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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)

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 doesnt change. only our perspective on it changes.

formulas:

https://www.physicsforums.com/showthread.php?t=263143

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.

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 doesnt change. only our perspective on it changes.

formulas:

https://www.physicsforums.com/showthread.php?t=263143

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.JesseM said: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

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.

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