# Is Time a vector quantity?

Ok I am not serious here. I was just wondering - as for us, time always moves forward; doesn't that mean time have a direction? So why not call it a vector? :-)

phinds
Gold Member
Mathematically, time is a vector (it can go forwards and backwards). You will hear the term "arrow of time". Yes, for US time always moves forward but math doesn't have to describe reality.

hilbert2
Gold Member
Mathematically even the number zero alone forms a (trivial) vector space, but in physics we mean at least a two-dimensional object when we talk about vectors.

how to multiply then?

Mathematically, time is a vector

Hmm, so time IS a vector. But what happens when we multiply time with a vector quantity like velocity? Will it be a cross or dot? (cross maybe? because displacement is a vector...) And what will be the angle between? I am confused.

hilbert2
Gold Member
You can calculate the cross product only for two 3-dimensional vectors and the dot product is defined for two vectors of same dimensionality. Multiplying with a 1D vector (like time) is called scalar multiplication.

Multiplying with a 1D vector (like time) is called scalar multiplication.
If we multiply velocity with time, we get displacement. How can we multiply two vector and get a vector? (I thought it is possible only under cross multiplication, but you reminded me that cross multiplication is only for 3D.) So how do we multiply the 'vector' time with any other vector?
Sorry if I am missing something.

Time is not a vector, not in the basic physics sense. It is a component of the 4-vector space-time, but that is not what you asked about. The fact that time can be positive or negative does not make it a vector.

Dale
Mentor
2020 Award
Ok I am not serious here. I was just wondering - as for us, time always moves forward; doesn't that mean time have a direction? So why not call it a vector? :-)
Do you think there is a difference between a 1D vector and a scalar? If so, what is the difference? If not, why bother?

Do you think there is a difference between a 1D vector and a scalar? If so, what is the difference? If not, why bother?
I am beginner intro physics student. So I might be lacking in concept. That said; when we study 1D kinematics don't we consider 1D quantity like 1D velocity. 1D acceleration as vectors?

D H
Staff Emeritus
Mathematically, time is a vector (it can go forwards and backwards).
Mathematically, time is a scalar. Scalars can go forwards and backwards. Since a 1D vector is equivalent to a scalar, one could also say that time is a vector. That's a bit tautological, though.

That's not what I think the OP was asking. I suspect it was more along the lines of "can time be multidimensional?"

a 1D vector is equivalent to a scalar
Sorry, but I dont understand this. For example: the unit vector along x axis is a 1D vector. Did you mean to say that its actually a scalar? And also,
when we study 1D kinematics don't we consider 1D quantity like 1D velocity. 1D acceleration as vectors?

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The important fact to learn is that scalarity, vectoriality and, more general, tensoriality of a physical quantity isn't a property of quantity itself but it has to be defined with the respect of what transformation we are acting, e. g. Velocity is a vector with the respect of rotation ( that is a particular kind of transformation) or mass of particle is scalar with the respect of rotations but also with the respect of Lorentz trasformations, and so on. They are very important topics for example for quantum field theory, Standard model of particle, and other subject. To learn more you can read about this on any book of Group Theory and Their Representation especially if they are written for physicists!

Actaully I dont understand this. For example: the unit vector along x axis is a 1D vector. Did you mean to say that its actually a scalar? And also,

In basic, intro physics (which is where you are at, right?) think of a vector as needing two or more number to describe. A scalar only needs one number to describe. The unit vector along the x-axis is a vector. In two dimensions you describe it by <1,0> where the first number is the x component and the second number is the y component. In three dimensions its described by <1,0,0> where we have a 1 in the x component and 0 in the y and z component. But if we were not considering two or three dimensions, if we are just considering one dimensional motion then there is no y or z component to be considered at all. In that case the unit "vector" lies along the only axis there is (no need to call it the x axis since its the only axis). And in this case we can just describe it with one lone number, 1 - a scalar.

when we study 1D kinematics don't we consider 1D quantity like 1D velocity. 1D acceleration as vectors?

I would guess that is done for convenience and/or as an aid into moving into 2 and 3 dimensions. Velocity in one dimension is not really a vector since it only needs one number to describe. You might as well cause it positive or negative speed (a scalar). Same with acceleration.

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Matterwave
Gold Member
I don't think it's productive to confuse the OP by giving him the mathematical definition of a vector space, or more abstract definitions of vectors in terms of coordinate transformations, etc., at this point. It seems to me that the OP is a beginner in physics who's simple concept of physical vector at this point is "a quantity with both magnitude and direction (in space as is usually implied)" whereas a scalar is something "only with magnitude (usually implied to be positive, like speed, or distance)".

In terms of those definitions, time is a scalar, it obviously doesn't have a direction in space. The passage of time we measure by the clocks that tick, and the number of ticks between two events is the "time between two events". As there is no absolute synchronization of time, we can specify time=0 arbitrarily, and really it is only the difference in time between two events that matters (and in fact this is all we can ever measure). As such, time is very much like distance. Between any two events, there is one unique number corresponding to time between events(I am working here only in the Newtonian framework of course, in special relativity there would be an infinity of different numbers), not an arrow that points from one event to the next.

What one should realize, though, OP, is that definitions are just that, definitions. They are useful only in so far as they help us conceptually or quantitatively. We should not allow the baggage of such definitions to hinder us. Given the definition you have learned, time is a scalar. But in this instance, it doesn't really help us all that much to pigeonhole ourselves and classify time into a "scalar". Really, we know what time physically is (that which is ticked off by clocks) and that should be enough for us.

Isn't correct! There are quantity which have three, but also quantity that have four or more ( also infinite) number of component but still remain scalar. There are quantities which have four components and are (called) spinor under Lorentz trasformations. For example the spinor field that describes particles called DIRAC FERMIONS (the electrons for example) with quantum spin 1/2, has four components and it is a spinor not a vector under rotations, under Lorentz trasformations, and under Poincaré transformations.

I would guess that is done for convenience and/or as an aid into moving into 2 and 3 dimensions.
Hey, thanks a lot. :-)

But I wonder why while writing intro physics books they never clarify this. I started this thread just for kicks but in the end I have learned something important.

D H
Staff Emeritus
Sorry, but I dont understand this. For example: the unit vector along x axis is a 1D vector.
No! You have to look at the space of which that vector is a member. Aside: "Space" here does not mean three dimensional space. It means the set of all possible vectors. There are many different vector spaces. The Euclidean plane, the three dimensional space you learn about in physics, and abstract spaces invented by mathematicians.

If the space is the number line, then yes, your unit vector along the x axis is a 1D vector. However, there is only one axis here, so why label it? By labeling it you are presumably talking about a higher dimensioned space such as the Euclidean plane or 3D space. If the mathematical space is the Euclidean plane, then your unit vector along the x axis is a 2D vector. You need two parameters to characterize that unit vector. If it's three dimensional space of classical physics, you need three parameters to characterize that unit vector. And so on.

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Is very important to understand that the notions of scalar, vector, tensor are meaningless whiteout specifying under which tradformation

There are quantity which have three, but also quantity that have four or more ( also infinite) number of component but still remain scalar.

Well, well... I think I should take a break. :-)

You have to look at the space of which that vector is a member.

That post clarifies a lot.

D H
Staff Emeritus
I don't think it's productive to confuse the OP by giving him the mathematical definition of a vector space, or more abstract definitions of vectors in terms of coordinate transformations, etc., at this point.
Agreed.

Writing about concepts such as spinors and tensors, Lorentz and Poincaré transformations, QED and general relativity, or group theory IS NOT HELPING.

Yes, I broke the rules of the forum by shouting. Every once in a while that's needed.

On the other hand, giving a slight inkling into the concept of vector spaces sometimes does help in straightening out beginner misunderstandings with respect to vectors (the physics 101 version of a vector, something with a magnitude and a direction). A full blown treatment? No. That doesn't help.

It seems to me that the OP is a beginner in physics who's simple concept of physical vector at this point is "a quantity with both magnitude and direction (in space as is usually implied)" whereas a scalar is something "only with magnitude (usually implied to be positive, like speed, or distance)".
Agreed. I take it that it's the sign that was confusing the OP rather than the dimensionality. If that's the case, pointing the OP to the paper "On determinism and well-posedness in multiple time dimensions" by Craig and Weinstein would fall in the category of "not helping".

If it's the sign that's the problem, Amio, consider temperature as an analogy. One way to express temperature is on an absolute scale, with zero meaning absolute zero. This makes all temperatures that will be encountered in an introductory physics class positive. However, telling someone that the high temperature tomorrow will be 308 K doesn't quite translate into "summer's here!". Telling them that the high will be 35 °C or 95 °F does. That Kelvin scale isn't as useful in the everyday world as are the Celsius or Fahrenheit scales. But now temperature can be negative.

One can in a way do that with time (please keep relativity out of this, everyone) by setting t=0 at the big bang. Doing that is not every useful in the everyday world.

Dale
Mentor
2020 Award
I am beginner intro physics student. So I might be lacking in concept. That said; when we study 1D kinematics don't we consider 1D quantity like 1D velocity. 1D acceleration as vectors?
There is no benefit to doing so. None of non-trivial concepts of vectors are relevant to 1D motion. That is precisely why 1D motion is taught, because you can treat all of the quantities that are vectors in higher dimensions precisely as though they were scalars.

For example, for a 3D vector y like velocity then there is a difference between the operations ##x\;\mathbf{y}## where x is a scalar and ##\mathbf{x}\cdot\mathbf{y}## where x is a vector. There is no such difference if y is a 1D "vector" like time.

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I think I got the idea. Thanks everyone.