What would be the form of this world line?

[student34]

OK, maybe some other suggestions then. Let’s say that you are building a deck and need to mark a straight line on the deck. How do you physically make a reasonably straight line on a floor? (Assuming the floor is flat)

Spoiler

Here is just one common method How does this simple device make a straight line?

Use a straight instrument?

 

[PeterDonis]

Use a straight instrument?

How do you know the instrument is straight?

 

[student34]

How do you know the instrument is straight?

I do not know.

 

[PeterDonis]

I do not know.

Then that's where you need to start. This is what you should be asking questions about.

 

[student34]

Then that's where you need to start. This is what you should be asking questions about.

Then I would very much like to know the answer to your question.

 

[Dale]

Use a straight instrument?

Yes, that is one possibility. A straight edge will let you know if your line turns left or right. That is one definition of a straight line, a line which doesn’t turn at any point.

Another definition is “the shortest path between two points is a straight line”. So for drawing a straight line between two points on a floor you can run a string between them and tighten the string to minimize its length. The resulting path is also a straight line.

It is possible to prove that these two definitions, no turning and shortest distance, are equivalent. So once you have a concept of “distance” you can get a concept of “straight”. And with “straight” you can also get “not straight”, “curved” or “angled”. Basically, all of geometry follows from the concept of distance.

Does that make sense?

 

[PeterDonis]

Then I would very much like to know the answer to your question.

The general answer is called "parallel transport", and the property (which has been mentioned before in this thread) that a geodesic--the generalization of "straight line" that we need--parallel transports its own tangent vector along itself.

In the simplest case, compare a line on a plane with a circle on a plane. Imagine an arrow (a vector) at some particular point on the line that points along (tangent to) the line, and an arrow (a vector) at some particular point on the circle that points along (tangent to) the circle.

Now imagine moving along the line or circle and seeing what happens to the arrow (vector) to keep it pointing in the right direction to stay tangent to the line or circle. In the case of the line, nothing at all has to happen to it; the arrows tangent to the line at every point are all parallel to each other, so transporting the arrow at one point along the line parallel to itself just gives you the arrow at any other point on the line. Thus, the tangent vector to the line is transported parallel to itself along the line. That is the invariant sense in which the line is "straight". (The equipment used by surveyors is an actual physical realization of what I have been describing: its intent is precisely to realize "straight lines" in this sense.)

When moving along the circle, however, parallel transporting the arrow at one point along the circle does not give you the arrow at another point. For example, suppose we start at a point on the circle where the arrow is pointing "up" (towards the "top" of the plane). Transporting the arrow parallel to itself means the arrow keeps on pointing "up". But the arrows tangent to the circle at other points on the circle do not point "up"; they point in different directions ("left" or "down" or "right"). So the tangent vector to the circle is not transported parallel to itself along the circle. That is the invariant sense in which the circle is "curved".

There is a lot more mathematics lurking beneath the above, which is needed in order to make everything rigorous and give mathematical conditions that can be checked for much more complicated cases where there is no easy way to visualize what is going on (such as general curved 4D spacetimes). Also, for spacetime in relativity, there is the additional physical rule that the reading on an accelerometer following a worldline indicates the worldline's curvature; in other words, that the accelerometer is the spacetime equivalent of the surveyor's equipment in ordinary space.

 

[PeterDonis]

Yes, that is one possibility. A straight edge will let you know if your line turns left or right.

But that only helps if you already know the straight edge is straight. Which just pushes the problem back to how you determine that.

 

[cianfa72]

Also, for spacetime in relativity, there is the additional physical rule that the reading on an accelerometer following a worldline indicates the worldline's curvature; in other words, that the accelerometer is the spacetime equivalent of the surveyor's equipment in ordinary space.

Basically the accelerometer at each point (event) along the given worldline "looks" on a small region around it and check if the tangent vector is actually parallel transported to itself or not.

 

[Dale]

But that only helps if you already know the straight edge is straight. Which just pushes the problem back to how you determine that.

Yes, which is why I included the subsequent sentence describing parallel transport in layman's terms. I wanted to focus on the "shortest distance" definition of straight, and just barely mention parallel transport and the fact that they are equivalent.

 

[student34]

Yes, that is one possibility. A straight edge will let you know if your line turns left or right. That is one definition of a straight line, a line which doesn’t turn at any point.

Another definition is “the shortest path between two points is a straight line”. So for drawing a straight line between two points on a floor you can run a string between them and tighten the string to minimize its length. The resulting path is also a straight line.

It is possible to prove that these two definitions, no turning and shortest distance, are equivalent. So once you have a concept of “distance” you can get a concept of “straight”. And with “straight” you can also get “not straight”, “curved” or “angled”. Basically, all of geometry follows from the concept of distance.

Does that make sense?

Yes, thanks

 

[student34]

The general answer is called "parallel transport", and the property (which has been mentioned before in this thread) that a geodesic--the generalization of "straight line" that we need--parallel transports its own tangent vector along itself.

In the simplest case, compare a line on a plane with a circle on a plane. Imagine an arrow (a vector) at some particular point on the line that points along (tangent to) the line, and an arrow (a vector) at some particular point on the circle that points along (tangent to) the circle.

Now imagine moving along the line or circle and seeing what happens to the arrow (vector) to keep it pointing in the right direction to stay tangent to the line or circle. In the case of the line, nothing at all has to happen to it; the arrows tangent to the line at every point are all parallel to each other, so transporting the arrow at one point along the line parallel to itself just gives you the arrow at any other point on the line. Thus, the tangent vector to the line is transported parallel to itself along the line. That is the invariant sense in which the line is "straight". (The equipment used by surveyors is an actual physical realization of what I have been describing: its intent is precisely to realize "straight lines" in this sense.)

When moving along the circle, however, parallel transporting the arrow at one point along the circle does not give you the arrow at another point. For example, suppose we start at a point on the circle where the arrow is pointing "up" (towards the "top" of the plane). Transporting the arrow parallel to itself means the arrow keeps on pointing "up". But the arrows tangent to the circle at other points on the circle do not point "up"; they point in different directions ("left" or "down" or "right"). So the tangent vector to the circle is not transported parallel to itself along the circle. That is the invariant sense in which the circle is "curved".

There is a lot more mathematics lurking beneath the above, which is needed in order to make everything rigorous and give mathematical conditions that can be checked for much more complicated cases where there is no easy way to visualize what is going on (such as general curved 4D spacetimes). Also, for spacetime in relativity, there is the additional physical rule that the reading on an accelerometer following a worldline indicates the worldline's curvature; in other words, that the accelerometer is the spacetime equivalent of the surveyor's equipment in ordinary space.

Interesting, thank you

 

[Dale]

Yes, thanks

Ok, so once you understand that physical geometry is about distance and with that one concept you can get straight, curved, angles, and all other geometric features, then you can go to coordinates.

My table top has a certain physical size, a length and width and an area. On the top of the table I can draw a grid, and I can tell if the grid lines are straight as follows: I can put a pin in any two points on one of the lines, and tighten a string between the pins. If the string follows the grid line then it is straight. Otherwise it is curved. So it is unambiguous if my grid lines are straight or not.

Furthermore, no amount of drawing or redrawing grid lines changes the length of a string along the edge of the table, or across the diagonal. And it also doesn’t change the area of the table top. Coordinates can be drawn, but they only control the numbers I use to locate objects on my desk, not the physical distance between those objects.

All of this stems from being able to measure the distance along any path. Clear?

 

[student34]

Ok, so once you understand that physical geometry is about distance and with that one concept you can get straight, curved, angles, and all other geometric features, then you can go to coordinates.

My table top has a certain physical size, a length and width and an area. On the top of the table I can draw a grid, and I can tell if the grid lines are straight as follows: I can put a pin in any two points on one of the lines, and tighten a string between the pins. If the string follows the grid line then it is straight. Otherwise it is curved. So it is unambiguous if my grid lines are straight or not.

Furthermore, no amount of drawing or redrawing grid lines changes the length of a string along the edge of the table, or across the diagonal. And it also doesn’t change the area of the table top. Coordinates can be drawn, but they only control the numbers I use to locate objects on my desk, not the physical distance between those objects.

All of this stems from being able to measure the distance along any path. Clear?

Yes that makes sense.

 

[Dale]

Yes that makes sense.

OK, last one then. Now we just move what we have already learned about geometry, lengths, and coordinates and we apply it to spacetime. For simplicity we will neglect gravity, so spacetime is flat, we are off in intergalactic space far from any significantly gravitating object (but do keep in mind that this all generalizes fairly easily to gravity).

Now, on the table, although you can use any coordinates you like, there do exist coordinates such that the distance in terms of the coordinates is given by the Pythagorean theorem: $ds^2=dx^2+dy^2$. Now, this distance is the same as the distance physically measured by a string or a ruler, and if you transform to any other coordinates the formula will change but $ds$ will stay the same. So $ds$ is what we call an invariant.

If we move to spacetime we have almost the same thing. Now, a point particle becomes a line in spacetime, and just like a line in space this line in spacetime has all sorts of geometric features, such as length, straightness, angles, etc. The only difference is that instead of using the Pythagorean theorem to figure out that geometry, we use the Minkowski metric $ds^2=-c^2 dt^2+dx^2 + dy^2 + dz^2$. This is very similar to the Pythagorean theorem, except that there is an extra term $-c^2 dt^2$ that is negative and is related to time. This is "distance" in spacetime and it is called the spacetime interval. This is the invariant measure of geometry that remains regardless of how you draw your coordinates.

Now, when $ds^2$ is negative we call the interval "timelike", and then $d\tau^2 = -ds^2/c^2$ is called the proper time. This is the time directly measured on a clock that moves along some path through spacetime. In other words, if an object's worldline is a string, then a clock measures the length of the string. Similarly, an accelerometer directly measures if the worldline turns any direction. So an accelerometer is equivalent to a straightedge, allowing us to measure bending of a worldline, and a clock is similar to a ruler, allowing us to measure the length of the worldline. We can use both to determine if a worldline is straight or not, as discussed above. A straight worldline doesn't turn anywhere (the accelerometer reads 0) and a straight worldline is the longest interval between two events. (note that for timelike paths a straight worldline is the longest interval instead of shortest interval because of the - sign in the metric).

 

[student34]

OK, last one then. Now we just move what we have already learned about geometry, lengths, and coordinates and we apply it to spacetime. For simplicity we will neglect gravity, so spacetime is flat, we are off in intergalactic space far from any significantly gravitating object (but do keep in mind that this all generalizes fairly easily to gravity).

Now, on the table, although you can use any coordinates you like, there do exist coordinates such that the distance in terms of the coordinates is given by the Pythagorean theorem: $ds^2=dx^2+dy^2$. Now, this distance is the same as the distance physically measured by a string or a ruler, and if you transform to any other coordinates the formula will change but $ds$ will stay the same. So $ds$ is what we call an invariant.

If we move to spacetime we have almost the same thing. Now, a point particle becomes a line in spacetime, and just like a line in space this line in spacetime has all sorts of geometric features, such as length, straightness, angles, etc. The only difference is that instead of using the Pythagorean theorem to figure out that geometry, we use the Minkowski metric $ds^2=-c^2 dt^2+dx^2 + dy^2 + dz^2$. This is very similar to the Pythagorean theorem, except that there is an extra term $-c^2 dt^2$ that is negative and is related to time. This is "distance" in spacetime and it is called the spacetime interval. This is the invariant measure of geometry that remains regardless of how you draw your coordinates.

Now, when $ds^2$ is negative we call the interval "timelike", and then $d\tau^2 = -ds^2/c^2$ is called the proper time. This is the time directly measured on a clock that moves along some path through spacetime. In other words, if an object's worldline is a string, then a clock measures the length of the string. Similarly, an accelerometer directly measures if the worldline turns any direction. So an accelerometer is equivalent to a straightedge, allowing us to measure bending of a worldline, and a clock is similar to a ruler, allowing us to measure the length of the worldline. We can use both to determine if a worldline is straight or not, as discussed above. A straight worldline doesn't turn anywhere (the accelerometer reads 0) and a straight worldline is the longest interval between two events. (note that for timelike paths a straight worldline is the longest interval instead of shortest interval because of the - sign in the metric).

Yeah thanks for that. I will try to keep all this in mind moving forward.