# B 4 four momentum energy component direction

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1. Jan 11, 2019

### 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?

2. Jan 11, 2019

### Staff: Mentor

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.

3. Jan 11, 2019

### Staff: Mentor

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.

4. Jan 12, 2019

### whatif

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.

5. Jan 12, 2019

### whatif

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.

6. Jan 12, 2019

### PeroK

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.

7. Jan 12, 2019

### Staff: Mentor

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.

8. Jan 12, 2019

### Staff: Mentor

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.

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.

9. Jan 12, 2019

### whatif

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?

10. Jan 12, 2019

### whatif

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.

11. Jan 12, 2019

### PeroK

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.

12. Jan 12, 2019

### Staff: Mentor

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.

Yes.

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

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.

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.

13. Jan 12, 2019

### Staff: Mentor

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.

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.

14. Jan 12, 2019

### whatif

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.

15. Jan 12, 2019

### whatif

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.
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.

16. Jan 12, 2019

### whatif

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

17. Jan 12, 2019

### Staff: Mentor

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.

18. Jan 12, 2019

### PeroK

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.

19. Jan 12, 2019

### Staff: Mentor

Last edited: Jan 12, 2019
20. Jan 12, 2019

### PeroK

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.