and latitude. I can look on my GPS receiver and see what my spatial coordinates are and you could do the same thing. We also have standard time here on the Earth (GMT) and we could use that for our time coordinate. My GPS receiver will tell me the local time but it is easy enough to calculate GMT. Since GPS has already done the work of synchronizing time for both of us, we don't have to worry about that. Note that if we define or common Frame of Reference this way, it will be very difficult to use the Lorentz Transform because we need both the time and the spatial coordinates to have a common origin. But my point is to show you the arbitrariness of establishing a Frame of Reference and to show that it doesn't have to be linked to either one of us.

On the other hand, you could use a different coordinate system defined by the table your computer is sitting on. You could say that the origin is the top front left corner of your table and +X extends to the right of the table, +Y extends to the rear of the table, and +Z extends upwards. Then you could start a stop watch on your cell phone and use the elapsed seconds as your time coordinate. Then you would probably say that the event (for the middle of your head) that describes when you read this (in [t,x,y,z] format with t in seconds and distances in feet) is something like [15,2,-1,1.5]. But note that we are not really very precise because you are using your cell phone's stop watch which probably has a resolution of a tenth of a second and light can travel a hundred million feet in that amount of time.

However, if you had some very precise electronic equipment and you wanted to set up an experiment involving light, you could actually be in a situation where details could matter. So let's say that on the left hand edge of your table, you have a very fast light strobe that can emit a very short (less than a tenth of a nanosecond) flash of light aimed at the right hand edge of your five-foot wide table where you have a mirror that reflects the light back to your strobe and right next to your strobe you have a light detector. You have wired up an electronic timer with a resolution of a thousandth of a nano second that starts when the strobe emits a flash of light and stops when the detector senses the reflected image. What do you think the timer will read when you do this experiment. Well, since the speed of light (for our purposes in this exercise) is one foot per nanosecond and the table is five feet wide and the light has to travel both directions, the timer will read 10.000 nanoseconds, correct?

But what if you wanted to measure how long it took the light to go from the left hand edge of your table to the right hand edge. Well that seems easy enough, you just put your detector on the right hand edge of your table and run a cable back to your timer to stop it when the light is detected, right? So you run your experiment and now what do you think you will get? Well if you said 5.000 nanoseconds, you'd be wrong because even if it did take 5 nanoseconds for the light to go from your strobe to the detector, it would take another 5 nanoseconds for the signal traveling in the cable to get from the detector to your timer.

So now you decide to put the timer next to the detector on the right side of the table so that you can use a very short cable to stop the timer but now you need a long cable going from the strobe to start the timer. Well you do your experiment again and what happens is that when the strobe flashes, the start signal travels down the cable right along side the flash of light so they both arrive at the other side of the table at the same time. The signal starts the timer and immediately the detector sees the flash of light and stops the timer so the reading is 0.000 nanoseconds.

So if you use your measurements to calculate the one-way speed of light, in the first case you will say that it appears the be 5 feet divided by 10 nanoseconds or 0.5 feet per nanoseconds and in the second case it appears to be 5 feet divided by 0 nanoseconds or infinite. Now these are actually the range of values that the one-way speed of light could be and there is no way to determine what it actually is.

So this is where Einstein's second postulate comes in. He simply says that whatever time it takes for the light to make the roundtrip, it takes exactly half that amount of time to make the one-way trip. Einstein says that unless you do something like this, you really have no basis for establishing the meaning of time at the right hand edge of your table just because you have a timer at the left hand edge.

So now what you can do, instead of having a stop watch to measure the time interval, you can actually use a pair of clocks with no wires in between and you synchronize them so that that when you make the one-way speed of light measurement, you will get 1 foot per nanosecond. This, of course, means that you have to go actually go through the process of synchronizing them.

One way to do this is to have a memory on each clock so that when it receives an external signal, it stores the current time. You do this at the strobe end and at the detector end. You do the experiment. Let's say the clocks have not yet been synchronized and the difference in the times on the two clocks is seven nanoseconds instead of five. Now you can set the detector clock back by two nanoseconds. The next time you do the experiment, you will get a difference in the clock readings of 5 nanoseconds. Now you can repeat the experiment with more clocks in other locations until you have a network of synchronized clocks at known locations. This, then, becomes your Frame of Reference.

Another question I have is do different reference frames depend on both motion and position or just motion?

Yes, but not just motion and postition but also directional orientation, although if two reference frames differ by only position or directional orientation (but not motion) then any events that are simultaneous in one will also be simultaneous in the others.

I imagine position changes the frame of reference as well. If my friend was sitting still next to me we would still have different frames of reference even though we are both stationary.

But like I said earlier, your friend sitting next to you is in whatever frame of reference you define and you are in what ever frame of reference he cares to define but you are both at rest in the frames you each define, then you both are at rest in both frames.

I also know that there really is nothing that is still. It's all relative. Relative to my floor I am still but relative to the Sun I am moving.

And relative to your floor, the Sun is moving. All states of motion are relative to something, whether that something be another object, a defined Reference Frame, or even a previous state of an object that has accelerated.

I know some of these questions might be a bit obvious to some but I just need to ask them and hope they might help others as well.

Thanks.

 

[goodabouthood]

The graphs are helping.

I am taking it that motion is what will really make people disagree on events and not spatial differences.

For instance if the two people sitting still always stay still they will always agree on the simultaneity of events. Am I correct in saying this?

Also considering time slows down as you speed up wouldn't the event at E3 be seen after for a moving observer than for the two people staying still?

 

[ghwellsjr]

How about this?

My friend and I are sitting still on the ground and we see a moving train come by at uniform motion. A flash of lightning hits each end of the train. Now me and my friend will still have different frames of reference even though we are both not moving. Right?

You don't have to have different Frames of Reference. You can both be at rest in a single frame of reference and differ by your location coordinates which never change. Let's say that you are sitting near where one flash of lightning strikes the front of the train (with an X coordinate of 500 feet) and your friend is sitting near the rear of the train (with an X coordinate of -500 feet) where the second flash of lightning strikes. Now you will each see the lightning that struck near you first and then later see the other one. So you will see the flashes in a different order. But this has no bearing on whether the two flashes were simultaneous in your chosen common Frame of Reference. What matters is what the pre-synchronized clocks read at the locations of the lightning strikes. If they read the same time, then the strikes were simultaneous, otherwise they were not simultaneous.

I know we would have different spatial differences but would we still agree on the simultaneity of the lightning because we are both still relative to the ground?

What I mean to say is would our time differences change as well even though we are both still?

If you had a common rest reference frame, then if your friend synchronized all the clocks to the one closest to him, then the one closest to you will also be synchronized to all the other clocks which include his.

If you each chose a different rest reference frame, there could be two sets of synchronized clocks, one set that he synchronized (let's say they are blue) and one set that you synchronized (let's say they are red) but at every location the blue and red clocks would differ by the same amount. It would be like having clocks that display two different time zones.

So if one of you determined that the lightning strikes were simultaneous in your FoR, the other one will also determine that the lightning strikes were simultaneous in his FoR.

But I want to emphasize once more, your two different rest frames could differ only in that your X origins were different or only in that your time coordinate is different and it would not make any difference for simultaneity.

 

[ghwellsjr]

The graphs are helping.

I am taking it that motion is what will really make people disagree on events and not spatial differences.

It's motion of Frames of Reference that matter, not of any people who may or may not be at rest or in motion in any particular frame. Remember, everyone and everything is in every Frame of Reference. Events are defined by Frames of Reference not by people observing things differently.

For instance if the two people sitting still always stay still they will always agree on the simultaneity of events. Am I correct in saying this?

Only in a Frame of Reference in which they are at rest. In other frames, the same events could happen at different times.

Also considering time slows down as you speed up wouldn't the event at E3 be seen after for a moving observer than for the two people staying still?

You have to quit thinking in terms of the events being seen by people remotedly located from the events. It has nothing to do with how people see things, it has only to do with the times on the synchronized clocks colocated with the events.

 

[goodabouthood]

I guess I just keep thinking about people because I am trying to visualize what simultaneity would be like for different people.

But I am realizing that no events are absolute. You can only say say when and where something happens according to its reference frame.

 

[bobc2]

The graphs are helping.

I am taking it that motion is what will really make people disagree on events and not spatial differences.

For instance if the two people sitting still always stay still they will always agree on the simultaneity of events. Am I correct in saying this?

Also considering time slows down as you speed up wouldn't the event at E3 be seen after for a moving observer than for the two people staying still?

Here are a few sketches to help with graphically visualizing motion in 4-dimensional space. Before (earlier post) we had the red guy sitting still with the black guy. Now, we put the red guy in motion. He is moving along the black guy's X1 axis. In the upper left corner sketch you can see that the farther red advances in time along the red time axis--the farther red advances along the black X1 axis. He advances to a point, XA, by the time black's clock is time, t. The speed calculation is shown below the sketch.

The next sketch to the right is similar, but the red guy is now moving faster, advancing along black's X1 direction to the position, XB, at the black time, t.

But, now comes the space-time diagrams corresponding to the first two sketches. The really mysterious and facinating thing about special relativity is that when red's X4 axis rotates relative to the black rest system, then red's X1 axis rotates also--rotating symmetrically with respect to the X4 rotation. The new rotated red X1 axis represents the cross-section of the 4-dimensional universe that red is now "living" in (we have supressed the red X2 and X3 coordinates for ease of diagramming, but they don't matter so much because they are in the same direction as the black X3 and X4 and do not add any insight).

Hopefully, this sets the stage for the main space-diagram below that shows the different simultaneous spaces at different times for black and red. Obviously, now the red guy no longer "lives" in the 3-dimensional simultaneous spaces depicted for the black guy. They each have different instaneous cross-section views of the 4-dimensional universe.

Recall from the earlier post that events E1 and E2 were simultaneous for the black guy at t = 1 second. E1 and E2 are certainly not simultaneous for the red guy. Rather, events E1 and E3 are now simultaneous for the red guy, but not for the black guy. The events E1 and E3 occur simultaneously for the red guy at his time, tA. At red's time tB (which is not 1 sec), his world includes the black guy at black's t = 1 sec. That means that red would "see" black with black's clock reading 1 sec. However, black is not "living" in red's 3-D cross-section of the universe at his 1 sec mark, so he certainly does not "see" red with red's clock reading 1 sec.

 

[goodabouthood]

Thanks.

But why do the red coordinates shift in the particular way they do?

Also what is the purpose of the photon?

Also do you mean that time at tB is one second for red?

Is it possible you could draw out the time and space numbers for both sets of coordinates? Thanks.

 

[BruceW]

I am taking it that motion is what will really make people disagree on events and not spatial differences.

Good question. The full symmetry of special relativity is the Poincare symmetry, which includes rotations, translations and boosts.
For beginners, you should only be concentrating on boosts. Don't worry about rotation or translation of the reference frame. (Anyway, the interesting stuff is all in the boost).
When you 'boost', the origin is the same, and the old axes are collinear with the new axes. So the only thing you are changing is the speed of the frame of reference.

 

[bobc2]

Thanks.

But why do the red coordinates shift in the particular way they do?

We started with the upper two sketches just showing how the rotated red X4 axis rotated in accordance with red's speed with respect to the black coordinates. I assume you get that part (if not let us know and someone will clarify that further I'm sure). So, the big question is why does the X1 axis rotate? Welcome to special relativity. You've just hit on the most mysterious aspect of nature (the QM double slit experiment may challenge that claim).

And it turns out that the laws of nature are the same in all worlds, provided that the coordinates representing those worlds have their X1 axis rotated symmetrically as we've been showing. It's as though the fabric of 4-dimensional space just took this very special unique form. Any other jumble of the 4-dimensional threads strung out through the 4th dimension would make no sense to observers (no recognizable comprehensible pattern formation). Only arrangements of 4-dimensional threads corresponding to our observable physics were put in place--only those that produced our physics for any observer whose X4 and X1 axes were rotated appropriately. Perhaps observers use the rotated coordinates just because its the only selection of coordinates that produce comprehensible observations, i.e., our physics.

But in any case, you have put your finger on one of the most puzzling aspects of nature. Let us know if you find an answer.

Also what is the purpose of the photon?

I just added in the 4-dimensional world line of a photon to illustrate how special it is and to suggest that in the limit of increasing speeds the X4 axis and X1 axis would converge on each other, i.e., become colinear. This would be a very strange coordinate system in which the X1 axis and the X4 axis were the same. It hints at an upper limit for the speed of light. But it also implies that the speed of light must be the same for all coordinate systems. Below is a sequence of ever increasing speeds with the X1 and X4 axes rotating closer and closer to each other with each increase in speed:

Also do you mean that time at tB is one second for red?

No.

Is it possible you could draw out the time and space numbers for both sets of coordinates? Thanks.

There are special hyperbolic calibration curves that are required to label the times along red's X4 axis and distances along red's X1 axis. I could put in those curves, but it makes the sketch very cluttered. However, there is a better way to do the space-time diagram to make obvious the numerical relations between red and black coordinates. Maybe I can work that up.

 

[goodabouthood]

Thanks.

I am still wondering why the other coordinate system takes on the position it does. I see as the coordinate system approaches the speed of light it closes in on each other I just don't understand why.

It also would help to see the numbers for both coordinate systems.

Thanks a lot for the help.

 

[goodabouthood]

http://en.wikipedia.org/wiki/File:Relativity_of_Simultaneity.svg

Take that image for example. It says the red reference frame is moving at .28c relative to green reference frame. Now why goes the graph take on that shape for the red reference frame?

 

[BruceW]

These graphs show us how the position and time of some event are interchangeable depending on the reference frame used.

In the original (green) reference frame, an event might be given as $(x,t)$ and in the new (red) reference frame, the same event might be denoted by $(x',t')$. So what we're saying is that the spatial coordinate and the time for an event are not absolute. Each can change depending on the speed of the reference frame used.

But one thing that is absolute is the quantity $x^2 - c^2t^2$. In other words, $x^2 - c^2t^2 = x'^2 - c^2t'^2$. (This is true because we are not changing the origin of the reference frame). The shape that the red graph takes is such that this quantity is the same when calculated by either graph.

 

[goodabouthood]

These graphs show us how the position and time of some event are interchangeable depending on the reference frame used.

In the original (green) reference frame, an event might be given as $(x,t)$ and in the new (red) reference frame, the same event might be denoted by $(x',t')$. So what we're saying is that the spatial coordinate and the time for an event are not absolute. Each can change depending on the speed of the reference frame used.

But one thing that is absolute is the quantity $x^2 - c^2t^2$. In other words, $x^2 - c^2t^2 = x'^2 - c^2t'^2$. (This is true because we are not changing the origin of the reference frame). The shape that the red graph takes is such that this quantity is the same when calculated by either graph.

Well now I am understanding what it means for events to be simultaneous but I am still having a bit of trouble understanding exactly why this happens.

I am trying to visualize in my head what each frame of reference would look like for the train example.

I'm finding lately that sometimes I feel I understand it and other times I am losing it. I do understand the concept that events happen at different times for different reference frames I just have a hard time understanding why this is.

I know it has to do with the speed of light and with the movement of one reference train relative to another I just haven't pieced it all together in my mind.

 

[goodabouthood]

I am pretty much saying I would like to see a visual reference of both the train and observers coordinates.

 

[bobc2]

I am pretty much saying I would like to see a visual reference of both the train and observers coordinates.

Here are the coordinates again for both observers. The black coordinates are the rest frame of the guy on the platform and the blue coordinates are the frame of reference for the guy sitting in the center of the train passenger car. Circles mark different events of interest.

The sketch in the lower right corner shows just black and blue coordinates without the clutter of the other details. It makes it clear how the blue coordinate grid lines are skewed.

 

[bobc2]

Thanks.

I am still wondering why the other coordinate system takes on the position it does. I see as the coordinate system approaches the speed of light it closes in on each other I just don't understand why.

Historically, it was just discovered (theoretically with Maxwell's equations and experimentally with Michelson-Morely experiments) that the speed of light was the same for all observers, no matter what their velocity. It turned out that the only way that could be possible was if the coodinates for different observers moving at different velocities relative to some reference system are rotated in the manner we've been showing in the space-time diagrams. And along with this was the discovery that the laws of physics are the same in all of these different coordinate systems. All of this is of course what special relativity is about.

So this peculiar way nature has of orienting the different 3-D cross-section views of the universe for different observers is something that physics has discovered. So far, no one has come up with a reason for this. Some may maintain that the coordinates are that way in order to make the speed of light constant for everyone and so that the laws of physics would be the same for everyone. That sounds kind of like mother nature said, "I think I'll make a 4-dimensional universe populated by 4-dimensional objects in a way that guarantees a constant speed of light and a set of uniform laws of physics." No one really knows why the universe is the way it is.

It also would help to see the numbers for both coordinate systems.

I will show you those numbers in my next post. But, first you should try to understand the sketches below. I am using high school level algebra to derive the Lorentz transformation equation that is used to compute time dilation.

It is very useful to use what we call a symmetric space-time diagram. You have two guys (red and blue) flying off in their rockets at relativistic speeds in opposite directions. They are moving with the same speeds, one to the left and one to the right with reference to the usual black rest coordinates. By having them going at the same speeds, you can easily compare the distances and times along each of their respective axes (to compare numerical times and distances with black you must use hyperbolic calibration curves--which we will do in a later post).

And by the way, you can always turn a problem for two observers moving with respect to each other into a symmetric diagram of this type. Just add in a rest system whose X4 axis bisects the angle between the two moving observers and display the space-time diagram as shown here.

Note below that when blue is at station 9 (you could label it as either a time or a distance along X4) he is in the 3-D cross-section of the 4-D universe that includes the red guy at station 8. And when the red guy is at station 9, the blue guy in his (red's) simultaneous 3-D space at station 8. (How can the red guy and blue guy each be at two different places at the same time?)

 

[goodabouthood]

I really appreciate the diagrams you are making. I can now see that the act of when the lighting hits and when the observer sees it are two different events in the frame of reference. Am I correct in saying that?

Also in regards to the last diagram you posted, is there anyway you could show me an example with actual numbers filled in for the variables in the equations? I think that would help me a lot.

Thanks.

 

[bobc2]

I really appreciate the diagrams you are making. I can now see that the act of when the lighting hits and when the observer sees it are two different events in the frame of reference. Am I correct in saying that?

You certainly are.

Also in regards to the last diagram you posted, is there anyway you could show me an example with actual numbers filled in for the variables in the equations? I think that would help me a lot.

You want numbers for the train example?

 

[BruceW]

I really appreciate the diagrams you are making. I can now see that the act of when the lighting hits and when the observer sees it are two different events in the frame of reference. Am I correct in saying that?
...
I am trying to visualize in my head what each frame of reference would look like for the train example.

Yes! The lightning hitting ground and observer seeing the light from that strike are two different events.

For the train example, the only difference between the frames of reference is the relative speed of the two frames of reference. So the origin and coordinate axes are the same in both cases, but the frames have a relative motion. One frame measures the two lightning strikes to have happened at the same time in different places, so the other frame must measure the strikes to have happened at different times.

This may seem weird because the origin and axes of the two reference frames are identical, but in special relativity, the relative speed of the reference frames also decides where/when events happen.

 

[bobc2]

Yes! The lightning hitting ground and observer seeing the light from that strike are two different events.

For the train example, the only difference between the frames of reference is the relative speed of the two frames of reference. So the origin and coordinate axes are the same in both cases, but the frames have a relative motion. One frame measures the two lightning strikes to have happened at the same time in different places, so the other frame must measure the strikes to have happened at different times.

This may seem weird because the origin and axes of the two reference frames are identical, but in special relativity, the relative speed of the reference frames also decides where/when events happen.

Nice observations, BruceW. Thanks.

 

[goodabouthood]

It would be nice to see what the equations you posted in your last diagram look like with actual numbers.

 

[goodabouthood]

I have a couple more questions as well.

What is meant when you say everyone travels at the speed of light on the time dimension?

Also, let us go back to the train example. I understand that the events of the lightning bolts take place at different times for each frame of reference but I still can't exactly understand why this happens.

Let say the observer on the platform sees them simultaneously at t=0. Now let say that the train passing is going at .5c. Would that mean the lightning bolt will hit a half second latter in the train's frame of reference?

The thing that is bothering me in my head is that I imagine the train situation and I imagine the observer on the platform and the observer in the train passing each other when the bolts hit simultaneously for the observer on the platform. What I don't understand is why they can't hit simultaneously for the train FoR. I know it's because the train of reference is in motion I just don't understand why it happens.

Thanks.

 

[LaurieAG]

I can now see that the act of when the lighting hits and when the observer sees it are two different events in the frame of reference. Am I correct in saying that?

I was on the phone to a friend during a thunderstorm and there was a huge flash of lightning. We both said wow at the same time and, as I was only 2 miles away from the strike and my friend was 7 miles away, we both heard the bang at my location after 2 seconds or so and then we both heard the same bang at his location after another 5 seconds. As we were both on landlines we decided to continue our conversation after the storm had passed.

I went outside approximately 5 minutes later, the air was still and thick with humidity, and took several photos from different angles of what looked like smoke coming from the direction of the strike. The smoke was actually coming from a chimney but the photos revealed a dark straight line leading to the tip of the chimney in both photos.

 

[BruceW]

What is meant when you say everyone travels at the speed of light on the time dimension?

The velocity of an object through spacetime is called its four-velocity. The equivalent magnitude of this vector is equal to the speed of light for all objects with mass.
If we take a reference frame fixed to the earth, then all humans are moving mostly in one direction through spacetime. If we call this direction the 'time dimension', then clearly, all humans are moving at nearly the speed of light through the 'time dimension'.

Let say the observer on the platform sees them simultaneously at t=0. Now let say that the train passing is going at .5c. Would that mean the lightning bolt will hit a half second latter in the train's frame of reference?

If the train is going at .5c, then the time difference between the events according to the train's FoR is:
$$\frac{\sqrt{3}}{3c} \Delta x$$
Where $\Delta x$ is the distance between the two strikes according to the platform's FoR. So clearly, it also depends on how far apart the two events were.

The thing that is bothering me in my head is that I imagine the train situation and I imagine the observer on the platform and the observer in the train passing each other when the bolts hit simultaneously for the observer on the platform. What I don't understand is why they can't hit simultaneously for the train FoR. I know it's because the train of reference is in motion I just don't understand why it happens.

From the platform's FoR, for the beams of light from each strike to reach the train at the same time, one strike would have to happen earlier, so its light could 'catch up' with the train. So if the events were simultaneous for the train's FoR, they can't be simultaneous for the platform's FoR and vice versa.

To be honest, making sense of it is the tricky part. The easy part is simply accepting that the laws of special relativity are better than Newton's laws, and then using special relativity to make predictions instead of using Newton's mechanics.

 

[bobc2]

It would be nice to see what the equations you posted in your last diagram look like with actual numbers.

I started to post another sketch with the numbers. But then I remembered that ghwellsjr has already done a nice job of presenting calculations in his posts #54 and #75. His post #75 is a particularly detailed discussion using numbers he calculated directly from the Lorentz transformations. You'll find his post #75 example on page 5. (good job, ghwellsjr)