# Notation for time in a relativistic graph

1. Jun 23, 2013

### cmcraes

I was just curious as a possible confusion that might occur when graphing a v vs t Graph.
Because velocity is a function of time, but time is also dependant on velocity, so how would we denote this on a graph? Hope i got my point across, thanks!

2. Jun 23, 2013

### ghwellsjr

Normally, when we want to plot velocity as a function of time, we do it on a graph that has position (or location or distance) along one axis and time on the other. Then the slope of line is the velocity.

Does that help?

3. Jun 23, 2013

### cmcraes

No I understand the simple concept of graphing velocity as a function of time on a cartesian coordinet system. my question concerns how we graph something like this as v approaches c. Because the time would appear to slow down for the observer i was wondering how this changed perception in time would affect the subjective V vs T graph because technically t is both the independent variable and a dependent variable

4. Jun 23, 2013

### Simon Bridge

Everything in relativity is done by considering the observer.
Same with this - plot the graph normally, but make the observer explicit.
You have to say whose clock you are using for t and whose ruler you are using for distance.

Aside:
There is a common misconception that time slows down for you when you travel at high speed. This is not correct. You are always stationary in your own reference frame. You don't see any difference to the way your clocks behave. It's everyone else that is different and that's reasonable since they are they ones moving.

Relativity only comes into play when two observer's results are compared.

Last edited: Jun 23, 2013
5. Jun 23, 2013

### cmcraes

Oh yes! I must of had a brain fart, i completely for got about length contraction. Well in that case could someone show/direct me to a Comparison between an observers reference frame and a secondary observer at rest relative to the person traveling at a relativistic velocity? Thanks

6. Jun 23, 2013

### Staff: Mentor

Time does not slow down for any observer. The only thing that happens is that all observers figure that clocks moving relative to them run more slowly. But nothing odd happens to their own clocks.

So if I watch you zoom by in a spaceship at some relativistic velocity, I have no problem graphing your speed - I look at my wristwatch, see what time it is, see where you are at that time (which may take a bit of calculation because I have to allow for light travel time between you and my eyes), repeat again a split second later to measure dx/dt at that time, and plot another point on my v/t graph. Your clock is running slow relative to mine, but that doesn't matter -I don't need it for my v/t graph.

And as for you? As far as you are concerned, you are at rest so your v/t graph is really easy to draw - v is zero for all values of t. You also see me moving away from you, so you will figure that my clock is running slow.

7. Jun 23, 2013

### cmcraes

Okay thanks

8. Jun 23, 2013

### ghwellsjr

There are two different times involved. I was talking about the Coordinate Time (and the Coordinate Position). You're also talking about the Proper Time for the traveling observer which is dilated (takes longer) than the Coordinate Time. I show this on all my diagrams as dots (or tick marks) that are spaced farther apart the faster the observer is traveling. You can see lots of examples if you do a search on my name for the word "diagram".

9. Jun 23, 2013

### Simon Bridge

Length contraction is not very relevant to the question you asked. I think you need to be more careful about what you are talking about. Until you get used to relativity, you need to be real pedantic about who observes what where.

If Alice travels at relativistic v, and Bob is at rest with respect to her, then Bob is also travelling at speed v. There must be another observer, Claire, who is measuring this speed. As far as Alice and Bob are concerned, they are not moving and Claire is the one with the high speed.

i.e. you need to be more careful with the way you set up the example.

10. Jun 23, 2013

### cmcraes

Okay people I appreciate the help but I already understand the concepts of relativity. I was just wondering if there was a notation/coordnet system that would properly define The observed vs perceived time in relation to some velocity ifthat makes any sence

11. Jun 24, 2013

### ghwellsjr

Sure, I'll show you. First, I'll draw a spacetime diagram of an observer (in blue) at rest in his own reference frame. He's in the front of a 500-foot long spaceship with the rear of the spaceship shown in red:

Now we want to see what it looks like in a frame in which he and his spaceship are traveling at 0.6c where c is 1 foot per nsec and gamma equals 1.25. All we have to do is use the Lorentz Transformation process to convert the coordinates of the significant events in the above diagram into a frame which is moving at -0.6c with respect to the original frame and we get:

Note that the dots, marking Proper Time intervals of 100 nsecs, are farther apart than the Coordinate Time axis grid lines. This shows a Time Dilation factor of gamma = 1.25.

Also note that the length of the spaceship, as indicated by the distance between the red and blue lines as marked off along any horizontal grid line, is 400 feet. This shows a Length Contraction factor of the inverse of gamma = 1/1.25 = 0.8.

Finally note that the beginning events for the red and blue lines are not at the same Coordinate Time like they were in the first diagram. This shows Relativity of Simultaneity.

Please note that all of these effects are caused by transforming to a different coordinate frame. We could transform to some other frame, moving at a different speed and get totally different Time Dilation, Length Contraction and Relativity of Simultaneity and this has nothing to do with any particular observer.

I don't know what the difference between observed and perceived is, they both connote the same thing to me so I'll show how we determine that by putting an observer (black) at rest in this last diagram. I've drawn in thin blue lines to show how images of the blue observer's clock propagate toward the black observer:

Note that before the front of the spaceship (blue) gets to the black observer (negative Coordinate Times), he sees the clock on the spaceship ticking at twice the rate of his own clock. After the blue observer passes the black observer, the black observer sees the clock on the spaceship ticking at one-half the rate of his own clock. These effects show the Relativistic Doppler factor and does not depend on the reference frame.

If the black observer has measured or otherwise knows the speed of the spaceship relative to him (or his rest frame), he can measure how long it takes for the spaceship to pass him (6.67 nsecs) and calculate the length of the spaceship = d = vt = 6.67 nsec * 0.6 feet per nsec = 4 feet.

Is this the sort of thing you were hoping to see demonstrated?

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Last edited: Jun 24, 2013
12. Jun 24, 2013

### cmcraes

Thank you ghwellsjr! :) Those helped