How to draw a spacetime diagram

[granpa]

here is my poor effort at teaching newbies how to draw spacetime diagrams (somebody has to do it):

first draw your x axis. this represents space. (one dimension of it anyway. one dimension is enough for most thought experiments)
the y-axis represents time. imagine there is a stationary observer at the origin. the y axis, therefore, represents that observers position over time.
now imagine a photon moving toward the right passing through the origin at time=0. draw a 45 degree angle line to represent that photons path. therefore if the y-axis represents seconds the x-axis will represent light seconds.

now imagine a spaceship moving toward the right passing through the origin at time=0. it can be at any speed you want. this is the moving observer. draw the path it would take over time. since, in this ships frame, the ship is stationary this line represents time in that coordinate system. call this y'.
flip this line over the 45 degree line to get x'. x' represents space at one simultaneous moment in the new coordinate system.

when 1 sec has passed for the stationary observer only 1/gamma sec will have passed for the moving observer (in the frame of the stationary observer). gamma=1/(1-(v/c)²).
in the new coordinate system the photon passing through the origin still makes the same 45 degree angle. so in 1/gamma sec (according to the new coordinate system) this photon will move 1/gamma light sec. use this to determine your x' units.remember:
2 events that occur at the same place and at the same time do so for all observers.
if event A causes event B then it will do so for all observers.
the 'proper time' and 'proper distance' between any 2 events will be the same for all observers.

its significance:
every event in this 1 dimensional universe has a specific location on our sheet of paper. we can change to any coordinate system without needing to move any of those events. only the numbers assigned to those events changes. coordinate systems are just arbitrary numbers assigned to events. albeit, the numbers used in our own coordinate system will always seem rather convenient since they agree with our own rulers and clocks.

therefore, changing frames (when we change velocity) is like changing perspective. the underlying reality doesn't change. only our perspective on it changes.formulas:
https://www.physicsforums.com/showthread.php?t=263143

if two clocks are synchronized and a distance L apart in their own rest frame, then in a frame where they are moving at speed v parallel to the axis between them, the clock in the rear will show a time that's ahead of the clock at the front by vL/c^2

the length of an object in a certain frame is the distance between the front and the back of the object at one simultaneous moment.

it is often convenient to imagine one line, for each observer, of evenly spaced clocks, stretching across the length of the space involved, all of which are perfectly synchronized with that observer. if they are one light sec apart then imagine them sending out radio pulses at one sec intervals.

 

[Dale]

That seems like a good step-by-step instruction. I would add one hint for new students. It is really convenient to use a spaceship with v=0.6c so that gamma is 1.25. It makes drawing things correctly pretty easy.

 

[granpa]

an example of using a line of synchronized clocks. (twin paradox):

The twin paradox is much easier to understand if one imagines a long line of stationary, evenly spaced, and synchronized clocks extending from the stationary twin to the point where the other twin turns around. Imagine that as these clocks pass the moving twins window a strobe flashes so he can read off the elapsed time. Even though the non-moving twins clock (and each of the stationary clocks) seems, to the moving twin, to be ticking at half the rate of his own clock, the elapsed time, as told by the strobe, is passing at twice the rate of his own. More importantly, just before he stops, in order to turn around, the line of clocks are, from his perspective, out of synch but the moment he stops the line of clocks will be perfectly synchronized again which means that the nonmoving twins clock now reads the same as the clock he is next to. That means that his calculation of what the nonmoving twins clock said jumps suddenly while he decelerates (which leads to general relativity).wikipedia refused to put any of it in the twin paradox page because they said it was 'original research'.
http://en.wikipedia.org/wiki/Twin_paradox

 

[granpa]

length contraction and time dilation are easy. what confuses all beginners is 'relativity of simultaneity'.

if you are a beginner and you are confused then there is a very good chance that it is 'relativity of simultaneity' that is responsible.

http://en.wikipedia.org/wiki/Relativity_of_simultaneityremember:
the length of an object in a certain frame is the distance between the front and the back of the object at one simultaneous moment.

 

[granpa]

one more common error that beginners make is to apply the fomulas for length contraction and time dilation to light itself. the formulas only apply to objects moving at less than light speed.

 

[granpa]

the special relativity way of looking at general relativity

Imagine a rocket starting at rest beside a long line of synchronized clocks one light sec apart (they tick simultaneously). As the rocket accelerates at 1G the clocks become more and more out of sync from the rockets point of view (due to relativity of simultaneity). That means that the clocks are running at different rates from the rockets point of view. If you do the math you will see that there is a point behind the rocket where time stops from the point of view of the rocket. A sort of black hole.

Gravity = acceleration hence gravitational time dilation

 

[oldman]

Granpa --- I'd like to add to this informative thread a comment:

Spacetime diagrams with one or two space dimension suppressed are helpful in many ways, but still puzzle me. In Special Relativity, spacetime diagrams clarify in a neat geometrical way the relationship between the time and space coordinates of events, as labelled by different moving observers; they allow one to sketch as world lines a continuum of events that are sequentially associated with physical objects; they distinguish with light or null cones nicely between events that happen in the past and future and, in General Relativity, they can be adapted to illustrate the characterestic features of spacetime near compact objects, like event horizons.

Convential spacetime diagrams are however restricted in some ways, usually to the perspective of a single observer (A, say) at the origin for ct and x, as measured by A. Other observer's world lines enter the usual picture only in "standard configuration" --- the observers or objects are assumed to move in one dimension (x) so that the origin of their coordinate systems coincide with A's at an instant when their clocks are synchronised to read zero. Such items then trace out 'world lines' inside A's light cones as time passes for A, so illustrating their history as observed by A.

When the perspective of other observers is of interest, it is illustrated by sketching stubs of their light cones, threaded like beads along a world line, or scattered all over a diagram (see for example diagrams in Penrose's The Road to Reality). Here the diagrams are of course not intended to be used for relating coordinates, as in SR.

But I'm puzzled about some aspects of spacetime diagrams. Why are less-restricted world lines that could describe simple situations not drawn in spacetime diagrams --- say that of the sun, from our perspective? The sun is stationary in an approximately (neglecting rotating and orbital motions << c ) inertial frame that we share, always a (nearly) constant distance R away from us. Its world line could for us be approximated as a line parallel to our time axis, at x = R , in our past light cone. It surely existed in our past. And we hope that it will continue along this line, projected into our future light cone. Do events in the gap along this line, between these cones have any significance? Or is this gap without events that have meaning, an artefact of the way 'now' is mapped in such diagrams? Sometimes they are just classified as "elsewhere". Puzzles me.

Or is it just convention?

 

[George Jones]

Do events in the gap along this line, between these cones have any significance? Or is this gap without events that have meaning, an artefact of the way 'now' is mapped in such diagrams?

I think my answer is "Yes." to both questions, but, to aovoid confusion about what I mean, I'll elaborate.

Forget coordinates, just draw the worldlines of the Earth and the Sun.

Pick any event on the Earth's worldline and label it A. From A, draw a past-directed lightlike worldline to an event, P, on the Sun's worldline. At event A we see the Sun as it was at event P. From A, draw a future-directed lightlike worldline to an event, Q, on the Sun's worldline. At event Q, the (someone on the) Sun sees us as we were at event A.

Now pick any R between P and Q on the worldline of the Sun, i.e., in the gap. From R, draw a future-directed lightlike worldline to an event, C, on the Earth's worldline. At event C, we see the Sun as was at event R. From R, draw a past-directed lightlike worldline to an event, D, on the Earth's worldline. At event R, the Sun sees us as were at event D.

Look at the finished diagram. The order of events on the Earth's worldline is DAC, and the order of events on the Sun's worldline is PRQ. Events DAC are all different, but equivalent, nows for us. PRQ are all different, but equivalent, nows for the Sun. Also note the symmetry between events A and R; neither the Earth nor the Sun is favoured in this coordinateless diagram. (I don't dislike the letter "B", I just unintentionally forgot about it.)

PS I haven't forgotten about the example I started some time ago. Hope to get back to it.

 

[Rasalhague]

Convential spacetime diagrams are however restricted in some ways, usually to the perspective of a single observer (A, say) at the origin for ct and x, as measured by A. Other observer's world lines enter the usual picture only in "standard configuration" --- the observers or objects are assumed to move in one dimension (x) so that the origin of their coordinate systems coincide with A's at an instant when their clocks are synchronised to read zero. Such items then trace out 'world lines' inside A's light cones as time passes for A, so illustrating their history as observed by A.

I thought the whole point of spacetime diagrams is that they allow you to compare two or more reference frames, and thus the perspectives of more than one observers moving relative to each other, whether their worldlines intersect or not. Examples where the world lines of two literal observers intersect and synchronise clocks are just a special case.

But I'm puzzled about some aspects of spacetime diagrams. Why are less-restricted world lines that could describe simple situations not drawn in spacetime diagrams --- say that of the sun, from our perspective? The sun is stationary in an approximately (neglecting rotating and orbital motions << c ) inertial frame that we share, always a (nearly) constant distance R away from us. Its world line could for us be approximated as a line parallel to our time axis, at x = R , in our past light cone. It surely existed in our past. And we hope that it will continue along this line, projected into our future light cone. Do events in the gap along this line, between these cones have any significance? Or is this gap without events that have meaning, an artefact of the way 'now' is mapped in such diagrams? Sometimes they are just classified as "elsewhere". Puzzles me.

Such a situation can be represented as a conventional spacetime diagram. Elsewhere just means outside of the light cone of a particular point in spacetime (a particular event). Following George's labels, suppose an event E happens outside the light cone of some event D on the earth, and therefore, if the sun is considered as approximately stationary with respect to the earth, outside the light cone of an event P at the sun which is simultaneous with the event on the Earth in the shared rest frame of Earth and sun. Does this event E have significance? As much significance as any other event. Its future light cone intersects the world lines of both Earth and sun, so it can influence events in the sun's future and that of the earth. Being in the "elsewhere" region just means that it can't have any causal influence on events D and P, and neither D nor P can have an effect on E.

 

[granpa]

I forgot to give the link to the relatistic rocket webpage.
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

a rocket accelerating at 1G as perceived by the people on the rocket has a 'rapidity' that is proportional to 'proper time'.

 

[oldman]

Thanks George, and Rasalhague, for taking the trouble to provide me with such clear replies. I'm reassured to hear that spacetime diagrams with non-intersecting world lines are conventional. I just hadn't seen any, and was puzzled by this. The comment that:

I thought the whole point of spacetime diagrams is that they allow you to compare two or more reference frames, and thus the perspectives of more than one observers moving relative to each other, whether their worldlines intersect or not. Examples where the world lines of two literal observers intersect and synchronise clocks are just a special case.

set me straight. And not all world lines need even be of relatively moving observers, I suppose.

I was also interested to learn that people regard events as "having light cones". Hadn't seen this association before. Very sensible.

 

[oldman]

Just a P.S. I've always wondered what I am. Now I know. I'm an event in spacetime. I'm so local in space and ephemeral in time, but nevertheless I'm equipped with a past and (I hope, a future) that I'm sure I qualify!

 

[granpa]

new improved version:

The twin paradox is much easier to understand if one imagines a long line of stationary, evenly spaced (say one light sec apart), and synchronized clocks extending from the stationary twin to the point where the other twin turns around. (It may help to imagine that they are simultaneously emitting radio pulses at one sec intervals)

If a rocket starting at rest near the stationary twin accelerates along the line of clocks to velocity v then stops accelerating then the clocks from his point of view will no longer be in synch. this is known as 'relativity of simultaneity' and this is what confuses most beginners.

If the rocket accelerates continuously at 1G the clocks become more and more out of sync from that rockets point of view (due to relativity of simultaneity). That means that the clocks are ticking at different rates from the rockets point of view. Some may even be running backwards. This is non-intuitive and is the source of the confusion surrounding the twins paradox. It also leads to general relativity and gravitational time dilation.

Further, imagine that as these clocks pass the moving twins window a strobe flashes so he can read off the time on that passing clock. Even though the non-moving twins clock (and each of the stationary clocks) seems, to the moving twin, to be ticking at half the rate of his own clock, the elapsed time, as told by the strobe, is passing at twice the rate of his own. More importantly, just before he stops, in order to turn around, the line of clocks are, from his perspective, out of synch but the moment he stops the line of clocks will be perfectly synchronized again which means that the non-moving twins clock now reads the same as the clock he is next to. That means that his calculation of what the non-moving twins clock said jumps suddenly while he decelerates.

 

[granpa]

Another thing beginners might want to know is that
general relativity is, at heart, just a way of explaining the equivalence principle
http://en.wikipedia.org/wiki/Equivalence_principle

according to the equivalence principle (and all experimental data to date) all objects fall at the same rate. This means that inertial mass must be proportional to passive gravitational mass.

 

[Mentz114]

There is a program ( for Windows :yuck:) that helps a lot. You can draw worldlines, define events and intervals between events. The whole diagram can be rotated to represent boosts.
Everything is properly scaled so numbers can be read off the diagram.

There's a link in my blog, and a link to that on the left.

 

[JDoolin]

new improved version:

The twin paradox is much easier to understand if one imagines a long line of stationary, evenly spaced (say one light sec apart), and synchronized clocks extending from the stationary twin to the point where the other twin turns around. (It may help to imagine that they are simultaneously emitting radio pulses at one sec intervals)

I now usually start any discussion of relativity with a description of superluminal motion. I cross the room toward a student in the back row with my hand just barely in front of my face. When I reach the student, of course, they think they saw me for the whole time, but NO the light just reached them! Then I hit them--well, not really. But it is a dramatic concept.

Then I describe the motion that they see when I walk away. I arrive at the wall opposite them, but when do they see me arrive at the opposite wall? Only after the light reaches them--so at most they can only see me retreating at half the speed of light.

But they saw me coming toward them at superluminal speeds for only a fraction of a second, and they saw me retreating for many, many seconds.

Then I try to get them to imagine things from my perspective. I see them coming toward me at superluminal speeds, and then retreating at half the speed of light for equal times! That means when I am looking forward to the wall I'm approaching it looks really far away, and coming at me really fast, but then when I look at the wall behind me, it is still close, and moving away really slowly.

Hopefully by this time, they can tell that somethings got to give, and then I introduce the relativity of simultaneity.

An animation like this one sometimes helps.

[URL]http://www.wiu.edu/users/jdd109/stuff/relativity/reflect30.gif[/URL]

Several years ago I made a http://www.wiu.edu/users/jdd109/stuff/relativity/Circle.swf", similar in some ways to Mentz114's. I tried to post a link to the interactive space-time diagram on Wikipedia, but was also told "No Original Research!"

 

[JDoolin]

Pretty cool.

They ought to have clocks on the front and back of the apparatus to bring it home.

 

[dimuthuROX]

A flash light at rest at the origin of the S' frame shines light along the y' axis. S' frame is moving at a speed 0.707c along the x-axis with respect to frame s which is at rest.what angle does the light make with the x-axis of the s frame?

 

[granpa]

sounds like homework

 

[dimuthuROX]

yaa kind of
do you no d answar?

 

[yuiop]

yaa kind of
do you no d answar?

It is against the rules to post homework questions here and even more against the rules to provide a direct answer to a homework question. I can provide a hint however. You know how far the light travels in the S frame in the x direction in one unit of time. You also know that by definition the light will travel unit of distance in one unit of time along the hypotenuse. Knowing the length of the hypotenuse and one of the other sides, it shouldn't be too difficult to figure out the length of the remaining side (how far the light travels in the y direction in one unit of time) and the angle.

 

[dimuthuROX]

Im sorry..Im just a new member here...
Hav no idea about rules...
Nway...thanx 4 da help...