Analyzing Spacetime Diagrams: Finding Coordinates and Checking Invariance

In summary, the two rockets send a light signal to x=0. The coordinates in the S frame (3.5, 2.75) are correct, but the coordinates in the S' frame (2.8, 1.1) are not correct. The invariant rule does not hold.
  • #1
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Homework Statement



Two rockets are sent off at t=0, one from x=0 and the other at x=4. The rocket leaving from x=0 is moving at .8c and the rocket leaving x=4 is moving at .2c. When the paths of the two rockets meet, they send a light signal to x=0. Read off the coordinates in the S frame and in the S' frame and check to see that the space and time differences between events 3 and 4 satisfy the invariant rule. Event 3 is the light signal being sent out and event 4 is the light signal arriving at x=0.

The S' frame is moving at .6c.

Homework Equations



x' = [tex]\gamma[/tex](x-vt)

t' = [tex]\gamma[/tex](t - vx/c^2)

invariant rule: (t4 - t3)^2 - (x4 - x3)^2 = (t'4 - t'3)^2 - (x'4 - x'3)^2

The Attempt at a Solution



So after drawing all world lines, I came up with the coordinates (3.5, 2.75) for event 3 and (0, 6.3) for event 4 in the S frame by looking at the graph. I am confident in these coordinates.

In the S' frame, I came up with (2.8, 1.1) for event 3 and (-6, 9.8) for event 4 in the S' frame. This is where I think there may be a mistake. These are just based off reading the graph, so they are approximate.

Now when I check to see if it satisfies the invariant rule,

(6.3-2.75)^2 - (0-3.5)^2 = (9.8-1.1)^2 - (-6 - 2.8)^2

.3525 = -1.75

Clearly this is not correct. I understand there will be some error since I am just eyeballing the coordinates from the graph, but this seems way off. Does anybody see where I went wrong?
 
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  • #2
What units are you using? In what direction do the spaceships move? Are the coordinates you're giving (x,t) or (t,x)?
 
  • #3
Each of the spaceships are moving towards each other. So the ship that launches from x=0 is moving towards x=4 and vice versa. I am giving the coordinates as (x,t).
 
  • #4
How can the spaceship travel from x=0 to x=3.5 when t goes from 0 to 2.75? Doesn't that mean the ship is moving faster than the speed of light? (I assume you're using units where c=1.)
 
  • #5
Yes, you're right. I had drawn my line incorrectly. Instead of drawing it as .8c I drew it as 5/4 c. I've got it now. Thanks.
 

1. What is a spacetime diagram?

A spacetime diagram is a visual representation of the relationship between time and space in a particular event or scenario. It combines elements of both space and time to show how objects move and interact in a specific frame of reference.

2. How do you read a spacetime diagram?

To read a spacetime diagram, you first need to understand the axes. The horizontal axis represents space, typically measured in meters, while the vertical axis represents time, usually measured in seconds. Objects are represented by lines on the diagram, with their position changing over time as they move. The slope of the line indicates the object's velocity, with steeper lines representing faster movement.

3. What is the significance of the speed of light on a spacetime diagram?

The speed of light, c, is a constant in the universe and has a special significance on a spacetime diagram. It is represented by a diagonal line with a slope of 1, dividing the diagram into two regions: the future and the past. Objects and information can only travel within the future region, as it is impossible to travel faster than the speed of light. This is a fundamental principle in the theory of relativity.

4. How do spacetime diagrams relate to Einstein's theory of relativity?

Einstein's theory of relativity is based on the idea that space and time are interconnected and relative to the observer. Spacetime diagrams visually represent this concept by showing how objects and events can be viewed from different frames of reference. They also illustrate the effects of time dilation and length contraction, which are key principles in Einstein's theory.

5. What can we learn from studying spacetime diagrams?

Studying spacetime diagrams can help us understand the fundamental principles of physics, such as the relationship between space and time, the speed of light, and the effects of relativity. They are also used in various fields of science, including astronomy, cosmology, and quantum mechanics, to model and predict the behavior of objects and events in the universe.

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