Do we live in 3 or 4 dimensions?

[Stricklandjr]

pleese don't critical on my little knowledge of this. a few of us have bull sessions sometimes and my cousin points front side up and says we live in 3 dimencions but the whole universe is 4 dimencions. this other guy always says my cousin is full of bull and einstine just used them for calculations. who is right?

also I'm new and read the rules for this forums. what did greg mean by you can't bring up ether theory?

 

[phinds]

spacetime is a 4 vector, so while we DO live in 3 SPATIAL dimensions, we also do live in 4 dimensions so they are both right. If one of them is saying we live in 4 dimension of space, then he misunderstands Einstein.

"Ether theory" is a theory you can find on Google. It was discredited something like 100 years ago and is not discussed on this forum.

 

[Stricklandjr]

spacetime is a 4 vector, so while we DO live in 3 SPATIAL dimensions, we also do live in 4 dimensions so they are both right. If one of them is saying we live in 4 dimension of space, then his misunderstands Einstein.

"Ether theory" is a theory you can find on Google. It was discredited something like 100 years ago and is not discussed on this forum.

sorry I don't understand when you first say we also do live in 4 dimensions and then if one of them is saying we live in 4 dimension of space he misunderstands Einstein.

 

[tensor33]

sorry I don't understand when you first say we also do live in 4 dimensions and then if one of them is saying we live in 4 dimension of space he misunderstands Einstein.

What phinds is saying is that we live in a 4 dimensional world where 3 dimensions are spatial, and the fourth is time. It is incorrect to say that we live in 4 spatial dimensions.

 

[Stricklandjr]

What phinds is saying is that we live in a 4 dimensional world where 3 dimensions are spatial, and the fourth is time. It is incorrect to say that we live in 4 spatial dimensions.

ok. thank you. I'm looking at some other threads and there is one by a guitar name about us moving through the 4 dimensional universe as fast as light. Does that mean that the universe is 4 dimensions (space and time) and we are just 3 dimensional moving through the 4 dimensions of the universe? So things like rocks and cars and people are three dimensions and the universe is 4 dimensions?

 

[HallsofIvy]

Physics deals with "events"- things that happen at a given point in space at a given time. In any coordinate system we need 3 numbers to specify the point and one to specify the time. That's what "four dimensional" means.

 

[PeterDonis]

I'm looking at some other threads and there is one by a guitar name about us moving through the 4 dimensional universe as fast as light.

We get a lot of threads here about that. It's not wrong, exactly, but it can easily lead to misconceptions. The basic physical fact underlying the statement is that every object (more precisely, every object with nonzero rest mass) has a 4-velocity vector whose length is the speed of light.

Does that mean that the universe is 4 dimensions (space and time) and we are just 3 dimensional moving through the 4 dimensions of the universe?

On the view I just described, where an "object" has a single 4-velocity vector, the object is idealized as point-like, so its path through 4-dimensional spacetime is a 1-dimensional line, called its "worldline". A real object, which is extended in space, would be modeled as a whole family of worldlines that stay together; this is sometimes called a "world-tube". But if individual parts of the extended object are moving with respect to each other, they will not all have the same 4-velocity, so you can't assign a single 4-velocity vector to the entire object. (Often you can assign a sort of "average" 4-velocity to the object because all of its parts are moving really slowly with respect to each other, compared to the speed of light. But that's an approximation.)

However, it's important to note that in all these models, the "object" (whether its an idealized point-like object or an object extended in space) is best viewed as its worldline or world tube. This is because of the relativity of simultaneity; different observers in different states of motion will "slice" 3-dimensional sections out of an extended object's world tube in different ways. So you can't really view the object as a 3-dimensional thing that moves through time; you have to look at its entire 4-dimensional world tube. When your cousin said that we live in 3 dimensions, he apparently wasn't taking that into account.

 

[Naty1]

Check out some descriptions here:

http://en.wikipedia.org/wiki/Spacetime

 

[Stricklandjr]

We get a lot of threads here about that. It's not wrong, exactly, but it can easily lead to misconceptions. The basic physical fact underlying the statement is that every object (more precisely, every object with nonzero rest mass) has a 4-velocity vector whose length is the speed of light.



On the view I just described, where an "object" has a single 4-velocity vector, the object is idealized as point-like, so its path through 4-dimensional spacetime is a 1-dimensional line, called its "worldline". A real object, which is extended in space, would be modeled as a whole family of worldlines that stay together; this is sometimes called a "world-tube". But if individual parts of the extended object are moving with respect to each other, they will not all have the same 4-velocity, so you can't assign a single 4-velocity vector to the entire object. (Often you can assign a sort of "average" 4-velocity to the object because all of its parts are moving really slowly with respect to each other, compared to the speed of light. But that's an approximation.)

However, it's important to note that in all these models, the "object" (whether its an idealized point-like object or an object extended in space) is best viewed as its worldline or world tube. This is because of the relativity of simultaneity; different observers in different states of motion will "slice" 3-dimensional sections out of an extended object's world tube in different ways. So you can't really view the object as a 3-dimensional thing that moves through time; you have to look at its entire 4-dimensional world tube. When your cousin said that we live in 3 dimensions, he apparently wasn't taking that into account.

Boy! You are really getting down to it. This is getting interesting even for a dumb guy like me. I took a course in physics in junior college and at the end of the semester he told us about relativity but never really talked about what you are saying, he did use some equations though. My cousin has had more college courses in physics and we all talk about it sometimes. Then do you mean the the objects you talk about are really 4 dimensional just like space (and time)? Are things really 4 dimensions and we just don't see all of the dimensions?

 

[PeterDonis]

Then do you mean the the objects you talk about are really 4 dimensional just like space (and time)?

Just like *spacetime*, yes. (Neither space nor time by themselves are 4-dimensional; only spacetime is.)

Are things really 4 dimensions and we just don't see all of the dimensions?

What makes you think we don't see all the dimensions? The 4th dimension is time; we "see" the time dimension by observing that objects don't just exist for an instant; they exist for some length of time.

 

[Naty1]

Strickland asks:

So things like rocks and cars and people are three dimensions and the universe is 4 dimensions?

Then do you mean the the objects you talk about are really 4 dimensional just like space (and time)? Are things really 4 dimensions and we just don't see all of the dimensions?

Here are two famous quotes that address your question in a humorous but interesting way:

"Time is what keeps everything from happening at once."

and I think it might have been Richard Feynman who said

"Space is what keeps everything from happening to me.'

I took a course in physics in junior college and at the end of the semester he told us about relativity but never really talked about what you are saying...

When you learned Newtonian physics the implicit assumptions used three dimensions of space, say (x,y,z) and an independent parameter time, t, which ticks along the same steady rate for everyone. And the assumed infinite speed of light allowed things to happen instantaneously...so it took no time for sunlight to get from the sun to us. Everybody measures the same distances and times. All this works pretty well when relative speeds are slow...like orbiting planets.

Einstein figured out that space and time are actually relative and depend on each other. 'Space' [distances] that appear fixed in our everyday slow speed existence can be different for different high speed observers. That is, the speed of light is finite and, crazy as it seems, everybody measures the same speed for light no matter their own local speed. So space and time vary by observer, related to their speeds, while the speed of light is finite and fixed.

In fact it was Einstein's college teacher, Minkowski, who realized that Einstein's early work meant that space and time should be treated equally, that events take place in four-dimensional space-time. Space and time were no longer to be considered separate, independent entities! So the three 'dimensions' of Newton (x,y,z) became four dimensions [spacetime] of relativity: (t,x,y,z).

edit: If I have stated things correctly here, it should all be consistent with what PeterDonis has posted. The inexpensive book FABRIC OF THE COSMOS by Brian Greene describes 'moving through spacetime at the speed of light' and a lot more without any math. I found that book to be fascinating reading when I started reviewing relativity.

 

[Stricklandjr]

Just like *spacetime*, yes. (Neither space nor time by themselves are 4-dimensional; only spacetime is.)

You are blowing me away with these ideas Dr. Donis. My mind is spinning. But I think I am beginning to catch on some.

What makes you think we don't see all the dimensions? The 4th dimension is time; we "see" the time dimension by observing that objects don't just exist for an instant; they exist for some length of time.

Now I think I am beginning to get it. I really liked taking computer drawing in jr college. With this program called I-Deas you could sketch a 2 dimensional surface and then just extend it into the 3rd dimension for whatever length you choose. So maybe relativity is a little like starting with a 3 dimensional thing and then extending it along the 4th dimension. Along a path you called a world tube? So if I see a rock on the ground I could imagine extending it along its world tube? This is heavy stuff. You are about the smartest physicist here. How could anyone ever figure out that was what is going on?

 

[Stricklandjr]

Strickland asks:





Here are two famous quotes that address your question in a humorous but interesting way:



and I think it might have been Richard Feynman who said





When you learned Newtonian physics the implicit assumptions used three dimensions of space, say (x,y,z) and an independent parameter time, t, which ticks along the same steady rate for everyone. And the assumed infinite speed of light allowed things to happen instantaneously...so it took no time for sunlight to get from the sun to us. Everybody measures the same distances and times. All this works pretty well when relative speeds are slow...like orbiting planets.

Einstein figured out that space and time are actually relative and depend on each other. 'Space' [distances] that appear fixed in our everyday slow speed existence can be different for different high speed observers. That is, the speed of light is finite and, crazy as it seems, everybody measures the same speed for light no matter their own local speed. So space and time vary by observer, related to their speeds, while the speed of light is finite and fixed.

In fact it was Einstein's college teacher, Minkowski, who realized that Einstein's early work meant that space and time should be treated equally, that events take place in four-dimensional space-time. Space and time were no longer to be considered separate, independent entities! So the three 'dimensions' of Newton (x,y,z) became four dimensions [spacetime] of relativity: (t,x,y,z).

edit: If I have stated things correctly here, it should all be consistent with what PeterDonis has posted. The inexpensive book FABRIC OF THE COSMOS by Brian Greene describes 'moving through spacetime at the speed of light' and a lot more without any math. I found that book to be fascinating reading when I started reviewing relativity.

I think this is just what Dr. Donis was saying. But I am going to the book store today and look for that book. And I have to think hard about how the 4 dimensions are not just space but what you and Dr. Donis call spacetime. When I think about the way I did drafting I can picture a rock extending into the 4th dimension but I have to think some more about how that could be something new called spacetime. Is it some kind of combination of space and time that is too hard for us to imagine what it is or do you and Dr. Donis know what it is?

 

[phinds]

I think this is just what Dr. Donis was saying. But I am going to the book store today and look for that book. And I have to think hard about how the 4 dimensions are not just space but what you and Dr. Donis call spacetime. When I think about the way I did drafting I can picture a rock extending into the 4th dimension but I have to think some more about how that could be something new called spacetime. Is it some kind of combination of space and time that is too hard for us to imagine what it is or do you and Dr. Donis know what it is?

Your statement of the question "what is it?" is not helpful. "WHAT" something is is just a name we give it. In this case we give it the name "spacetime". There is no other answer to WHAT it is. Science is about describing the properties of the things we give names to and the properties of spacetime are discussed in general relativity.

A really excellent example of this terminology issue is the confusion that was experienced early in the days of quantum mechanics when people insisted on calling a photon either a "wave" or a "particle" because they were hung up on WHAT it is. In reality, a photon is NEITHER a wave nor a particle but rather a "quantum object" that sometimes has the properties of a wave and sometimes has the properties of a particle.

So "spacetime" is not exactly "space" added to "time", it is its own thing called spacetime and it has characteristics that can be discussed although discussing them in English leads to problems. The proper language of science is math and if you want to REALLY describe spacetime you're going to have to learn a lot of math.

 

[Fredrik]

Does that mean that the universe is 4 dimensions (space and time) and we are just 3 dimensional moving through the 4 dimensions of the universe? So things like rocks and cars and people are three dimensions and the universe is 4 dimensions?

A car is 3-dimensional, but its entire existence from when it's built to when it's destroyed, is 4-dimensional. If you want to understand this better, you should learn about spacetime diagrams (also called Minkowski diagrams).

Are things really 4 dimensions and we just don't see all of the dimensions?

Relativity doesn't say anything like that.

 

[Naty1]

I think this is just what Dr. Donis was saying.

Yes. It was supposed to be the same...Getting different descriptions, even though equivalent, is often needed to gain perspective...

I'll give you a rough analogy for space and time being considered 'spacetime' :

If you consider the ocean, say, one, and air the other, they seem pretty distinct, right? ...especially when it's flat calm... Easy to tell them apart...Then the wind picks up and waves are seen...where did the air go?? Now it's 'curved' too just like the surface of the water...[Space, time and spacetime also 'curves' by the way] then at around 50 or 60 mph, the surface of the water is picked up in a 'spume'...like rain droplets driven from the surface of the water...and it may extend three feet or more above the surface...you can no longer, perhaps, see the waves...'air and water' now have no clear boundary...they 'merge' into each other as if they maybe were a single entity...I've seen this a number of times when boating...and reading physics...

Well, in an analogous way, 'flexible' space and time also change from one to the other based on frames of reference [different observer perspectives]. Space and time 'conspire' together in such a mathematical form as to keep the speed of light the same for all observers.

By the way, if you start buying books, see Amazon Books or other online sources for less costly used copies...

 

[Naty1]

Strickland posts

I'm looking at some other threads and there is one by a guitar name about us moving through the 4 dimensional universe as fast as light.

BRAVO! I forgot to complement you on doing some searches... just about everything has been discussed numerous times. The trick is to figure out who knows what they are doing...the 'experts' I call them...and then you can just read those posts, because some threads go on and on and on...Few if anybody, I think, knows it ALL...

 

[Stricklandjr]

Your statement of the question "what is it?" is not helpful. "WHAT" something is is just a name we give it. In this case we give it the name "spacetime". There is no other answer to WHAT it is. Science is about describing the properties of the things we give names to and the properties of spacetime are discussed in general relativity.

A really excellent example of this terminology issue is the confusion that was experienced early in the days of quantum mechanics when people insisted on calling a photon either a "wave" or a "particle" because they were hung up on WHAT it is. In reality, a photon is NEITHER a wave nor a particle but rather a "quantum object" that sometimes has the properties of a wave and sometimes has the properties of a particle.

So "spacetime" is not exactly "space" added to "time", it is its own thing called spacetime and it has characteristics that can be discussed although discussing them in English leads to problems. The proper language of science is math and if you want to REALLY describe spacetime you're going to have to learn a lot of math.

Thank you Mr. phinds. I can handle it not knowing what spacetime really is as long as no one really knows, I just thought maybe everone here knows what spacetime is. I won't study on that one any more.

 

[Stricklandjr]

However, it's important to note that in all these models, the "object" (whether its an idealized point-like object or an object extended in space) is best viewed as its worldline or world tube. This is because of the relativity of simultaneity; different observers in different states of motion will "slice" 3-dimensional sections out of an extended object's world tube in different ways. So you can't really view the object as a 3-dimensional thing that moves through time; you have to look at its entire 4-dimensional world tube. When your cousin said that we live in 3 dimensions, he apparently wasn't taking that into account.

I wanted to come back to this because it really puts your finger on something really cool. In my drafting class we would picture a solid and then draw a line across it with arrows pointing in the direction for a cross-section view. We would do this for two or three different cross sections and label them Sec A-A, Sec B-B, Sec C-C, and so on. So you are telling me that we could imagine my rock extended into the 4th dimension of spacetime (not space, I'm getting that part now) and different people would see different cross sections of the 4 dimensional rock? What determines what their cross section view will be?

 

[Stricklandjr]

A car is 3-dimensional, but its entire existence from when it's built to when it's destroyed, is 4-dimensional. If you want to understand this better, you should learn about spacetime diagrams (also called Minkowski diagrams).


Relativity doesn't say anything like that.

First you said a cars existence is 4 dimensional just like Dr. Donis said. But then why did you say relativity doesn't say anything like that?

 

[PeterDonis]

So you are telling me that we could imagine my rock extended into the 4th dimension of spacetime (not space, I'm getting that part now) and different people would see different cross sections of the 4 dimensional rock?

For an appropriate sense of "see", yes. However, that appropriate sense of "see" is *not* the usual sense. What we are calling the 3-D "cross section" of a 4-D object is *not* what any observer actually sees, visually, because of the finite speed of light. Each 3-D cross section is really an abstract construction, not a direct observation.

What determines what their cross section view will be?

Their state of motion relative to the rock.

 

[PeterDonis]

First you said a cars existence is 4 dimensional just like Dr. Donis said. But then why did you say relativity doesn't say anything like that?

I think he was responding to the part where you said "we don't see all of the dimensions", which is indeed incorrect, as I (and others, I think) have already posted.

(Btw, I'm not actually a Dr. in either sense--medical or doctoral degree.)

 

[Stricklandjr]

I think he was responding to the part where you said "we don't see all of the dimensions", which is indeed incorrect, as I (and others, I think) have already posted.

(Btw, I'm not actually a Dr. in either sense--medical or doctoral degree.)

Thank you for clearing that up. And after looking at other threads you are definitely a Dr. of physics in my book.

 

[Stricklandjr]

For an appropriate sense of "see", yes. However, that appropriate sense of "see" is *not* the usual sense. What we are calling the 3-D "cross section" of a 4-D object is *not* what any observer actually sees, visually, because of the finite speed of light. Each 3-D cross section is really an abstract construction, not a direct observation.



Their state of motion relative to the rock.

Im going to try doing a drawing kind of like I would do in drafting class. I had to make it up in Paint since I don't have I-deas on my computer. I remember in jr. college physics drawing diagrams of things moving along the horizontal direction when time is changing so I hope this picture is kind of like you are talking about for a cross section view. Please tell me where I have gone wrong. I labeled the cross section as A-A like we did in drafting.

 

[Stricklandjr]

A car is 3-dimensional, but its entire existence from when it's built to when it's destroyed, is 4-dimensional. If you want to understand this better, you should learn about spacetime diagrams (also called Minkowski diagrams).

I have to thank you for that one Mr. Fredrik. I am now looking at the spacetime diagram subjects and you really put me on to something there. I should have done this before drawing that picture for Dr. Donis in the last post because I see it is not right. I can see that the world tube for someone in motion doesn't have the space direction perpendicular to his time direction. That's something I will have to really study on. How do you know how much to slant the space axis compared to the angle of the moving world tube?

 

[PeterDonis]

How do you know how much to slant the space axis compared to the angle of the moving world tube?

For 2-dimensional spacetime diagrams like the ones you're drawing (i.e., 1 dimension of space and 1 dimension of time), the space and time axes both make the same angle with the worldlines of light rays, which are 45-degree lines. So in your diagram, since the world tube of the person looking at the rock tilts up and to the right, the lines of simultaneity for that observer (i.e., the "slices" of the cross sections) will tilt to the right and up, so that they and the world tube are symmetrical about 45-degree lines going up and to the right.

 

[Fredrik]

How do you know how much to slant the space axis compared to the angle of the moving world tube?

See pages 5-8 in Schutz. (Especially fig. 1.4 on p. 8).

The early parts of that book should be readable even if you don't have much of a mathematical background. The book by Taylor & Wheeler is an alternative with less math.

 

[Stricklandjr]

See pages 5-8 in Schutz. (Especially fig. 1.4 on p. 8).

The early parts of that book should be readable even if you don't have much of a mathematical background. The book by Taylor & Wheeler is an alternative with less math.

Thank you Mr. Fredrik. I'm going to see about getting that book.

 

[Stricklandjr]

For 2-dimensional spacetime diagrams like the ones you're drawing (i.e., 1 dimension of space and 1 dimension of time), the space and time axes both make the same angle with the worldlines of light rays, which are 45-degree lines. So in your diagram, since the world tube of the person looking at the rock tilts up and to the right, the lines of simultaneity for that observer (i.e., the "slices" of the cross sections) will tilt to the right and up, so that they and the world tube are symmetrical about 45-degree lines going up and to the right.

I really think I am getting closer now. I put in a 45 degree line and then made the moving space direction the same angle and found the section A-A. But why is the light ray at 45 degrees and why did the moving guy's view slant to the same angle? Either you are beginning to make a genius out of me or else a little knowledge is going to be a dangerous thing for me. I'm studying on the spacetime diagram subject suggested by Mr. Fredrick and am seeing some things I have to study hard on before knowing enough to ask my next questions about what happened to the scaling of the distances when I rotated the moving guy's space direction.

 

[WannabeNewton]

The speed of the light ray is given by $\frac{\Delta x}{\Delta t} = 1$ implying $\frac{\Delta t}{\Delta x} = 1$ which is the slope of the light ray as depicted on the diagram. Recall that this is then equal to $\tan\theta$ where $\theta$ is the angle that the ray makes with the horizontal. $\tan\theta = 1\Rightarrow \theta = \frac{\pi }{4}$ in the first quadrant.

 

[PeterDonis]

I really think I am getting closer now. I put in a 45 degree line and then made the moving space direction the same angle and found the section A-A.

Yes, this looks correct.

But why is the light ray at 45 degrees

Because we are using units in which the speed of light is 1; for example, if the space axis is in meters, the time axis is in meters of light-travel time, i.e., one unit of time is about 3.3 nanoseconds, since that's the time it takes for light to travel 1 meter. (Or we could have time in seconds and space in light-seconds = 300,000 km, or time in years and space in light-years; whatever is appropriate to the problem.) I slipped those units in on you without telling you. But those are the most commonly used units in relativity, so it's good to get used to them.

and why did the moving guy's view slant to the same angle?

There are several ways to answer this, but it basically comes down to the fact that that's what makes the math work consistently. The reason it has to be different from the ordinary Euclidean case, where the two axes stay perpendicular, is that there is a minus sign in the spacetime interval formula; this is the spacetime analog to the Pythagorean theorem:

$$s^2 = x^2 - c^2 t^2$$

Notice the minus sign, where in the ordinary Pythagorean theorem we have a plus sign: $s^2 = x^2 + y^2$. The minus sign leads to a number of key differences from the Euclidean case, and the way the space axis tilts for a moving observer is one of them.

 

[Stricklandjr]

The speed of the light ray is given by $\frac{\Delta x}{\Delta t} = 1$ implying $\frac{\Delta t}{\Delta x} = 1$ which is the slope of the light ray as depicted on the diagram. Recall that this is then equal to $\tan\theta$ where $\theta$ is the angle that the ray makes with the horizontal. $\tan\theta = 1\Rightarrow \theta = \frac{\pi }{4}$ in the first quadrant.

Got it. This is really important. I'm going to mull this over a lot. The 4 dimensional light ray is always at 45 degrees from time and space directions. Now it's really getting in touch with nature and the scheme of things. Lot's of thinking to do. Thanks a lot Mr. WannabeNewton (you must already be a Newton).

 

[WannabeNewton]

Thanks a lot Mr. WannabeNewton

Haha it sounds so awkward when you call me Mr because I'm still a young'un. You can call me scrappy

(you must already be a Newton).

Sadly, a feat only Newton himself could achieve

 

[Stricklandjr]

Yes, this looks correct.

Thanks only to you and the others.

Because we are using units in which the speed of light is 1; for example, if the space axis is in meters, the time axis is in meters of light-travel time, i.e., one unit of time is about 3.3 nanoseconds, since that's the time it takes for light to travel 1 meter. (Or we could have time in seconds and space in light-seconds = 300,000 km, or time in years and space in light-years; whatever is appropriate to the problem.) I slipped those units in on you without telling you. But those are the most commonly used units in relativity, so it's good to get used to them.

I'm really starting to get on to this. But I am overwhelmed with the idea that the universe is organized with the ray of light always at the 45 degree angle for the guy with perpendicular time and space directions and that the time and space angles for the moving guy are the same on either side of the light ray. That just seems really awesome for some reason I can't really fathom right now. I can see how the moving world tube must be slanted due to its speed, but then for the moving guys space direction in spacetime to know just how much to rotate up to make those angles equal is something I will study very long and hard on.

There are several ways to answer this, but it basically comes down to the fact that that's what makes the math work consistently. The reason it has to be different from the ordinary Euclidean case, where the two axes stay perpendicular, is that there is a minus sign in the spacetime interval formula; this is the spacetime analog to the Pythagorean theorem:

$$s^2 = x^2 - c^2 t^2$$

Notice the minus sign, where in the ordinary Pythagorean theorem we have a plus sign: $s^2 = x^2 + y^2$. The minus sign leads to a number of key differences from the Euclidean case, and the way the space axis tilts for a moving observer is one of them.

O.K. Dr. Donis I will really start studying on this because it seems like a really big point in all this. I will even get out my algebra book and review the Pythagorean theorem.

One question I have is it looks like when I try to follow the internet articles on spacetime diagrams that something strange is going on with the scaling of space distances between perpendicular directions for time and space compared to the slanted moving guy's distances. I notice the length is different for different cross sections like they are in drafting. Is that really what happens?

 

[Fredrik]

something strange is going on with the scaling of space distances between perpendicular directions for time and space compared to the slanted moving guy's distances. I notice the length is different for different cross sections like they are in drafting. Is that really what happens?

Check out page 15 in Schutz. Curves on which $-t^2+x^2$ is constant look the same in all coordinate systems. So if you pick a point on the t axis that's 1 second away from the origin, and follow such a hyperbola from that point to the point where it intersects the t' axis, you have found an event that is 1 second away from the origin in the other coordinate system.