B 4 four momentum energy component direction

[whatif]

I have been reading about four momentum. There are four component vectors, three spatial, momentum, components and a time, energy, component. They each have a direction. I understand direction for the momentum components, being in the direction of the respective spatial components of the velocity of an object. I understand that as a component vector the energy has direction. However, I do not understand how to interpret the direction of the energy component and I have not seen an explanation. Does it mean that energy is in the direction of time and if so, is that meant to be so obvious that no one seems to explain it? Alternatively, is it just a mathematical attribute that works and has no particular physical association?

 

[Nugatory]

Yes, the mass-energy component points in the timelike direction - although that's actually sort of backwards - it's probably better to say the we identify the timelike component with the mass-energy.

An easy way to see that the mass-energy component points in the timelike direction is to consider an object at rest. The four-momentum is tangent to the object's worldline and the three momentum components are zero; therefore the fourth component must point in the direction of the worldline.

 

[PeterDonis]

There are four component vectors, three spatial, momentum, components and a time, energy, component. They each have a direction.

No. The components of a vector are not vectors. They are numbers. They don't have a direction.

You seem to be confusing components of a vector with basis vectors. When we choose a coordinate system to describe spacetime, that coordinate system defines four basis vectors at each point. These are (at least for the case of flat spacetime and inertial coordinates that we're discussing) four mutually orthogonal unit vectors, i.e., they each have length 1 (note that we are here using units where $c = 1$, which are the natural units to use in relativity) and they point in four mutually orthogonal directions. It's fairly simple to show that three of those directions must be spacelike and the fourth must be timelike. The three spacelike basis vectors point in three mutually orthogonal spatial directions, like the basis vectors of a standard Cartesian coordinate system in ordinary 3-dimensional Euclidean space. The timelike basis vector points into the future.

Given a set of four basis vectors, any vector at all can be expressed as a weighted sum of the four basis vectors. The weights of each basis vector in the sum are the components of the vector. (Note that this means that a vector's components depend on the coordinates you choose.)

The 4-momentum is a vector, so it can be expressed this way just like any other vector. The three spatial components (i.e., the coefficients that multiply the three spacelike basis vectors) are the components of the 3-momentum (i.e., ordinary momentum), and the time component (i.e., the coefficient that multiplies the one timelike basis vector) is the energy.

 

[whatif]

Yes, the mass-energy component points in the timelike direction - although that's actually sort of backwards - it's probably better to say the we identify the timelike component with the mass-energy.

Reference https://www.physicsforums.com/threads/4-four-momentum-energy-component-direction.964058/

Thank you. I understood that it is in the timelike direction. My difficulty is interpreting what that physically means, especially coming from thinking that energy is a scalar quantity. I do not have the same problem with momentum.

 

[whatif]

You seem to be confusing components of a vector with basis vectors.

I do not think so. It is a question of language. I would think that a weighted vector is a vector. I am not arguing. If the word ‘components’ of vector must conventionally strictly refer to weights of the unit basis vectors then I will go along with that.

 

[PeroK]

Thank you. I understood that it is in the timelike direction. My difficulty is interpreting what that physically means, especially coming from thinking that energy is a scalar quantity. I do not have the same problem with momentum.

If you accept the concept of spacetime, as a 4D continuum of time and space, then time has a direction. And vectors representing physical quantities will have four components. The four-momentum, for example.

You can think of four-vectors as a mathematical trick, but I think that's an overly Newtonian viewpoint. You cannot convert momentum from one frame to another without using the energy (time component). And, in fact, the conservation of the momentum four-vector encapsulates the conservation of mass, energy and momentum, which are three separate conservation laws in Newtonian mechanics. From that point of view, it's a unification of energy and momentum.

 

[PeterDonis]

If the word ‘components’ of vector must conventionally strictly refer to weights of the unit basis vectors then I will go along with that.

Yes, that is what the word "components" means. I don't think there is a conventional term for the "weighted vectors", but since your main concern seems to be with direction, not magnitude, you can just think of the directions of the basis vectors and it will come to the same thing.

 

[PeterDonis]

My difficulty is interpreting what that physically means

It means that the timelike basis vector points into the future, just as I said before. "Into the future" is a direction since we're talking about spacetime.

especially coming from thinking that energy is a scalar quantity

So are each of the components of momentum. But energy times the timelike basis vector is, as you note, a vector quantity (for which there is not, as I noted before, a conventional name that I'm aware of), just as each momentum component times its corresponding spatial basis vector is a vector quantity. All of these things work the same in relativity.

 

[whatif]

You can think of four-vectors as a mathematical trick

I am neither claiming it is not discounting that it may be (and I would not mean that disparagingly). It may be that it is too difficult for me to interpret what that physically means but it is noteworthy to me that no one attempts to explain it like some do with momentum.

You cannot convert momentum from one frame to another without using the energy

You are implying that the energy component must be treated as a vector. I have to take your word for that because I am not experienced to know the intricacies of different coordinate systems. On the face of it, at my basic level of understanding, mass (rest mass) is invariant between frames, spatial velocities can be transformed so that momentum (spatial component) can be transformed and time always points in the same direction so that energy can, in principle, be treated as a scalar quantity. Is that just wrong?

 

[whatif]

"Into the future" is a direction since we're talking about spacetime.

That corresponds to a yes to my first question. I accept that. My question was prompted because energy having a direction into the future is a somewhat nebulous concept to me, together with that not being explicitly addressed when introduced, prompted my question.​

 

[PeroK]

I am neither claiming it is not discounting that it may be (and I would not mean that disparagingly). It may be that it is too difficult for me to interpret what that physically means but it is noteworthy to me that no one attempts to explain it like some do with momentum.You are implying that the energy component must be treated as a vector. I have to take your word for that because I am not experienced to know the intricacies of different coordinate systems. On the face of it, at my basic level of understanding, mass (rest mass) is invariant between frames, spatial velocities can be transformed so that momentum (spatial component) can be transformed and time always points in the same direction so that energy can, in principle, be treated as a scalar quantity. Is that just wrong?

You really need to sort out the difference between a component of vector, which is a number, and a vector, which is an array of numbers.

As has been said before, energy is the zeroth or time component of the energy momentum four-vector, but that does not make energy a vector. Nor does it give energy a direction.

 

[PeterDonis]

That corresponds to a yes to my first question.

No, it doesn't. The timelike basis vector points into the future. That doesn't mean the energy points into the future. You could say that the vector you get when you multiply the energy times the timelike basis vector points into the future, but that vector is not the energy. Energy doesn't have a direction, and "the direction of time" (i.e., "into the future") is not the energy, or the direction of the energy.

mass (rest mass) is invariant between frames

Yes.

spatial velocities can be transformed so that momentum (spatial component) can be transformed

Not by themselves; you have to transform all four components of a 4-vector together. See further comments below.

time always points in the same direction

No, it doesn't. More precisely, the timelike basis vector doesn't always point in the same direction. Nor do the spatial ones. It might help to work through this in more detail.

In ordinary Cartesian coordinates in ordinary 3-dimensional space, you can rotate the coordinates, which changes the directions in which the basis vectors point. The components of all vectors change when you transform them from one set of coordinates to the other.

In 4-dimensional spacetime, a Lorentz transformation--a change of inertial frames--works similarly: it changes the directions in which the basis vectors point. But now we have to include the timelike basis vector in this; in other words, changing inertial frames changes which way the timelike basis vector points, as well as the spatial ones. Actually, a pure Lorentz transformation--what you'll often see called a "boost" in the literature--only changes the direction in spacetime of one spatial vector, the one that points in the direction of relative motion between the two frames.

But there is a key difference, intuitively speaking, with a Lorentz transformation. Note that I said above "direction in spacetime", not "direction in space". A pure boost does not "mix" directions in space: it only changes the direction in spacetime of one spatial basis vector, leaving the other two spatial basis vectors alone. But it also changes the direction in spacetime of the timelike basis vector. You can see an illustration of how this works here:

https://en.wikipedia.org/wiki/Minkowski_diagram#Minkowski_diagrams_in_special_relativity

If you look at the first diagram on the right in that section of the article, the black axes are the $t$ and $x$ axes (the $x$ direction is the direction of relative motion) before the transformation, and the blue axes are after the transformation. Note that the axes "tilt" towards each other, instead of staying at 90 degrees in the diagram as a normal spatial rotation would do. That's because the geometry of spacetime is not Euclidean, it's Minkowskian (the minus sign in the metric). But the point is that both axes change direction in spacetime (spacetime here is just the diagram as a whole) in the transformation, so both the $t$ and the $x$ components of vectors have to change.

so that energy can, in principle, be treated as a scalar quantity

The term "scalar quantity" is ambiguous. Strictly speaking, a "scalar" in relativity (more precisely, a "Lorentz scalar") is a quantity that's just a number (no direction) and doesn't change when you change frames. So rest mass is a scalar, on this strict definition, but energy is not. Energy is just one component of a 4-vector, and transforms like all vector components when you change frames.

 

[Dale]

Does it mean that energy is in the direction of time and if so, is that meant to be so obvious that no one seems to explain it?

Yes, energy is in the direction of time. Often it will be described as the timelike component of the four momentum or the 0th component where previously that component has been associated with time.

time always points in the same direction so that energy can, in principle, be treated as a scalar quantity.

In different reference frames the time basis vector points in different directions. So energy is not a scalar, it is a component of a vector. In relativity, the term scalar is reserved for things that are invariant under a change in the coordinates. So the components of a four-vector are not scalars since they do change as coordinates change. Clearly energy is different in different frames, so it is not a scalar.

 

[whatif]

You really need to sort out the difference between a component of vector, which is a number, and a vector, which is an array of numbers.

I apologise. I am just so used to treating vectors as things with magnitude and direction, and a vector being comprised of other vectors added to together (components), a common definition and usage, and not having the restriction on what is meant by the word component.On the other hand, a book about spacetime physics by Taylor and Wheeler that I think a colleague of yours suggested I read, seems to use language somewhat aligned to the way I am using it especially under the title “What is a 4-Vector?” and in a diagram that uses arrows to show the direction of the momenergy vector being the direction of the world line; including the case when momentum is zero.

 

[whatif]

Energy doesn't have a direction, and "the direction of time" (i.e., "into the future") is not the energy, or the direction of the energy.

I am glad to hear it and I think I am getting the picture. I asked the question because other literature seemed to suggest otherwise and I did not know how to interpret it.

If you look at the first diagram on the right in that section of the article, the black axes are the t'>tt and x'>xx axes (the x'>xx direction is the direction of relative motion) before the transformation, and the blue axes are after the transformation. Note that the axes "tilt" towards each other, instead of staying at 90 degrees in the diagram as a normal spatial rotation would do.

I get the Minkowski diagram. I get the point you are making about transforming 4 vectors. It did not quite address the point I was trying to make and raises more questions but that energy does not have direction is enough.

Thank you.

 

[whatif]

Yes, energy is in the direction of time.

Thank you, but I am clearly getting different opinions about that.

 

[Dale]

I am just so used to treating vectors as things with magnitude and direction, and a vector being comprised of other vectors added to together (components)

As you go from flat geometry to curved geometry you start to have to distinguish more carefully some distinct concepts that can get mixed up.

A vector is a geometric object which is a member of a vector space. A vector space has notions of addition of two vectors and multiplication of a vector by a real number to get a vector and multiplication of two vectors to get a real number. A vector space can be equipped with (orthonormal) basis vectors and then any vector in the space can be formed as a linear combination of the basis vectors. The real numbers that multiply the basis vectors are called components. Sometimes, sloppily, the product of the component and the respective basis vector is also called a component, but I believe that should be called the projection. Because the usage I s sloppy it can be hard to tell if someone is referring to the real number or the product of the real number and the basis vector.

 

[PeroK]

Thank you, but I am clearly getting different opinions about that.

I suggest the acid test is a measurement of energy which gives a single number of Joules or electron-volts. That's not a vector.

A measurement of momentum, however, has three components (or projections) and therefore momentum is a three-vector.

I've never heard of energy being called a one-vector, although I guess there's nothing to stop someone coining that term.

 

[Dale]

Thank you, but I am clearly getting different opinions about that.

From https://phys.libretexts.org/Bookshe...rowell)/4:_Tensors/4.2:_Four-vectors_(Part_1) there is a whole section titled “Energy is the timelike component of the four momentum”. I am sure that many other references state something similar, but this one is an easily accessible and very good book by an author I trust.

 

[PeroK]

RFrom https://phys.libretexts.org/Bookshelves/Relativity/Book:_General_Relativity_(Crowell)/4:_Tensors/4.2:_Four-vectors_(Part_1) there is a whole section titled “Energy is the timelike component of the four momentum”. I am sure that many other references state something similar, but this one is an easily accessible and very good book by an author I trust.

On the other hand, the energy of a particle can be defined as the scalar product of the particle's four-momentum and the observer's four-velocity. And a scalar product has no direction.

 

[whatif]

From https://phys.libretexts.org/Bookshe...rowell)/4:_Tensors/4.2:_Four-vectors_(Part_1) there is a whole section titled “Energy is the timelike component of the four momentum”. I am sure that many other references state something similar, but this one is an easily accessible and very good book by an author I trust.

That reference does not seem to have anything to say about energy having direction.

 

[whatif]

I suggest the acid test is a measurement of energy which gives a single number of Joules or electron-volts. That's not a vector.

That presupposes energy does not have a direction and so side steps my question. As a novice to relativity I am not making that assumption. There are explanations that indicate that that energy has direction. If I were to make that assumption it would seem to contradict anyone that expresses that four momentum points in the direction of the world line; including Taylor and Wheeler.

On the other hand, there is a kind of logic with using arrows to show momentum and direction. I imagine this is deceptive. I assume momentum has direction, but the energy ‘arrow’ is only representative of magnitude.

I've never heard of energy being called a one-vector, although I guess there's nothing to stop someone coining that term.

That also goes somewhat to my question. To a novice, like myself, it may appear energy is being treated as a one vector and I ask myself what the point of that would be. That goes to why I do not discount a mathematical ‘trick’.

 

[lomidrevo]

That presupposes energy does not have a direction and so side steps my question. As a novice to relativity I am not making that assumption. There are explanations that indicate that that energy has direction. If I were to make that assumption it would seem to contradict anyone that expresses that four momentum points in the direction of the world line; including Taylor and Wheeler.

You should read again all the responses given to you. You are still confused about the difference between a vector, which has a magnitude and a direction, and a component of a vector, which is just a single number. Energy is the timelike component of the momentum 4vector. No mystery there...

 

[whatif]

You are still confused about the difference between a vector, which has a magnitude and a direction, and a component of a vector, which is just a single number.

I do not think so and I do not see how that has any relevance to what I wrote about the “acid” test. I am ready to accept that energy does not have direction. Let me tackle this another way:

Taylor and Wheeler write that 4 momentum has the same direction as the world line of an object. A reference frame shows the world line of that object. Where/”while” that object is spatially stationary it only goes in the time direction. The object has no spatial momentum and only rest mass energy. Taylor and Wheeler use arrows to show 4 momentum and, in this circumstance, they show an arrow pointing in the time direction only and associate the rest mass energy with the arrow. Why does 4 momentum just point in the time direction in this circumstance? Are Taylor and Wheeler right or wrong?(e.g. perhaps 4 momentum points in the direction of time and the just the magnitude of 4 momentum is just the energy of 4 momentum in this circumstance).

 

[PeroK]

I Taylor and Wheeler use arrows to show 4 momentum and, in this circumstance, they show an arrow pointing in the time direction only and associate the rest mass energy with the arrow. Why does 4 momentum just point in the time direction in this circumstance? Are Taylor and Wheeler right or wrong?

(e.g. perhaps 4 momentum points in the direction of time and the just the magnitude of 4 momentum is just the energy of 4 momentum in this circumstance).

$E \ne (E, 0, 0, 0)$

The left-hand-side of that equation is defined as the energy. You could define the energy of a particle as $(E, 0, 0, 0)$, but that would not be the standard definition of energy. That would be your own definition of energy. Whenever any other physicist talks about "energy", you would say "magnitude of energy" etc. It would simply be different terminology.

 

[Ibix]

Taylor and Wheeler use arrows to show 4 momentum and, in this circumstance, they show an arrow pointing in the time direction only and associate the rest mass energy with the arrow.

The length of that arrow is always the rest mass. In a frame where an object is at rest, this is also the total energy of the object.

Why does 4 momentum just point in the time direction in this circumstance?

It always points in a time direction - the direction used by its rest frame. In the case where the diagram shows its rest frame, then, it points in the direction the diagram uses as time.

PeroK's last post sums it up neatly. In most coordinate systems the energy is the number multiplying the time-like basis vector. It is not itself a vector. It is always positive, so does not even have a direction in the sense of possibly being negative.

 

[lomidrevo]

Why does 4 momentum just point in the time direction in this circumstance? Are Taylor and Wheeler right or wrong?

They are correct. But energy is still only the 0th component of that 4-vector. I don't think that they would claim in text that whole 4 momentum ("including direction") represents energy of the particle.

(e.g. perhaps 4 momentum points in the direction of time and the just the magnitude of 4 momentum is just the energy of 4 momentum in this circumstance).

You are on the right track here...

 

[whatif]

The length of that arrow is always the rest mass. In a frame where an object is at rest, this is also the total energy of the object.

That clears it up nicely.

 

[whatif]

I don't think that they would claim in text that whole 4 momentum ("including direction") represents energy of the particle.

That is correct, I never said they did and going back they actually wrote the right thing. However, they had diagram that was suggestive of energy being a vector and they did treat some things as obvious that to me, were not so obvious.

Now, I could ask what sense is to be made of kilograms moving in the time direction. If I am taking things to be obvious and not requiring explanation then 4 momentum has units of mass. The component of the time direction is the energy of the object; energy being the sole contributor to the time component. Momentum and energy were united as a 4 vector by mathematically giving them the same units. However, relativity theory still distinguishes between momentum and energy (as it does between time and spatial distance). While I appreciate the distinction between a component and a vector direction, lack of an explanation somewhat blurs the distinction between the vector time direction and its component, in my mind, (not whether it is a component but whether energy is a vector direction of 4 momentum). Energy is not momentum just as spatial distance is not time, despite distance and time being measured it the same units for the purpose of unification. The derivation of energy as the time component has nothing to say that it is not a vector, as far as I can see. It may justify energy being called a one vector, not that I am advocating it, or it may justify calling it a mathematical trick.

This might lead us back to square one so just saying and let's not go there.

 

[Dale]

That reference does not seem to have anything to say about energy having direction.

It says “Timelike component”. How you want to interpret that wrt your question is up to you. It is reasonable to say that the timelike component is a real number with no direction and it is reasonable to say that a timelike component is a projection on the timelike basis vector so it points in the direction of time. Choose your preference as you will.

The four momentum is $P=(E,p_x,p_y,p_z)=E\hat t+p_x \hat x+p_y \hat y + p_z \hat z$. You can say that $E$ points in the $\hat t$ direction just as you might say that $p_x$ points in the $\hat x$ direction.

 

[lomidrevo]

Energy is not momentum just as spatial distance is not time, despite distance and time being measured it the same units for the purpose of unification. The derivation of energy as the time component has nothing to say that it is not a vector, as far as I can see.

Energy and momentum are different quantities, of course. The reason why 4 vectors are introduced in relativity, is not just to have a common units. What is far more important is that time and space individually do not transform properly under Lorentz transformation. You have to transform them together. Hence it becomes natural to talk about spacetime, and to define position 4-vector, where the 0th component represent the timelike coordinate of an event. But you still keep in mind that space and time are not the same, as you know intuitively. The same applies to energy and momentum. They cannot be properly transformed individually, but if you define a 4-vector, which you construct using the classical 3D momentum of a particle plus using its energy as the 0th component, the whole new beast will transform properly under Lorentz transformation. I don't see any reason why we should regard energy as having any direction...

 

[pervect]

Thank you. I understood that it is in the timelike direction. My difficulty is interpreting what that physically means, especially coming from thinking that energy is a scalar quantity. I do not have the same problem with momentum.

If you understand spatial basis vectors, then specifying three spatial basis vectors, for instance the spatial basis vectors of an observer "at rest" in some coordinate system in the Minkowskii space of special relativity, uniquely define one time-like vector that is orthogonal to all three spatial vectors.

Hopefully the notion of orthogonality is familiar. Formally, two vectors are orthogonal if their dot-product is zero. And hopefully the notion of the dot-product of four-vectors is familiar. If not, I would guess a review is in order, I'm not sure how to explain things further without using these concepts.

A note on terminology here, to avoid confusion. We usually say "space-like" vectors as an adjective when we talk about four-vectors rather than "spatial" vectors. Then a "purely spatial" four-vector would be a four-vector with no component in the time "direction".

The mathemtics then are that specifying three space-like vectors mathematically defines a unique time-like vector that's orthogonal to all of the space-like vectors. Similarly, if we define a time-like vector, we can determine three space-like vectors that are orthogonal to the time-like vector, and to each other. However, this set of space-like vectors with this property is not unique because of the existence of spatial rotations. Performing purely spatial rotations of the three space-like vectors will generate a different set of space-like vectors that meet our orthogonality conditions.

Thus specifying a set of three space-like vectors automatically specifies a time-like vector that's orthognal to all of them.

You hopefully recall that simultaneity is relative in special relativity. If not, you need to study the issue, it's important. The fact that different observers, those moving with different velocity, have different notions of simultaneity is closely related to the fact that the time-like four-vectors that are associated with these observers are different vectors. The name of the time-like four-vector associated with an observer may also be useful, it's called the observers four-velocity vector.

Making a vectors purely spatial basically implies that we set the time-component of the vectors to zero if we assuming Minkowskii coordinate conventions. As a consequence of defining what it means for a vector to have no time component, we have implicitly defined the notion of what a time-component is, and how to set that time component to zero. This specifies a notion of simultaneity, and the notion of simultaneity depends on the observer, in particular the state of motion of the observer.

Mathematically, the notion of dot-product comes first, and we use the notion of dot-product of an arbitrary vector with a basis vector to define the meaning of a component of a vector. A components of an arbitrary vector, accordign to the mathematical defintion, is the dot product of the arbitrary vector with the basis vector.

Your intuition is probably working backwards from the formal mathematical definition, assigning physical significance to the "vanishing" of the time component. This is exactly backwards from the mathematical approach, which treats vectors as abstract entities with abstract properties, and the "frame of reference" of an observer is a physical interpretation of the mathematical representation of a set of mutually orthogonal basis vectors associated with the observer.

The relativity of simlultaneity means that the notion of making the "time-component" of a vector "vanish" depends on the choice of basis vectors used. The mathematical description starts with the basis vectors, and allows one to define an observer as a set of basis vectors. The intuitive notion usually goes backwards, one defines an "observer" intuitively first, then finds set of basis vectors associated with that observer.

 

[Herman Trivilino]

I apologise. I am just so used to treating vectors as things with magnitude and direction, and a vector being comprised of other vectors added to together (components), a common definition and usage, and not having the restriction on what is meant by the word component.

You no doubt saw declarations like $\vec{A}$ has components $A_x$ and $A_y$.

But you never saw expressions like $\vec{A}=A_x+A_y$.

Instead, you'd see $\vec{A}=A_x\hat{i}+A_y\hat{j}$.

The reason is because you can't, as you claim above, add components to get vectors.

You have vectors, you have components, and you have magnitudes.

 

[whatif]

It says “Timelike component”. How you want to interpret that wrt your question is up to you. It is reasonable to say that the timelike component is a real number with no direction and it is reasonable to say that a timelike component is a projection on the timelike basis vector so it points in the direction of time. Choose your preference as you will.

The four momentum is $P=(E,p_x,p_y,p_z)=E\hat t+p_x \hat x+p_y \hat y + p_z \hat z$. You can say that $E$ points in the $\hat t$ direction just as you might say that $p_x$ points in the $\hat x$ direction.

You appear to be using the word component in the sense that I originally used it, which lead others to claim that I was confusing a component (magnitude) with a vector. I was taking that on board when writing that the reference did not seem to have anything to say about direction. I am not yet convinced that you are wrong but it is not because I am confusing a component with a vector.

 

[Dale]

I don’t think there is a wrong or right in this. I think that it is sloppy or not. The term “component” to my knowledge when used precisely refers to the real number that multiplies the basis vector. But it is pretty common sloppy language to refer to the product of the real number and the basis vector the same way. It is sloppy, but pretty reasonable and not too problematic. So in that sense we can say that energy points in the time direction, referring to $E\hat t$ rather than $E$.

Certainly, if you asked in which direction $p_x$ points most people would be ok saying the $\hat x$ direction.

 

[whatif]

Energy and momentum are different quantities, of course. The reason why 4 vectors are introduced in relativity, is not just to have a common units. What is far more important is that time and space individually do not transform properly under Lorentz transformation. You have to transform them together. Hence it becomes natural to talk about spacetime, and to define position 4-vector, where the 0th component represent the timelike coordinate of an event. But you still keep in mind that space and time are not the same, as you know intuitively. The same applies to energy and momentum. They cannot be properly transformed individually, but if you define a 4-vector, which you construct using the classical 3D momentum of a particle plus using its energy as the 0th component, the whole new beast will transform properly under Lorentz transformation. I don't see any reason why we should regard energy as having any direction...

… for consistency.

Langauge m2 = E2 – p2 is a relationship between the components of 4 momentum. The timelike component is called energy, not timelike momentum component. For the purposes of the equation we are distinguishing between energy and momentum. The 4 vector does not make this distinction, in the sense that we may not call the timelike part of the vector, energy (mathematically it works). That is OK, so long as we keep track of what is being done.

Does the 4 momentum vector have units of kilograms? I ask because I would think so but you might say no. If so, then these units are a measure of energy in the spacelike direction and momentum in the spatial directions.

How is the timelike part of the 4 momentum vector to be interpreted (what does it mean to have kilograms moving in the time direction)? The answer to this question is left hanging. You might be able to tell me. You might say you do not know, which would be OK. However, it should have meaning. If you cannot tell me then, although it would still difficult to make sense of, I suggest energy moving forward in time is the “obvious” interpretation. It would seem that making the timelike part of the 4 momentum vector, energy, would put 4 momentum energy on a par with how timespace is treated in terms of language and interpretation.

 

[whatif]

It is sloppy, but pretty reasonable and not too problematic.

I agree with you. However, I did it unwittingly and it seems to be extremely problematic to others.

I don’t think there is a wrong or right in this.

Others strongly disagree and insist that energy does not have direction. There has to be a definitive answer for the purpose of agreement. I think the answer is in explaining what it is that is moving forward in time (i.e. why it is different from kilograms of energy). It seems to me that, in effect, an decision has been to define energy as non directional.

 

[whatif]

The reason is because you can't, as you claim above, add components to get vectors.

I made no such claim in terms of what you understand "component" to mean. What I did in the first instance was to unwittingly use the word 'component' in a way that I did not realize had a specific restricted meaning. The way I meant it, using general usage, does not disagree with the principles that you are conveying to me. The way you mathematically try to explain it does not directly convey the distinction.
I willingly concede that there is good reason to use common language.

 

[PeterDonis]

Langauge m2 = E2 – p2 is a relationship between the components of 4 momentum.

Yes, but all of the quantities in that equation are scalars. None of them are vectors. So if you are going to call those quantities "components", then you are implicitly defining "components" to be scalars, not vectors.

Does the 4 momentum vector have units of kilograms?

If you choose that system of units, yes. Note that in that case, you will either need to use the same units for energy and momentum, or put in factors of $c$ where needed to make the units come out right.

It seems to be more common in the literature, or at any rate the classical relativity literature, to use either momentum or energy units for 4-momentum; in that case, again, you either need to pick the same units for all the components, or put in factors of $c$ as required.

In quantum field theory, the natural units of energy and mass are inverse length (because Planck's constant is taken to have no units).

How is the timelike part of the 4 momentum vector to be interpreted (what does it mean to have kilograms moving in the time direction)?

It's the energy in mass units. More precisely, it's the energy that the object would be observed to have by an observer at rest in the inertial coordinates being used.

You should not get hung up on the units; they're irrelevant to what we're discussing.

If you cannot tell me then, although it would still difficult to make sense of, I suggest energy moving forward in time is the “obvious” interpretation.

No, the "obvious" interpretation is that energy is 4-momentum in the time direction. And momentum is 4-momentum in the spatial directions. Momentum doesn't move. It's a property of the motion of something. Energy is just momentum in the time direction, so it doesn't move any more than ordinary momentum does. It just describes the "motion in time" of an object, the way ordinary momentum describes its motion in space.

 

[whatif]

So if you are going to call those quantities "components", then you are implicitly defining "components" to be scalars, not vectors.

I understand that and I was trying to make it clear that I understand that. Are you trying to say that I have defined energy to not be a vector? It is others that have implicitly defined energy to not be a vector and I am just showing that I understand. Where I am going is that energy could be treated as the timelike part of the 4 momentum vector and there is reason to do so.

Energy is just momentum in the time direction, so it doesn't move any more than ordinary momentum does.

Yes, OK, it does not move. However, I expect that you meant energy is just the magnitude of momentum in the time direction because otherwise you seem to be saying that energy does have direction.

No, the "obvious" interpretation is that energy is 4-momentum in the time direction.

That does not really explain anything meaningful because while it is correct it ignores that a distinction is made between momentum and energy, and is why I should get hung about units.

You should not get hung up on the units; they're irrelevant to what we're discussing.

Why are units irrelevant? Just because the same units are used does not mean the timelike part has the same character (if I may use that word) as the spacelike parts. A meter of time is different from a meter of space despite being measured in the same units. The speed of light, c, provides the relationship that facilitates the analysis of spacetime. Similarly, whatever units are used, a unit of that energy is different from that unit of momentum.

 

[lomidrevo]

(what does it mean to have kilograms moving in the time direction)?

What do you think it means for an object to move in the time direction?

I agree with you. However, I did it unwittingly and it seems to be extremely problematic to others.

Quite contrary, you look like the only one in this thread who is having troubles with the interpretation of the energy. Instead you would accept the more natural interpretation provided in the previous posts, you insist that it might have a direction. OK, it is free country, you can do it obviously, but: What would be the interpretation of the "direction"? Would be such definition anyhow helpful for solving problems? As far as the concept is not compatible with observations/experiments, how it can be useful? It is meaningless...

 

[whatif]

What do you think it means for an object to move in the time direction?

As PeterDonis pointed out having direction does not mean it moves. The unit kilogram is a measure of energy so it makes sense to say energy is pointing in the direction of time. It makes more sense, to me, to regard the timelike component 4 momentum as energy than simply stating that the "obvious" interpretation is that energy is 4-momentum in the time direction (because a kilogram of energy is not the same as a kilogram of momentum, or whatever units you use).

Your second quote comes from a post in reply to someone who treats energy as having direction. Also, you are taking the second quote out of context.

What would be the interpretation of the "direction"? Would be such definition anyhow helpful for solving problems? As far as the concept is not compatible with observations/experiments, how it can be useful? It is meaningless...

The direction is already defined by the timelike part of 4 momentum. It is completely compatible with observations/experiments as far as I can see. It is a matter of interpretation and it would be useful by way of consistency with the relationship of time with space and incorporating that a unit of energy is not the same as that unit of momentum.

Your intuition is probably working backwards from the formal mathematical definition, assigning physical significance to the "vanishing" of the time component. This is exactly backwards from the mathematical approach, which treats vectors as abstract entities with abstract properties, and the "frame of reference" of an observer is a physical interpretation of the mathematical representation of a set of mutually orthogonal basis vectors associated with the observer.

Maybe, I do not know but I am not convinced that is the source of a problem. However, it implies the physical interpretation must mean that energy must be treated as it is. I think that the physical interpretation, as is, incorporates an implied definition that energy is a component and nothing more (probably to simply correspond with the classical idea of energy being a scalar). I can live with that and I have already written so. Since then it has not been that I am insisting that energy has direction but putting a physical interpretation that seems to be reasonable (and more consistence, in my view).

 

[whatif]

It makes more sense, to me, to regard the timelike component 4 momentum as energy than simply stating that the "obvious" interpretation is that energy is 4-momentum in the time direction (because a kilogram of energy is not the same as a kilogram of momentum, or whatever units you use).

I made two mistakes here:

1. Old habits die hard. I used component where I should not have.
2. PeterDonis wrote:

   No, the "obvious" interpretation is that energy is 4-momentum in the time direction.

   I think he meant to write: 'No, the "obvious" interpretation is that energy is the magnitude of 4-momentum in the time direction.' Otherwise, he appears to be agreeing with me.

I meant to write that it makes more sense, to me, to regard the timelike part of the 4 vector, 4 momentum as energy than simply stating that the "obvious" interpretation is that energy is the magnitude of the 4 vector, 4-momentum in the time direction (because a kilogram of energy is not the same as a kilogram of momentum, or whatever units you use).

 

[PeterDonis]

Are you trying to say that I have defined energy to not be a vector?

In the particular statement you made that I responded to, yes.

Where I am going is that energy could be treated as the timelike part of the 4 momentum vector and there is reason to do so.

Both viewpoints are reasonable, and neither one is "right" or "wrong". You can adopt whichever one seems to be most helpful for a particular problem. There is no reason why one viewpoint must always be taken.

Just because the same units are used does not mean the timelike part has the same character

That's correct, and that's exactly why you shouldn't get hung up on units: because the difference between "timelike" and "spacelike" is not that we usually use different units to measure them. We can use the same units to measure them and they're still different. So the difference must be something else other than the units; hence, you shouldn't get hung up on the units because they're irrelevant to figuring out what the difference is.

 

[PeterDonis]

I think he meant to write: 'No, the "obvious" interpretation is that energy is the magnitude of 4-momentum in the time direction.'

No, I made the statement ambiguous on purpose, to illustrate that both interpretations are useful, and neither one is "right" or "wrong".

 

[PeterDonis]

the difference between "timelike" and "spacelike" is not that we usually use different units to measure them

both interpretations are useful, and neither one is "right" or "wrong".

In fact, to emphasize that both interpretations are useful, consider the following: the difference between "timelike" and "spacelike" is not units, and it's also not that we normally consider energy a scalar and momentum a vector. We can consider energy to be a vector pointing into the future, i.e., "momentum in the time direction", and the difference between timelike and spacelike is still there. So the difference can't be related to the difference between scalar and vector either.

 

[whatif]

Energy doesn't have a direction, and "the direction of time" (i.e., "into the future") is not the energy, or the direction of the energy.

No, I made the statement ambiguous on purpose, to illustrate that both interpretations are useful, and neither one is "right" or "wrong".

I know there have been problems with my language but I do not see the ambiguity because it specifically gives energy direction.
Anyway, thank you, that explains why you have appeared to me to agree and disagree in different messages.

the difference between "timelike" and "spacelike" is not units, and it's also not that we normally consider energy a scalar and momentum a vector. We can consider energy to be a vector pointing into the future, i.e., "momentum in the time direction", and the difference between timelike and spacelike is still there. So the difference can't be related to the difference between scalar and vector either.

That is an abstract mathematical explanation that effectively allows two valid but different physical interpretations (in one energy cannot be a vector by implicit or explicit definition and in the other energy is a vector).

Also, some have been saying that the component (scalar) is just a number. If it is just a number then units have to come into it in another way. If the units are considered to be part of the component (scalar), then it is not just a number. Either way, the timelike momentum has a direction and the direction is in the direction of time (and kilograms associated with a vector is not the same as kilograms only associated with a scalar).

No, the "obvious" interpretation is that energy is 4-momentum in the time direction.

If the timelike momentum is not energy then what is it (rhetorical question)? Just calling it momentum in the timelike direction is nebulous. Both the timelike and spacelike parts are called momentum even though the same units of timelike momentum have a different meaning when applied to spacelike momentum.Anyway, attaching a scalar to a vector gives meaning to the vector, including units. I still have to get my head around kilograms in the timelike direction. However, I get it and in the end my initial understanding of was valid but language was a hangup.Thank you.

 

[whatif]

I don't think that they would claim in text that whole 4 momentum ("including direction") represents energy of the particle.

I am not sure what is meant by whole 4 momentum. The discussion has been about timelike momentum (the energy part).

Actually, Taylor and Wheeler call 4 momentum, 'momenergy' and describe energy as the "Time Part" of momenergy. They do so in headings and the text. From Taylor and Wheeler text:

Energy is only the time part of the momenergy 4-vector.

So, in the text, they actually do treat energy as a vector part, the timelike part, and that includes direction.

That would be your own definition of energy. Whenever any other physicist talks about "energy", you would say "magnitude of energy" etc. It would simply be different terminology.

That comment came in response to a reference to Taylor and Wheeler. Taylor and Wheeler refer to energy as timelike.
Anyway, I get it.

 

[lomidrevo]

I fully agree with the statement: "Energy is only the time part of the momenergy 4-vector." That is what I claimed in my previous posts.

So, in the text, they actually do treat energy as a vector part, the timelike part, and that includes direction.

But I don't agree with this conclusion. Even though you can construct a vector as a linear combination of basis vectors, the component itself is not a vector. It is just a number representing the weight (or magnitude if you wish) of the corresponding basis vector.

In my opinion, another reason to look at energy as it is just a number (you may notice I am not saying scalar, because it is not invariant under Lorentz transformation): In a low speed limit, the special relativity must reduce to Newtonian mechanics. Where the "direction of energy" disappeared? Why you now got just a real number instead of a vector? It is inconsistent...

Energy is very closely related to time coordinate: conservation of energy is related to the symmetry of physics under translation in time. Similar relationship is between conservation of momentum and the space-translation symmetry. So that may give you a hint why in the 4-momentum you can find the energy as the time-like component and components of the classical 3D momemntum as the space-like components. But again, this doesn't make the energy being a vector quantity.

In summary, I don't see any benefits of considering the energy as a vector quantity. And probably that is the reason why you never see symbols like $\vec E$ to represent energy in textbooks.

That comment came in response to a reference to Taylor and Wheeler. Taylor and Wheeler refer to energy as timelike.

That's OK, we all agree on that.

 

[whatif]

"Energy is only the time part of the momenergy 4-vector." That is what I claimed in my previous posts.

That agrees with Taylor and Wheeler but says that energy if a part of a 4 vector (it misses the word 'component').

Energy is the timelike component of the momentum 4vector.

That is was you wrote in a previous post and is not the same thing as my quote from Taylor and Wheeler. The Taylor and Wheeler quote does not have the word 'component'.

Even though you can construct a vector as a linear combination of basis vectors, the component itself is not a vector.

Knowing the meaning of the word 'component' as used in this context that is a correct statement. At the same time, that is not the point; the point being whether you treat energy as a vector, being the timelike momentum. I understand and see the point of those who have said that neither interpretation is right or wrong (but at the moment I would regard the timelike momentum to be energy whether it is called that or not).

In my opinion, another reason to look at energy as it is just a number

Energy has units. It is not just a number.

In a low speed limit, the special relativity must reduce to Newtonian mechanics.

At low speeds, Newtonian mechanics approximates relativistic mechanics. Also, timelike parts (timelike vectors) are irrelevant in Newtonian mechanics and need to be given a physical interpretation.

In summary, I don't see any benefits of considering the energy as a vector quantity.

The benefit in my view is that it is a better physical interpretation. Others have said that each interpretation is useful.