Causality

## Graphing Spacetime Events

Up come now, we have been representing the place of a spacetime occasion relative to the spatial axes the Ann and Bob, if representing time by drawing the axes at various positions, v the collection of axes staying fixed being the coordinate system of the observer from who perspective we space seeing the event. Right here we will represent spacetime occasions in a more efficient and useful (though perhaps a bit an ext abstract) manner.

You are watching: Make a spacetime diagram and draw worldlines for each of the following situations.

We will continue to be with simplified situations where all the activity occurs along the $$x$$-axis, which way that we have actually room for something else. Specifically, us will draw the place (along $$x$$-axis), and also time axes because that an observer in one inertial framework whose view we space viewing from. Through this construction, a spacetime occasion is just a point located in the airplane of the two axes, and the coordinate position and coordinate time because that the observer we space viewing from can be review off the axes directly.

In 9HA we functioned with 1-dimensional graphs v the position represented top top the upright axis, and the time on the horizontal axis. In relativity, the is traditional (perhaps just to alarm the leader to the reality that relativity is being considered) to use time together the upright axis. What is more, this diagrams offer both axes the exact same units by scaling the upright axis through the speed of light, $$c$$. The resulting representation of spacetime occasions is called a spacetime or Minkowski diagram.

Figure 2.1.1 – Spacetime events on a Minkowski Diagram

The times shown by the values on the upright axis room those measured by synchronized clocks in the framework of the observer represented by these axes. If the two events happen to be aligned vertically, climate the two events occur in ~ the same position in this frame, which way the difference $$t_2 - t_1$$ is a appropriate time $$\Delta\tau$$. Moreover, due to the fact that we have assumed the the observer is in one inertial frame, this time span is the spacetime interval. It is conventional to definethe spacetime interval together the distance between these two occasions in the spacetime diagram, which means multiplying the time span by $$c$$. That is, rather of saying that the spacetime interval between events equals the appropriate time between those occasions when measure up in one inertial frame, us say that it is a distance that is proportional to it:

\<\Delta s = c\Delta \tau\;,\;\;\; \Delta \tau \text measure up in an inertial frame\>

## World Lines

While the is pretty to be able to draw two separate occasions on the very same diagram, rather than illustration multiple drawings as we did before, over there is an also greater benefit to the tool of the Minkowski diagram – we have the right to represent spacetime trajectories, additionally known as world lines. Intend we monitor a flashing beacon together it moves follow me the $$x$$-axis. Each flash documents a spacetime event with the place it is located and also the time of the flash, so we get a string of several points top top the spacetime diagram. If the flashing frequency is increased to infinity, climate the spacetime points kind a continuous curve the tracks the spacetime history of the moving object.

Let"s think about a few special people lines. Once someone observes a stationary object, its trajectory in the Minkowski diagram should be such the the position doesn"t adjust (but of course the time does). If this person witnesses an item moving at a continuous speed, climate the world line is sloped, positively when the motion is in the $$+x$$ direction, and also negatively when the motion is in the $$-x$$ direction.

Figure 2.1.2 – Some simple World Lines

attract a spacetime chart in Ann"s recommendation frame showing the people lines the both Bob and Chu, and also label the essential spacetime events along these worldlines. Use the diagram to recognize the time ~ above Ann"s clock in her spaceship (not at the lattice allude in her reference frame) once she sees through her telescope the Chu has readjusted speed. Usage the chart to recognize the time on Bob"s clock in his spaceship once he sees with the home window of his spaceship the Chu has adjusted speed. Solution

a. The diagram listed below has light minutes as units on both axes. Both start the race at the same suggest in spacetime ($$x=0,\;t=0$$ in Ann"s frame), and end at the same suggest in spacetime ($$x=40$$ light-minutes, $$80$$ minutes later). Chu transforms speed in ~ $$x=10$$ light-minutes, after $$t=40$$ minutes. Bob move at a consistent speed of half the speed of light, for this reason the steep of his people line is $$2$$. Chu moves at one-quarter the rate of light and also then three-quarters the speed of light, therefore the slopes of his people line segments room $$4$$ and also $$\frac43$$, respectively.

b. Ann doesn"t check out the occasion of Chu an altering speed until light native that event reaches her. The human being line that this light has a steep of $$-1$$ since it is coming earlier to Ann (in the $$-x$$-direction). Illustration this right into the diagram reflects that she discovers Chu has adjusted speeds in ~ time $$t=50$$ minutes, because that"s when the light world line intersects with Ann"s world line (which is the vertical axis, since she beginning at she origin and never moves).

c. Currently we desire to understand when Bob sees light from the occasion of Chu transforming speed. This time the irradiate goes in the $$+x$$-direction to get to Bob, therefore it has actually a steep of $$+1$$. This enables us to discover the intersection allude of the light human being line through Bob"s world line, but then deriving from that the time on Bob"s clock calls for a little much more work. The simplest means to obtain this number is to uncover what the time is ~ above Ann"s clock, climate use the time dilation formula to acquire Bob"s time. Looking in ~ the diagram below, we view that the moment of this intersection in Ann"s framework is $$60$$ minutes, which in Bob"s frame translates to:

\

## Minkowski Spacetime

Let"s represent on spacetime diagrams a comparison of what Ann sees come what Bob sees once they witness the very same object"s motion. Special, let"s speak they both watch the exact same clock, which provides off 2 light flashes at various times, and also records those times. Let"s more have the clock remain stationary in Ann"s frame between these flashes.

With the clock at remainder in Ann"s frame, the civilization line because that it looks favor the first graph in the figure above. Through Bob moving in the $$+x$$-direction loved one to Ann in ~ a speed $$v$$, that sees this very same object relocating in the $$–x$$-direction at a rate $$v$$, which way the worldline the sketches for the thing looks favor the 3rd graph in the figure above.

As we confirmed above, the spacetime interval in between the two occasions is the length the the segment the the people line connecting the two occasions in Ann"s spacetime diagram. When we very first discussed the three types of time, we made a allude of noting the all suitable times – the spacetime expression in specific – room invariants, which way that everyone actions the exact same value. This would seem to indicate that if Bob measures the size of the segment that the people line he draws between the two events, that should gain the very same result. In reality this is true, however as we will certainly see, over there is a surprise twist.

We discovered that the moment dilation result gives the comparison in between time intervals to it is in (Bob is the primed structure here):

\

But wait, due to the fact that $$\gamma_v>1$$, this method that $$ct"_2 - ct"_1 > ct_2 - c t_1$$, and also when us look in ~ the spacetime diagrams because that Ann and also Bob, we watch this way that the vertical readjust in the graph is higher for Bob than it is because that Ann.

Figure 2.1.3 – comparison of Spacetime Intervals because that Ann and also Bob

The calculation of $$\Delta s$$ for the relocating clock in the noninertial structure (middle graph) is the same as soon as computed by any kind of observer. The calculation of $$Delta s$$ because that the moving clock in the inertial structure (middle graph) is the same once computed by any kind of observer. Don"t confuse this because that saying they space the same as each other! the time elapsed on the clock is proportional to the path length $$\Delta s$$ v spacetime, and the path length is different for a right line compared to a curved one.

Example $$\PageIndex2$$

Once again returning to the believed experiment for the twin paradox entailing Ann, Bob, and also Chu at the finish of section 1.4, use the spacetime diagram developed in component (a) of example 2.1.1 above to answer the following:

Compute the (Minkowski) size of the people line because that Bob to gain the time elapsed for him during the race. Repeat the procedure in part (a) for Chu. Solution

a. The $$x$$ and also $$ct$$ components of Bob"s world line are $$40$$ and $$80$$ light-minutes (lm), respectively, therefore the Minkowski length of his civilization line is:

\<\Delta s = \sqrt80^2 - 40^2\;lm \approx 69 \;lm \nonumber\>

Dividing $$\Delta s$$ by the speed of light provide the time, which way that 69 minutes elapse for Bob. This agrees v the an outcome we got in section 1.4.

b. To compute the size of the civilization line because that Chu, we should compute the lengths of two different segments and then include them together.

\<\left. \beginarrayl \Delta x_1 = 10\;lm\;,\;\;c\Delta t_1 = 40\;lm && \Rightarrow && \Delta s_1 = \sqrt40^2-10^2\;lm \approx 38.7\;lm \\ \Delta x_2 = 30\;lm\;,\;\;c\Delta t_2 = 40\;lm && \Rightarrow && \Delta s_2 = \sqrt40^2-30^2\;lm \approx 26.5\;lm \endarray \right\}\;\;\; \Delta \tau_Chu = \dfrac\Delta s_1 +\Delta s_2c \approx 65min\nonumber\>

Again, this is in covenant with our result before.

space-like separation: \(\Delta s^2

For this case, not also light can connect the two events with a people line. If we imagine an occasion occurring since someone in ~ the spatial place of the event pushes a switch to cause a irradiate to flash, then the human being responsible for two occasions separated in this means have no idea the the other event exists, as there is no human being line to bring a message around one occasion to the other event.

This critical point around space-like separated events brings up crucial concept – that of causality. The study of princetoneclub.orgics is all around cause-and-effect – the is a net pressure that causes an acceleration, for example. We see currently that we deserve to relate two occasions according to even if it is one can cause the other. If over there is no way, also in principle, for a message to obtain from one event to another, climate there is no method for the first event to cause the other.

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We deserve to characterize what occasions can it is in causally-related nicely by making use of a spacetime diagram. Start by picking a spacetime event. Then note that to it is in causally-related come this event, a 2nd event should be positioned such that a straight civilization line drawn in between the two has actually a slope that is no much less than 1 (and if it has a steep of 1, they should be associated through a signal that moves at the rate of light). This limits the region of the 2nd event loved one to the very first into what is referred to as the light cone of that an initial event.