4 four momentum energy component direction

[whatif]

"Energy is only the time part of the momenergy 4-vector." = "Energy is the timelike component of the momentum 4vector."

So you are insisting that ‘time part’ equates to ‘timelike component’ and there is no such thing as a timelike vector. Is that defined anywhere, in particular, anywhere that I should think that is what Taylor and Wheeler meant?

I haven't said it is a dimensionless number. It was said to you several times before, that units are irrelevant in the context of this discussion.

Just a number is just a number. Numbers do not mean anything without application. Units are irrelevant to the abstract mathematics, so long as the units have been made alike beforehand (e.g. kilograms for both energy and momentum). So long as the concepts are understood, the mathematics is made simpler. After the abstract mathematics is done, units have to be reapplied for physical meaning, and it has to be understood that a kilogram of energy is not the same as a kilogram of momentum (and a kilogram of the space part of the 4 momentum, 4 vector, is not the same as a kilogram of the time part of part of the 4 momentum, 4 vector).

I think you mean timelike component of a vector. So you are claiming that energy is irrelevant in Newtonian mechanics?

By your interpretation of timelike, the word ‘component’ is redundant here (see first quote). Just saying (you might just be reinforcing the point).

I do not mean timelike component of a vector and I am not claiming energy is irrelevant to Newtonian mechanics. I mean that ‘timelike’ is a notion that applies to relativistic mechanics and not Newtonian mechanics. Ideas of Newtonian mechanics are revised in relativistic mechanics.

 

[whatif]

"Energy" is just a word. You can't change the physics by changing the meaning you assign to a word.

You are right, they can be called apples and oranges and all kinds of fruit and it makes no difference to the mathematical relationships. Likewise, there are not really 2 interpretations, each of which has its own use. We just choose words that help keep a sense of what we are talking about.

Perhaps this is part of your problem. The whole 4-momentum, a 4-vector, has two important properties:

(1) It points in the direction in spacetime that lies along the object's worldline, i.e., its direction, in spacetime, is the same as the direction in spacetime in which the object's worldline points (more precisely, the future direction of the worldline).

(2) Its magnitude is the rest mass (also called the invariant mass, which is more relevant in this discussion) of the object.

That is what I thought. Given that understating, “whole 4 momentum” did not seem helpful in the context in which it was raised.

 

[whatif]

I'm surprised that the term component vector hasn't come up in this thread. See "Pitfall Prevention 3.2" here: https://books.google.com/books?id=XgweHqlvtiUC&pg=PA59

I started with that term and it was the source of a big problem. I found the term in your reference. Where I found it in the text of your referrence, it was written in italics to distinguish it from its other use of the word ‘component’. The term makes good sense, however, the word ‘component’ here is restricted to correspond with the other meaning of the word ‘component’ used in your reference. In other words, the term 'component vector' and its meaning is effectively not acceptable here.

 

[PeterDonis]

In other words, the term 'component vector' and its meaning is effectively not acceptable here.

It's not that the term is "not acceptable" here, it's that we didn't understand what you meant by it earlier in the thread. It helps a lot if you give a reference that uses the term the way you are using it, as @SiennaTheGr8 did. As you note, that reference explains exactly the distinction between "component vectors" and "components" that was discussed earlier in this thread.

 

[Dale]

I'm surprised that the term component vector hasn't come up in this thread.

I am glad you posted that. I didn’t know the term.

 

[whatif]

It helps a lot if you give a reference that uses the term the way you are using it, as @SiennaTheGr8 did.

It is the very first term used. I then used the word 'component' in that context, which created an issue. Why would I give a reference when I do not know that there is a necessity (especially when I thought it was commonly used term).

(Actually, the meaning is acceptable here but I would try and avoid the term if I knew beforehand the problem that would cause with the word 'component'.)

 

[PeterDonis]

It is the very first term used.

In that particular textbook, yes. I have not seen it in other textbooks I have read. (Apparently @Dale hasn't either given his post a little bit ago.)

Why would I give a reference when I do not know that there is a necessity

The fact that you were running into problems getting others to understand what you were saying might have helped to indicate that a reference would be helpful.

(especially when I thought it was commonly used term).

The fact that you were running into problems getting others to understand what you were saying, particularly since three of them were Mentors, might have helped to indicate that the term was not as commonly used as you thought it was.

 

[whatif]

I'm surprised that the term component vector hasn't come up in this thread.

It is the very first term used. I then used the word 'component' in that context, which created an issue.

In that particular textbook, yes.

Not in the textbook but the very first term I used at the start of this thread (/conversation/in the first message). I have never known about that textbook before now. It is also a term that in common English conveys the correct concept for the topic.

I fully appreciate the importance of using common terminology, that my language has been a problem and that I have made mistakes. I tried to not use the word when I understood the problem. On the other hand, I would expect the experts to pick up where I was using a word that made sense in common English and used appropriately in that sense, but had a special meaning to them rather than just saying I was wrong and/or confused. (Its a bit like computer program writers writing the help for a computer application that is not much help to anybody but other computer programmers).

Anyway, it has been instructive to me, if that is any consolation.

Thank you.

 

[SiennaTheGr8]

In that particular textbook, yes. I have not seen it in other textbooks I have read. (Apparently @Dale hasn't either given his post a little bit ago.)

Just had a quick look at some other popular university physics texts. Young/Freedman (13th) use the term component vector. Halliday/Resnick (9th) use vector component (as opposed to scalar component, which is what they always mean by plain component).

While we're on the subject of terminology...

The "parallel" / "antiparallel" vector nomenclature bugs me. Better would be "codirectional" / "contradirectional," so that "parallel" can be used unambiguously as a catchall for both. Here's a snippet from p. 76 of that Young/Freedman edition, regarding the resolution of the acceleration vector into components parallel and perpendicular to the velocity vector:

If the speed is decreasing, the parallel component has the direction opposite to $\vec v$ ...

This is doubly sloppy. First, they use component where they mean component vector. Second, they're using parallel as a catchall, but their earlier definition of parallel vectors was restricted to vectors that point in the same direction. According to their own nomenclature, the component vector in question is antiparallel to $\vec v$, not parallel.

 

[PeterDonis]

Not in the textbook but the very first term I used at the start of this thread

Ah, got it.

 

[PeterDonis]

Just had a quick look at some other popular university physics texts.

Yes, but bear in mind that this is the relativity forum, not the general physics forum. When I referred to not seeing the term "component vector" in textbooks, I was referring to relativity textbooks. The terminology in relativity is not always the same as that in pre-relativity physics, which is what the introductory physics textbooks you refer to generally start out with. If they even cover relativity at all, it will usually be towards the end of the book, at least the ones that I have seen (but it's been a while).

 

[SiennaTheGr8]

Yes, but bear in mind that this is the relativity forum, not the general physics forum. When I referred to not seeing the term "component vector" in textbooks, I was referring to relativity textbooks.

Yep, and I don't recall seeing that (useful) terminology in relativity books, either.

It also wouldn't surprise me if the terminology were relatively new to the general physics texts.

 

[Dale]

When I referred to not seeing the term "component vector" in textbooks, I was referring to relativity textbooks.

I don’t recall having seen it in Newtonian physics either. Until this thread I hadn’t even worried about it because I can usually tell from context if it is the vector or real number version that is intended.

 

[whatif]

It also wouldn't surprise me if the terminology were relatively new to the general physics texts.

It is a long time since I dealt with vectors, but I learned the term a long time ago.

And on the subject of terminology, when the same units are used for the spacelike and timelike parts they are the same in name only. For example, a meter of time is different than a meter of space and a kilogram of spacelike momentum is different than a kilogram of timelike momentum. It may be preferred that I write that a timelike meter is different than a spacelike meter and timelike kilogram is different than spacelike kilogram.

 

[Nugatory]

when the same units are used for the spacelike and timelike parts they are the same in name only. For example, a meter of time is different than a meter of space

A kilogram of gold is different than a kilogram of lead... but if I take a sealed box containing one or the other and drop it on your toes the effect will be similar in more than just name.

Phrased that way, it seems at first that I'm just being flippant. But there's a real point here: a kilogram is a kilogram, whether it's measuring lead or gold. If it weren't, we wouldn't (for example) be able to recognize $F=ma$ as a valid statement about what forces do to the objects they act on and we wouldn't have classical mechanics.

Time and distance work the same way. They are as different from one another as lead and gold, but just as quantities of lead and gold are both measured in the same units of mass, the intervals between points in space and the intervals between points in time are both measured in units of length.

 

[whatif]

But there's a real point here: a kilogram is a kilogram, whether it's measuring lead or gold.

With respect, no analogy is perfect but that does not serve the analogy well. I will take on board that physicists treat the application of the same units applied to timelike and spacelike properties to be the same in all respects and not just name. However, I need a far better explanation.

Time and space are put on an equal footing but they are not the same nature. Lead and gold are made of the same particles that have the same nature to which kilograms is applied.

One meter of time is the time it takes for light to travel a meter of space; it is not a meter of space, in my view. One second of space is the distance light travels in a second; it is not a second of time, in my view. Other timelike units are made the same as spacelike units by treating speed as a ratio with the speed of light, thereby eliminating the units of speed which serves a mathematic purpose but, in my view, modifies the meaning of the units that remain; because timelike and spacelike properties have different natures.

 

[Ibix]

One meter of time is the time it takes for light to travel a meter of space; it is not a meter of space, in my view.

Of course, someone using another frame won't say it's a meter of time. He'll say it's a mix of time and space. So an interval that you say is just time is a mix of time and space for someone else. How can they be "not the same nature" when they're the same interval?

 

[whatif]

How can they be "not the same nature" when they're the same interval?

From Taylor and Wheeler;

Equal footing, yes; same nature, no.

While I am quoting Taylor and Wheeler, I am not saying that Taylor and Wheeler do not agree with you about the units being the same in all respects. The may and they may not. I am not sure they are committal in what I have read.

I do not have a problem with them being of different natures when they have the same timespace interval. That is the nature of measuring timespace in different frames. It would just mean that the speed of light, c, determines a relationship between measured spacelike and timelight properties; each being of a different nature. I see no incompatibility with timespace being a mix of time and space. The way the maths deals with the units tells me that they are of a different nature. It seems to me that the units are only irrelevant to the mathematics after the same units in name, but not nature, have been applied and before the units are reapplied.

I will take on board that physicists treat the application of the same units applied to timelike and spacelike properties to be the same in all respects and not just name. However, I need a far better explanation.

 

[Dale]

timelike and spacelike properties have different natures.

For example, they are measured using different physical devices.

 

[Ibix]

For example, they are measured using different physical devices.

I'd say that's arguable. A light clock is just a radar set with a target at a fixed distance.

 

[SiennaTheGr8]

Of course, someone using another frame won't say it's a meter of time. He'll say it's a mix of time and space. So an interval that you say is just time is a mix of time and space for someone else. How can they be "not the same nature" when they're the same interval?

From Taylor and Wheeler;

Equal footing, yes; same nature, no.

What precisely "the same nature" means is up for debate, but that's just semantics.

That said, I think Taylor and Wheeler would agree with the sentiment of @Ibix's post. If I remember correctly, the thrust of their very first chapter is that space and time are related in much the same way that forward/backward is related to right/left—i.e., they're not the "same thing," but what distinguishes one from the other is a matter of perspective.

I also agree with @Nugatory's point on units. Distinguishing a "meter of space" from a "meter of time" is akin to distinguishing a "kilogram of gold" from a "kilogram of lead." The unit is the unit, regardless of what you're measuring.

A light second is a second. A light meter is a meter. The dimensional analysis is pretty simple:

$[x/c] = \dfrac{ [\textrm{meters}] }{[\textrm{meters}] / [\textrm{seconds}]} = [\textrm{seconds}]$

$[ct] = \dfrac{ [\textrm{meters}]}{[\textrm{seconds}]} \times [\textrm{seconds}] = [\textrm{meters}]$.

 

[whatif]

What precisely "the same nature" means is up for debate, but that's just semantics.

To say that it just semantics is to predetermine the conclusion. If "the same nature" is just a matter of semantics, then why do we experience time and space differently? The experience of dropping a box with a kilogram of lead in it on our toes is the same as the experience of dropping the same box with a kilogram of gold in it on our toes.

The dimensional analysis is pretty simple:

In my view those equations do nothing more than show a relationship, c = xt, where c has the units meters per second. In Newtonian mechanics the speed of light in being treated as having units and in relativistic mechanics the speed of light is unitless by means of using the ratio of speed with respect to the speed light. Without sufficient explanation the mathematical tool is assumed to be more than a mathematical tool.

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

E = mY (where m is mass and Y is meant to be gamma)
p = mvY (where m is mass, v is speed, magnitude of velocity, and y is meant to be gamma)

The units of E are only the same as p because v is the ratio speed with respect to the speed of light.

At low speed it approximates Newtonian mechanics but for rest mass, which is an invariant (I think am using invariant correctly). In Newtonian mechanics:
E = ½ mv2
(a half mass times speed squared, and the units are kilogram meters squared per second squared)

p = mv
(mass times speed and the units are kilogram meters per second)

If we are to take the relativistic interpretation that the units have exactly the same meaning then kilogram meters squared per second squared means the same thing as kilogram meters per second.

However, they only have the same name, kilograms, in relativistic mechanics when relativistic mechanics uses the ratio of speed with respect to the speed of light, c, thereby mathematically eliminating meters per second.

In my view, saying that the units, kilograms, mean exactly the same thing for the timelike momentum and spacelike momentum is to predetermine that the relationship c = x/t is more than just a relationship.

 

[Dale]

I'd say that's arguable. A light clock is just a radar set with a target at a fixed distance.

Sure, it is arguable, but I think there will always be a “with a ...” that makes it a slightly different device.

 

[pervect]

In my view, saying that the units, kilograms, mean exactly the same thing for the timelike momentum and spacelike momentum is to predetermine that the relationship c = x/t is more than just a relationship.

If I'm understanding you correctly, the usual description of what you're talking about is called "using geometric units".

In geometric units, velocities are dimensionless number, fractions of "c".
c is considered to be a units conversion constant, it converts time units to distance units, distance = c * time

When velocities are considered to be dimensionless, mass, mass*velocity, and mass*velocity^2 all have the same units.

In general relativity, we go a bit further than that with geometric units, but I think it'd be confusing to go into this at this point.

Geometric units make things a lot easier, and I _think_ Taylor & Wheeler use them in "Spacetime physics", though I'm not positive and don't have the book available to check.

To use geometric units, one picks a base unit, perhaps a distance unit (the centimeter is a common choice), or a time unit (I'm not sure I recall this choice being made in a textbook, but I know it can be convenient).

If one picks the centimeter as the base unit, one needs to use a time unit of 1cm /c = 33.3 picoseconds
If one picks the second as the base unit, one needs to use light seconds = 299792458 meters as the distant unit.

The math is basically the same, geometric units, once one gets used to them, make calculations much simpler, at the expense of having to do a units conversion back from geometric units to more familiar units at the end of the calculation.

If you're studying from a textbook that uses geometric units, I'd advise you to learn about them. Maybe you don't really like them, but if you want to use that text, you have to come to terms with them enough so that you can understand the text.

If you don't like them - well, have fun keeping tract of all those factors of "c". And you'll need a text that also uses standard units. You may find it hard to follow posts, talk with people, or read papers or textbooks that use geometric units if you're not willing to learn about them.

There's no real reason to avoid learning about them - they're quite useful and simplify a lot of details that are ultimately not very important.

 

[PeterDonis]

I need a far better explanation.

An explanation of what? Of why we experience time and space differently? Nobody knows, beyond our knowledge of the fact. Of why we normally measure time with clocks and space with rulers? Nobody knows, beyond our knowledge of the fact.

It might be that some more comprehensive theory of physics than those we have now will explain these things in terms of some more fundamental difference that we don't currently understand. Or it might be that some more comprehensive theory of physics than what we have now will tell us that these apparent differences are illusions, artifacts of our particular human perceptions, and have no fundamental significance. We won't know which until we have the more comprehensive theory.

 

[whatif]

If you don't like them

It is not that I do not the geometric units. It is about the conclusion of what the mathematics is saying.

well, have fun keeping tract of all those factors of "c".

Everything comes at a cost. The cost is realising the units mean for different things for timelike timespace and spacelike timespace; but that is easy.

 

[whatif]

An explanation of what?

An explanation of why the speed of light provides is more than just a relationship between time and space and makes them the same nature. And yes, an explanation of why we experience time space differently.

Nobody knows, beyond our knowledge of the fact. Of why we normally measure time with clocks and space with rulers? Nobody knows, beyond our knowledge of the fact.

If nobody knows then it is not known as a fact. That we experience time and space differently is evidence. That we measure time with clocks and space with rulers is evidence.

 

[PeterDonis]

If nobody knows then it is not known as a fact.

What isn't known as a fact? As you yourself say, we know by evidence that we experience time and space differently and that we measure time with clocks and space with rulers. We also know that the speed of light in vacuum is a universal constant, independent of the observer's state of motion. And we know that the spacetime model based on all these facts makes predictions that have been confirmed to many decimal places in thousands of experiments. So what, exactly, isn't known as a fact?

 

[whatif]

What isn't known as a fact?

It is known as a fact that the speed of light provides a relationship between time and space. It is not known as fact that the same units in name for the timelike and spacelike mean the same thing.

As you yourself say, we know by evidence that we experience time and space differently and that we measure time with clocks and space with rulers.

That we experience time and space differently is evidence that the units do not mean the same thing for timelike and spacelike parts. That we use rulers to measure space and clocks to measure time is evidence that the units do not mean the same thing for timelike and spacelike parts. These are evidence that timelike and spacelike do not have the same nature and that it is not just a matter of semantics.

There is a relationship between time and space provided by the speed of light. That does not make time and space (or timelike and spacelike) the same thing or the same units in name have the same meaning. It can be said say they are the same thing but if it cannot be explained why they are the same thing then it not a known fact that they are the same thing. It might be claimed that they are the same thing (and it might be said who cares it works) but I need an explanation, not an assumption. The analogy with the property mass in association with gold and lead does not work.

 

[Dale]

It is not known as fact that the same units in name for the timelike and spacelike mean the same thing.

The thing is that units are a convention. As such, we can decide exactly what they mean. If we decide to use units such that they are the same thing then they are the same thing and clocks and rulers measure the same thing by definition. And we can change our mind as often as we like and use different units with different rules whenever we feel like it.