Space Time Diagrams? And lorentz Transformation?

[sizle95]

I am starting to learn Special and General realitivity by reading through Bernard F. Schutz's book "A First Course in General Realitivity". However I can't seem to grasp the relationship between two reference frames as compared with a Space-Time diagram. I understand the geometry of the diagrams but how are the axis on the primed reference frame positioned with relation to the standard axis at right angles?


Also I don't under stand the lorentz transformation and the consequences of applying this transformation to the equations.


If anyone could answer my questions (however broad they may be) I would greatly appreciate the help.



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[Simon Bridge]

Welcome to PF;
The relationships all depend on the speed of light being the same for all observers.
A step-by-step construction of space-time diagrams is given in the more accessible FAQ:
http://www.physicsguy.com/ftl/html/FTL_intro.html
... you will want to read through it carefully.

Introductory special relativity is typically 4-5 lectures at college level but forms a whole semester to go into the ins and outs of it. General relativity tends to occupy a whole course at post-grad level. Don't expect to get everything right away. If you get stuck on something specific, we can help - but this is not the place for a whole course.

 

[vanhees71]

I've always asked myself, what are the space-time diagrams good for in understanding relativity? For me it was always easier to use the formulas with the Lorentz transformation formula than to construct things like length contraction, time dilation, relativity of simultaneity in space-time diagrams.

 

[dauto]

I am starting to learn Special and General realitivity by reading through Bernard F. Schutz's book "A First Course in General Realitivity". However I can't seem to grasp the relationship between two reference frames as compared with a Space-Time diagram. I understand the geometry of the diagrams but how are the axis on the primed reference frame positioned with relation to the standard axis at right angles?


Also I don't under stand the lorentz transformation and the consequences of applying this transformation to the equations.


If anyone could answer my questions (however broad they may be) I would greatly appreciate the help.



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Your question is not clear. What do you mean by right angles? Do you mean the angle between t' and x' as seen in a t-x diagram? If that's the case, than the answer is that they are not at a right angle (or at a acute angle or at an obtuse angle). That angle is not an invariant.

 

[DrGreg]

I am starting to learn Special and General realitivity by reading through Bernard F. Schutz's book "A First Course in General Realitivity".

If you've never studied Special Relativity before, you need to spend some time studying that before beginning a book on General Relativity. Schutz's book assumes you have already "studied special relativity, including the Lorentz transformations and relativistic mechanics" to quote the preface.

 

[robphy]

I've always asked myself, what are the space-time diagrams good for in understanding relativity? For me it was always easier to use the formulas with the Lorentz transformation formula than to construct things like length contraction, time dilation, relativity of simultaneity in space-time diagrams.

Special Relativity isn't just about applying the Lorentz Transformation.
Euclidean geometry isn't just about applying the rotation formulas.

In my opinion, a good diagram (whether it be a spacetime diagram, position-vs-time diagram, free-body diagram, PV-diagram, etc...) tries to help organize one's thoughts in an abstract calculation... especially in a subject that is counterintuitive.

 

[robphy]

Your question is not clear. What do you mean by right angles? Do you mean the angle between t' and x' as seen in a t-x diagram? If that's the case, than the answer is that they are not at a right angle (or at a acute angle or at an obtuse angle). That angle is not an invariant.

If you are talking about "Euclidean angles", then I agree.
However, in Minkowskian geometry, there is still the invariant concept of orthogonality between those axes, although one might not be able to assign a corresponding Minkowskian-angle measure (rapidity) between a spacelike-vector and a timelike-vector.

 

[dauto]

If you are talking about "Euclidean angles", then I agree.
However, in Minkowskian geometry, there is still the invariant concept of orthogonality between those axes, although one might not be able to assign a corresponding Minkowskian-angle measure (rapidity) between a spacelike-vector and a timelike-vector.

I think (might be wrong) the OP is talking about the angle between the t' and x' axis as seen on a graphing paper where the t and x-axis have been plotted orthogonal to each other. That angle is not an invariant (not even in Galilean relativity).

 

[Simon Bridge]

I think (might be wrong) the OP is talking about the angle between the t' and x' axis as seen on a graphing paper where the t and x-axis have been plotted orthogonal to each other.

That is what I got too.
I think the question has been well answered by now.

 

[pervect]

I am starting to learn Special and General realitivity by reading through Bernard F. Schutz's book "A First Course in General Realitivity". However I can't seem to grasp the relationship between two reference frames as compared with a Space-Time diagram. I understand the geometry of the diagrams but how are the axis on the primed reference frame positioned with relation to the standard axis at right angles?Also I don't under stand the lorentz transformation and the consequences of applying this transformation to the equations.If anyone could answer my questions (however broad they may be) I would greatly appreciate the help.
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I'm not sure I understand what you don't understand, but I can make some remarks that might be helpful. Do let me know if they are - or are not.

The point of a space-time diagram is that there is a one-one correspondence between points on the space-time diagram, and events in space-time.

Thus for every event, there is exactly one point on the diagram, and for every point on the diagram, there is exactly one event.

Space-time diagrams are maps, or representations, of space-time, they are not space-time themselves. So there are multiple maps, or representations, of the same physical reality Note that for "ordinary" maps, we have a similar situation - maps that are translated, or rotated, are equivalent maps that represent the same physical reality.

The abstract existence of the Lorentz transform can be described as follows. Observer O uses variables t,x to describe the points he uses on his space-time diagram. (I'm going by the 2d case here, because most space-time diagrams are 2d, you'll occasionally see a 3d one though.)

Observer O' uses variables (t', x') to describe the points he uses on his, different space-time diagram.

Given the fact that there is a one-one correspondence between the points on Observer O's map of space time and space time itself, and there is also a one-one correspondence between the points of Observer O' map of space time and space-time itself, there must be a one-one correspondence between observer O and observer O'.

When observer O' is moving relative to observer O, we call this one-one correspondence the Lorentz transform.

At this point you may or may not know how it was derived, but it's significance should be clear. You just have

t' = some function (t,x)
x' = some other function (t,x)

Because it's a one-one correspondence you can also write

t = yet another function (t', x')
x = a different function(t', x')

I could write the functions out, but I hope the argument is more clear this way. (Feedback on this point would be useful).

As far as axes go, the t-axis for observer O is just the set of points where x=0. So if you want to graph that set of points on Observer O' space-time diagram, you just find the set of points represented by (t=variable, x=0), convert each point separately by using the lorentz transform, i.e. t' = some function of (t,0) and x' = some other function (t,0) for all t, and you plot them.

You then have two different views of the same set of events, one view for observer O, and another view for observer O'. The points themselves are independent of coordinates, the Lorentz transform is just the set of equations that provides a onene mapping from obsever O's coordinates to observer O' coordinates.

The actual carrying out of this process is more or less mechanical. If you are having difficulty applying it to a specific example, you can ask again - I'm not sure if I can help with that, but maybe someone else can.

[add]I guess I can go into even more excrutaiting detial!

Take the following set of points (t,x), and plot them on a graph paper, appropriately scaled:

(-10,0) (-9,0) (-8,0) (-7,0) (-6,0) (-5,0) (-4,0) (-3,0) (-2,0) (-1,0) (0,0) (1,0) (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (8,0) (9,0) (10,0)

Connect all the dots by lines. You should, if you've done this right, get the t axis, with x=0.

Now transform each and every point by using the Lorentz transform to transform (t,x) into (t',x'). This involves evaluating two functions of two variables at every point.

Connect all the dots by lines. You now have the t axis plotted on observer O's space-time diagram.