4 four momentum energy component direction

[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.

 

[whatif]

As such, we can decide exactly what they mean.

There should be consistency and it should reflect the physics. I do not think that is happening in this instance. I have explained my position and I will not labour the issue further.

 

[Dale]

There should be consistency and it should reflect the physics.

The reason that units are conventions is that there is more than one way which is both self consistent and also consistent with the physics.

 

[PeterDonis]

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

"Mean the same thing" is ambiguous unless you can suggest how we could test experimentally whether it is the case or not. Can you suggest such a test? You suggest things as "evidence", but if that evidence is not conclusive enough, what further evidence would you need?

 

[whatif]

The reason that units are conventions is that there is more than one way which is both self consistent and also consistent with the physics.

That applies when, for example, you decide what is positive or negative.

It does not matter whether we drive on the right hand side of the road or the left hand side of the road, so long as we all drive on the same side of the road.

On the topic at hand, units are removed from the speed of light by a mathematically tool without consideration of the physics.

 

[whatif]

You suggest things as "evidence", but if that evidence is not conclusive enough, what further evidence would you need?

We disagree about what constitutes sufficient evidence. I would say that:

• timelike properties are clearly different from spacelike properties, for reasons I have given.
• others claim timelike and spacelike properties are the same despite those differences and having to measure and treat them differently. If you claim they are the same, then you preclude testing experimentally that they are different because of the relationship between them and the mathematics used.
• others have taken the unification of space and time into spacetime too literally for no good reason. That is, the time direction is not a fourth space direction.

So be it.

 

[PeterDonis]

We disagree about what constitutes sufficient evidence.

We do? As far as I can tell, you are saying that you need a better explanation, and I'm saying that we don't currently have one. So I don't see how what you are saying disagrees with what I am saying. I just think we are emphasizing different aspects.

 

[pervect]

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

My impression is that you are arguing philosophy and semantics, but if you can suggest an experiment that can resolve exactly how "timelike timespace and spacelike timespace" (which aren't really standard terms and seem a bit awkwards, but I think I can guess what you mean) are "different things", I'm willing to listen.

If you're arguing about something that isn't even in principle experimentally testable, I don't think there's ever going to be a scientific resolution. Issues that are not testable are not in the realm of science, but philosohpy. Philosophical discussions on PF are currently implicitly discouraged, as they are "permitted at the discretion of the mentors." according to the global guidelines. Presumably the mentors currently no objection since the thread is still open, personally though I don't see where the discussion is going anywhere.

I do have some personal bias (for lack of a better term) against philosphy which can probably be traced to the writings of Feynman, see for example "Is Electricity Fire". This bias aside, it is certainly possible to have good philosophical discussions with references to the philosophical literature. PF forums, though, don't seem to be conducive to such high-quality discussions. Frequently we have people stating closely-held personal opinions without much in the way of logical justifications or references to what other people in the field of philosophy have written. This is the primary reason for the limitations on philosophical discussions mentioned in the PF global guidelines.

 

[whatif]

My impression is that you are arguing philosophy and semantics

My impression is that the use of geometric units is turning the notion of units on its head, by giving the same unit to different physical properties. If you call that philosophy and semantics then so be it. My impression is that it is inconsistent.

 

[Dale]

On the topic at hand, units are removed from the speed of light by a mathematically tool without consideration of the physics.

The SI units, for example, are not found in physics. They are the result of a committee that meets to decide what those units mean. People realize that the size of the units is a matter of convention, that the BIPM committee can arbitrarily choose how large a meter is and how large a kilogram is.

What is not as well appreciated is that the dimensionality of a quantity is also a matter of convention that can be decided by the committee. For example, in SI units electrical charge has dimension $Q$ and units $C=A\;s$. However, in cgs units electrical charge has units $statC=g^{1/2}cm^{3/2}s^{-1}$ and dimension $M^{1/2}L^{3/2}T^{-1}$. So not only the size of the units is a matter of convention but also the dimensionality.

There are systems of units that consider mass and charge to have the same dimensionality. There are systems that consider energy and length to have the same units. There are systems that consider horizontal distance and vertical distance to be different.

All of these systems of units can be used to do physics, and there are good well-considered physics reasons for each of them. Each one is consistent with the outcome of all physical experiments. So each one is equally valid physically. It is purely a matter of convention.

 

[Nugatory]

My impression is that the use of geometric units is turning the notion of units on its head, by giving the same unit to different physical properties. If you call that philosophy and semantics then so be it. My impression is that it is inconsistent.

If your impression were correct then either general relativity, which is based on this equivalence, is also inconsistent or it should be possible to construct a consistent theory of gravity that does not require this equivalence.

Both of these possibilities are unlikely enough that you might want to give more weight to another possibility: Your thinking about time and distance is based on a lifetime of experience at non-relativistic speeds, and this experience is misleading you.

 

[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).