B 4 four momentum energy component direction

[whatif]

So each one is equally valid physically. It is purely a matter of convention.

That is OK, so long as it is understood, that implicitly by convention, the unit of a property does not necessarily have anything to say about the property. That is, the expression of a number, together with units, does not necessarily distinguish between the physical properties to which it is applied.
I do not think that is a matter of semantics.

 

[whatif]

Both of these possibilities are unlikely enough that you might want to give more weight to another possibility

It would be harder but not impossible to construct a consistency theory without using geometric dimensions.

Everything is understood, so long it is understood that the expression of a number, together with dimensions, does not necessarily distinguish between the physical properties to which it is applied. (I hope my use of 'units' did not cause confusion in the above message).​

 

[Nugatory]

It would be harder but not impossible to construct a consistency theory without using geometric dimensions.

Can you produce such a theory? Or failing that, do you have a plausible suggestion for how such a theory might be constructed?

 

[Dale]

implicitly by convention, the unit of a property does not necessarily have anything to say about the property. That is, the expression of a number, together with units, does not necessarily distinguish between the physical properties to which it is applied.

I am not sure I understand what this means, but I think maybe I agree.

 

[whatif]

Can you produce such a theory?

I could not have produced the theory of relativity in the first place.

In principle, it is just a question of accounting. For example, to start off, I would copy the theory but rather than using meters for time I would use seconds. That means that whenever time was used used in calculations it would require the use of the constant for the speed of light, c. The principal is simple, the application is tedious, which makes it more prone to mistakes.

From Taylor and Wheeler:

The parable of the surveyors cautions us to use the same to measure both time and space.

If you know of the context and understand the reference, then I would say that to abide by that caution is not a necessity. Also, the caution does not quite fit with the parable (no analogy is perfect). Time is a different property than space. Timelike is different than spacelike. If everyone used seconds for time and meters for space then they would still agree.

Everything is understood, so long it is understood that the expression of a number, together with dimensions, does not necessarily distinguish between the physical properties to which it is applied (using seconds for time and meters for space distinguishes between time and space).

 

[Ibix]

using seconds for time and meters for space distinguishes between time and space

How do you distinguish between time and space? What I call "time" another frame will call "a bit of time and a bit of space". Should that other frame use seconds to describe intervals in my time direction, even though it includes distance in space?

I think a sensible answer to this will conclude that you mean to distinguish between timelike and spacelike, not between time and space. And distinguishing between timelike and spacelike is what the metric does. Why, then, would you need your unit system to do it too?

The whole argument feels to me like arguing that we ought to use fathoms for vertical distances and meters for horizontal ones because you can't measure horizontal distance with a plumb line. You can, of course, measure any distance with a ruler - you just hold it pointing horizontally or vertically. Similarly you can measure intervals with a radar set - just say you know the distance and it's a light clock, or say you know the time and it's a radar set.

 

[whatif]

Why, then, would you need your unit system to do it too?

You don't.

I am used to dealing with measurements that do reflect the properties to which they are applied and I dare say that applies to many people new to relativity. It just needs to be realized that a meter of time is not the same as a meter of space. "a bit of time and a bit of space" is a mix of two different properties. Timelike and spacelike are different properties.

The whole argument feels to me like arguing that we ought to use fathoms for vertical distances and meters for horizontal ones because you can't measure horizontal distance with a plumb line.

That is not a good analogy, because it is using different units of measurement for the same property.

In my view, I am not arguing semantics.

 

[Ibix]

"a bit of time and a bit of space" is a mix of two different properties.

To you. In my frame it's just time. So is it a mix of different properties or not?

That is not a good analogy, because it is using different units of measurement for the same property

But height and width aren't the same property. You can't use a plumb line to measure width. If you use a ruler to measure height, you can't use it to measure width without rotating through 90°.

 

[m4r35n357]

Is it just me or is this thread veering away from relativity altogether?

 

[Ibix]

Is it just me or is this thread veering away from relativity altogether?

Depends on your unit system.

 

[whatif]

To you. In my frame it's just time. So is it a mix of different properties or not?

You can regard it as a mix with zero spacelike part or not a mix as you choose. If you choose the latter then it does not have to be a mix.

But height and width aren't the same property. You can't use a plumb line to measure width. If you use a ruler to measure height, you can't use it to measure width without rotating through 90°.

Timespace has one timelike part and three spacelike parts. Height and width have the same property of distance between the points of measurement, as does depth.

You seem to now be heading towards indicating that the units of measurement of timelike and spacelike parts do, indeed, relate to a specific property.

 

[whatif]

Is it just me or is this thread veering away from relativity altogether?

I will make no more contribution.

 

[PeterDonis]

It just needs to be realized that a meter of time is not the same as a meter of space.

@Ibix's comment about "timelike" and "spacelike" being more appropriate here is valid. "Time" and "space" are relative, so it makes no sense to say that a "meter of time" is always a meter of just time, since in a different frame it will, as @Ibix says, be a mixture of time and space. But whether a given spacetime interval is timelike, spacelike, or null is invariant, independent of your choice of frame. So it makes sense to say "a meter of timelike interval" is distinct from "a meter of spacelike interval". Note that that, in itself, does not mean it must be true that a meter of timelike interval is fundamentally not the same as a meter of spacelike interval; it simply means it makes sense to formulate the question and consider it (whereas it doesn't even make sense to formulate the question of whether "a meter of time" is fundamentally different from "a meter of space", since "time" and "space" have no invariant meaning).

Timespace has one timelike part and three spacelike parts.

More precisely: any orthogonal set of basis vectors in spacetime must have one timelike basis vector and three spacelike basis vectors. But there is nothing that requires you to pick an orthogonal set of basis vectors; the only requirement for basis vectors is that they have to be linearly independent, which is a much weaker condition.

Also, not all vectors in spacetime are timelike or spacelike; some are null. Does that mean spacetime has to have a "null part" as well as one timelike and three spacelike parts? How can that be when spacetime has only four dimensions?

 

[sweet springs]

ake the components (magnitudes) of the 4 momentum, 4 vector (I am assuming it is known where these equations come from):

p^\mu=mcu^\mu where 4-velocity u is
u^\mu=\frac{dx^\mu}{cd\tau} where tau is proper time.

It seems x^\mu, u^\mu and p^\mu are perpendicular from this relation.