Spacetime Diagrams Lab Exercise The spacetime diagram (also known as a Minkowski diagram) is a tool that can used to graphically describe complex problems in special relativity. In many cases, with a properly drawn spacetime diagram you can estimate an answer without working any complex problems. 1. Spacetime Diagrams and Events: A spacetime diagram is a tool that can show how physical events relate to one another. Making measurements of time and distance in relativity can be very complex so we have to use clear definitions. An event is is a physical occurrence that happens at a definite place in space, and a definite instant in time. For our purposes, we will restrict ourselves to 1D motion, so an event will be defined by a pair of coordinate, the time, t, and position, x. In a spacetime diagram we plot the time along the vertical axis, and the position along the horizontal axis. This will be different the motion diagrams you used in your first physics courses where the axes are usually swapped. In this orientation, we can represent time as moving upward to show the sequence of events. Open the spacetime diagram program. At the bottom there is a slider called beta, slide this all the way to zero for the first exercise. On the right, there is are checkboxes for a set of events from Event A to Event F. Click the checkbox to activate one of the events. You can change the coordinates of the event by entering the values into the text boxes under the Home Frame heading (ignore the Other Frame for the moment.) Or you can click and drag the event to move it to a new set of coordinate values. When an event is selected, the program will show the coordinate traces illustrating how the coordinate values are determined from the axes. A note about units: the time and position axes are scaled with the same units, in this case time. You can think of the x-position values as representing the distance in light-seconds, the distance that light can travel in one second. Specific units have been left out so that the diagram can represent light-seconds, light-minutes, light-years, or whatever scale is appropriate to a given problem. 2. World-lines: We can use a spacetime diagram to show the motion of object. Consider an object moving at the speed of light, it moves 1 light-second (or ls) for each 1 second of time. Note that the diagram has a yellow dotted line drawn with a slope of 1 that represents the speed of light (this is why we use light-seconds and seconds instead of meters and seconds on a spacetime diagram.) The speed of light is particularly important in special relativity, so it is convenient to display it on the spacetime diagram. a) Now consider an object moving ½ the speed of light, or 0.5c. How many light-seconds will it move each second of time? b) Now think of the object having a blinking light on it, that blinks every second (as measured by a stationary observer, don't make it too complicated yet.) If the object starts at the origin at t = 0, and is moving to the right, where will it be at 1s? c) In the spacetime diagram program, place an event for each light blink as the object moves in the +x direction. Think of a line connecting each of the evens. What is the slope of this line in the spacetime diagram? d) If the object was moving at 0.8 c, what would be the slope of the line? e) A world-line is a line showing the position of the object as a function of time. The velocity of the object is 1/slope of the world-line on the spacetime diagram. If the object is stationary at x = 2 ls, what will its world-line look like?
3. Spacetime Interval and Non-Euclidean nature of spacetime. In Euclidean geometry (flat, parallel lines always stay parallel, angles of a triangle add up to 180 ) we define the distance d between two points as d 2 = x 2 y 2, where x is the difference between the x coordinates of the two points, and y is the differences between the y coordinates of the two points. The distance is invariant under coordinate transformations, i.e. any other coordinate system will give the two points different values for the x and y coordinates of the two points, but the distance will always come out the same. a) On diagram 1 below, sketch the set of all points for which d =2. What shape do these points make? b) On the diagram 2 below, there is a rotated coordinate system. Place a point at x =1, y = 3 in the unrotated coordinate system. What are the coordinates of the point in the rotated (x', y') coordinate system? c) Calculate the distance between the origin and point in each reference frame. You should get the same result for each. Diagram 1 Diagram 2 Spacetime, is non-euclidean and we must define a new invariant quantity for measuring the distance between two points, the spacetime interval: s 2 = t 2 x 2. Note that we are using time units for both position and time (think of Δ x = 1s as 1 light-second, the distance that light travels in one second.) d) The spacetime diagram program has a tool for calculating the spacetime interval between two events. Click the Measure Δ s button. Then click and drag inside the diagram. A line with two arrows (and coordinate traces on the endpoints) will show how the spacetime interval is calculated between two points. The text fields will show the Δ t,δ x, Δ s 2, and Δ s values for the two points chosen. The spacetime interval calculator should snap to the origin, or any events that are already drawn in the diagram to make it easier.
e) One the spacetime diagram place the first point of the Δ s calculator on the origin and find the points for which Δ s=2s. Move events to mark those points. Try to cover a fairly large range of x-values, perhaps every 1 light-second along x. Take a snapshot of the screen and put it in a document. f) What shape does this make? 4. World-lines and the t' time axis : Since an object moving at constant velocity is in an inertial frame, we can say that the world-line of each object is the position of the origin for a coordinate system being carried by each moving object. The origin marks where x ' = 0 in the other coordinate system, and therefore the world-line becomes the t' axis for the moving object. Along the t' axis then, x ' = 0, and therefore t ' = s as measured from the origin along the t' axis. This gives us the scaling for the t' axis. a) Keep the curve from the previous exercise for s=2s. Adjust the slider for beta below the diagram. Increasing beta, increases the relative velocity of an object moving along the t' axis. Where this curve intersects the t' axis for each object shows us where t ' =2s. Note that as we adjust the value of beta, the tic mark for t '=2s always coincides with the curve of events drawn in the previous exercise. b) What happens to the scaling of the t' axis as an object's velocity increases? c) What happens as the velocity approaches the speed of light? 5. Two-observer Spacetime Diagrams: We can use a diagram with two coordinate systems setup to describe a scenario from the perspective of two observers in inertial reference frames. The first step is to create the coordinate system for the observer moving to the right with respect to the first. By convention, the reference frame of an observer moving to the right, the Other Frame. The Home Frame is the one that measures the observer moving to the right (which would be moving to the left from the perspective of an observer moving in the Other Frame) a) Set the slider to = 0.6c. We can use an alternate to the hyperbola from parts 1and 2 to calibrate the scale for the t' axis. Use the relationship Δ t = γ Δ t ', where = 1 1 2. b) Calculate for our example and check that it agrees with the value shown next to the slider: c) Now place an event on the world-line of the object at t '=1s tic-mark. Read off the t coordinate of the event from the home frame spacetime coordinates: d) Calculate t = t ', with Δ t ' = 1s and compare to the t coordinate of the event: Drawing the x'-axis. To keep the speed of light constant to all observers in inertial reference frames, the x' axis rotates up from x-axis the same angle as the t' axis rotates away from the t -axis. Essentially, the x' axis will have a slope of, while the t' axis will have a slope of 1/. We can use a similar means to scale the x' axis as well, Δ x = γ Δ x '
e) Place an event on the x '=1 tic mark. Read off the x coordinate of the event from the home frame spacetime coordinates: f) Check that it agrees with x = x' Reading Event Coordinates in both frames: When you select an event, the coordinate traces that for the event are parallel to the t and x coordinate axes, which is how we normally read coordinates off of a graph. This is how we read the spacetime coordinates for an event in the Home Frame. For spacetime coordinates measured in the Other Frame it is a bit more complex because the t' and x' coordinate axes are not perpendicular. But the process is similar, we trace a line from the event parallel to the x' axis back to where it hits the t' axis. This gives the t' value. Similarly, a line parallel to the t' axis is drawn from the event back to the x' axis to give the x' value. It should be noted, that these results are consistent with the Lorentz transformation equations that allow you to convert spacetime coordinates from the Home Frame to the Other Frame (and the Inverse transformations that do the reverse). Lorentz Transformations: Inverse Lorentz Transformations t ' = t x t = t ' x ' x ' = t x Δ x = γ (βδ t ' +Δ x ') g) Place an event on the spacetime diagram that is not directly on any of the coordinate axes, but is between the angle between t' and c=1 line. Read off the coordinates of the point in the Home Frame coordinate system (t, x): h) Now click the Other Frame button at the top right. The coordinate traces should switch to showing how the coordinates are calculated from the t' and x' axes by drawing lines parallel to the t' and x' axes. Record the coordinates: i) Check that the coordinates as shown in both frames agree with the Lorentz transformation equations: 6. Simultaneity is Frame Dependent : One of the consequences of special relativity, is that the concept of simultaneity (that two event occur at the same time) is frame dependent. Two observers, moving relative to one another will not agree as to whether or not two events happen at the same time. This can easily be shown on a two-observer spacetime diagram. a) Place events A and B on the spacetime diagram so that they occur at the same time with respect to the home frame, at t = 2s, but at different positions, with A being somewhere to the left of B. Record the spacetime coordinates for the two events. b) Set the beta slider to 0.6, and switch to the Other Frame. Now record the spacetime coordinates of the two events in the Other Frame: c) Do the events occur simultaneously in both reference frames? Which event occurs first in the Other Frame? d) Adjust the beta slider to change the relative velocities, and you change the order of the two events?
Exercise (adapted from Tom Moore, Six Ideas that Shaped Physics Unit R) Allan sits at the origin of the an inertial reference situated inside an 8.0-μs-long train tunnel at rest on the earth. We will call this frame the Home Frame. The east end of the tunnel is at x = +6.0 μs and the west end is situated at x = -2.0 μs. Sally operates a reference frame in a train racing east across the countryside at a constant speed of β = 0.6 c. At a certain time (call it t ' = 0 ) Sally passes Allan. At that same instant, Allan flashes a strobe lamp (call this event A), which sends bursts of light both east and west. Betty, who is standing at the west end of the tunnel, receives the west-going part of the flash (call this event B), and a bit later (according to clocks in the station) Carter, who is standing at the east end of the tunnel, receives the east-going flash (call this event C). e) When do events B and C occur in the tunnel frame? Who sees the flash first according to the clocks in the tunnel, Betty or Carter? f) Use the spacetime diagram program to draw a two-observer spacetime diagram of the situation. Locate and label events A, B and C as points on the diagram. Take a snapshot of the spacetime diagram. g) When and where do events B and C occur in Sally's frame? Sally claims that Ellen sees the flash first in her frame. Is this true? Use the spacetime program to check her claim. Take a snapshot showing how you checked this. h) Is there a velocity at which Sally could be moving so that events B and C happen simultaneously? Check it with the spacetime diagram. i) Can events B and C be causally linked?