Ask Ethan: What Is A Scalar Field? (2023)

Scalars, vectors, and tensors come up all the time in science. But what arethey?

One of the major goals of science is to describe our reality as accurately as possible. If you give us a setup — and tell us what the conditions of a system are — and our best scientific theories are powerful enough, science will be able to predict for you precisely how that system will evolve into the future. If we can measure and know the properties of whatever it is we’re dealing with, from atoms to humans to planets to stars and galaxies and more, a useful scientific theory will be able to predict what they will be like a finite time from now. But sometimes, understanding what a scientific theory is, does, or even means requires that we learn some terms we’re unfamiliar with, including ones that have their roots in mathematics and often aren’t intuitive. That’s what’s tripping up Elen Sentier, who asks:

“Help! I’m getting somewhere with your piece on string theory but I haven’t a clue what a scalar field is. I have no maths and no physics but I love the ideas and concepts. Can you explain a scalar field in words of 2 syllables please?”

This is a perfectly reasonable request, but one that’s a challenge even for a seasoned scientist or science communicator. From hereon out, let’s teach you what a scalar field is, and why it matters, in the simplest terms we can find.

Ask Ethan: What Is A Scalar Field? (1)

Let’s say you want to describe our planet: Earth. There are a number of things we can choose to look at and study. For instance, we can decide to look solely at Earth’s surface, and ask questions about the terrain at every point on the globe. In an instant, you can start thinking of certain things you might like to know. They include:

  • Where are we looking, right now, in terms of place?
  • When is the instant in time that we’re concerned about?
  • What is our height above the “sea level” of Earth’s surface?
  • If I put down a ball, which way will it roll, and how fast will it roll down that slope?
  • Is there any stress or strain on the Earth at that point?
  • If I put down a large amount of water, how will the water flow? What route will it take and how fast will it go? Will it develop eddies or a vortex anywhere?

The Earth itself is just one object to think about, but thinking about the surface of our planet gives us a great way to think about what a field is, as well as the different types of fields that matter for science.

Ask Ethan: What Is A Scalar Field? (2)

Let’s start with the question of height. If the Earth were perfect, smooth, and didn’t rotate, it would form an exact sphere. Because the Earth rotates, that shape gets compressed at the poles and bulges in the middle, forming a shape known as an oblate spheroid. Still, there are ups and downs all along the surface, with the oceans, seas, lakes, and rivers filling some of the deep depths with water.

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Everywhere on the surface, then, we can ask a question like “what is our height above Earth’s sea level,” where sea level is the height at which every point on Earth would be covered in ocean if there were no land masses that rose above it. So if you want to describe “what is your height above sea level” at every point on the surface of Earth, how would you do it?

That’s exactly the case that would call for a scalar field.

Ask Ethan: What Is A Scalar Field? (3)

A scalar field is, to be frank, the most simple type of field you can have. What it says is, if you give the values that tell you “where” and “when” you are — where you are in space and when you are in time — the scalar field will give you one and only one value that describes the “amount” of stuff you’re trying to measure. If that thing you’re asking about is “height above sea level,” then the scalar field can tell you that height. Not just on average, or over the whole surface, but at each and every point. If the height of the Earth were something that changed over time (and over long enough timescales, it does), a scalar field could capture that, too.

But that scalar field won’t tell you all you’d ever want to know about the Earth’s surface. It just tells you “what is the value of the thing I’m asking about at any point in space and/or at any moment in time?” If you wanted, instead, to know the answer to some of the other questions, like “which way will water flow down this surface,” a scalar field just isn’t enough.

For that, you’d need a slope map instead, and that’s not a scalar field, but a vector field.

Ask Ethan: What Is A Scalar Field? (4)

So what is a vector field, then, and how is it different from a scalar field?

A vector field doesn’t just tell you what the “value” of something is at each and every point in space and time, but tells you a value and also which way that value points in some sense. A river will always flow, at any point, at a certain speed, but speed alone is not enough to describe its motion in full. The river also flows a certain way along a certain heading: we need to know which way it goes, not just how fast it’s going.

There’s an extra thing that we can do with a vector field that we cannot do with a scalar field: we can have a vector field give rise to a “curl,” which describes how objects move around a certain point in space. In math, the curl of a scalar field is always zero, so if all we used were scalar fields, we could never have a vortex, a whirlpool, a twister, or motion that describes going around in a circle. If you point your thumb one way and look at how your fingers want to wrap around your hand, that wrapping motion you’ll try to make is one way to conceive of a curl.

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Ask Ethan: What Is A Scalar Field? (5)

In our concrete, real world, scalar fields can get us very far, but they can’t get us any old thing we can dream up. To account for motion, we need to know which way things are going, and that means a vector field. To explain forces, and hence, how motion changes over time, we need not just the amount of force, but also which way that force points. For spinning motions, as things rotate or revolve around other objects, we also need vector fields; do things curl the way your fingers curl around your right hand or your left?

Think about all the different traits that an object might have that you might want to know, measure, or use to predict the outcome of a system that’s set up a certain way. Almost all of them can be described fully by either a scalar (just knowing the amount is enough) or vector (where the amount and also which way it points is important) field.

  • Mass is a scalar.
  • Speed is a scalar.
  • Height is a scalar.
  • Distance is a scalar.
  • Time elapsed is a scalar.
  • Slope is a vector.
  • “Which way” is a vector.
  • Torque is a vector.
  • Force is a vector.

Well, mostly on that last one.

Ask Ethan: What Is A Scalar Field? (6)

In the eyes of Newton, a force is always a vector. It has a strength and it goes along a certain heading, and that’s enough to describe it fully. Between two charged objects, that force is a vector. Inside an atom’s core, those forces — between protons and neutrons and even inside a proton itself — are all vectors.

But in the eyes of Einstein, when it comes to the most famous force of all (the one that occurs between all massive objects, but that has too many word-parts to use here), force is neither a scalar nor a vector, but requires something even more complex to describe: a tensor.

So what’s a tensor, then?

Picture a solid object like a cement pillar. You have it, you watch it, and you subject to a lot of real-world factors. It gets hot and cold. It has weight placed atop it and removed. People push it, pull it, or lean against it. The masses around it all pull (or push) on it. If you could map out all the different forces acting inside the pillar, including things like strains and stresses, you’d find that they not only varied with time and in which ways they pointed, but that even a vector field wasn’t enough to describe it. Instead, you’d need something even more broad, that can include things scalars and vectors couldn’t. That’s when you need a tensor.

Ask Ethan: What Is A Scalar Field? (7)

If you were to push on something along a precise heading, you’d expect the force to go that same way: along that distinct axis that you pushed it. But sometimes — and you can poke a congealed Jell-O mold if you want to see the effect in action for yourself — a starting force that points one way can create forces within an object (or on an object) that point along different axes than the initial action that started it all. This creates forces along lines you could not explain if you only worked with scalar or vector fields.

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This was the key to Einstein’s great idea. If you can, from any point of view that you choose, tell us:

  • where all the masses, photons, and other quanta are,
  • what their mass and mass-like values are,
  • how they are placed,
  • and how they are moving at any one point in time,

Ask Ethan: What Is A Scalar Field? (8)

Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard!

then Einstein’s theory can tell you, at every point in space and time, how space will curve, and how space will tell matter and photons and every other quantum how to move.

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Ask Ethan: What Is A Scalar Field? (9)

This theory — the biggest science exploit of Einstein’s life — is purely a tensor theory. There is no scalar part; there is no vector part. In fact, there are very strong limits on how much a “scalar” or “vector” part can contribute to how spacetime curves. If we want to get the cosmos that we know and observe, we cannot have scalar or vector parts to the law that governs spacetime.

And that is one big problem with string theory. String theory doesn’t give you 3D space (or 4D spacetime), but rather six extra ones that you have to get rid of. It doesn’t give you a tensor theory that tells you how mass curves spacetime, but rather a theory with both scalars and tensors, and you have to purge the theory of all the scalar ones. Put simply, it gives you “extra stuff” to your cosmos that our cosmos does not have.

One of the toughest tests comes from LIGO, which has seen ripples in spacetime from more than 50 events as of today. The way they deform the fabric of space shows a purely tensor nature, with very little wiggle room for scalar or vector parts to even exist; the constraints have gotten very tight.

Ask Ethan: What Is A Scalar Field? (10)

All in all, a scalar field can only give you an “amount” of something, but it can give it to you at every point in space at any time that you choose. If you want to add in something more, like which way an amount of something points, you need to upgrade to a vector field. And if you have something even more complex, like:

  • space that is curved,
  • stresses and strains,
  • or effects that point along different headings from the force that begat it,

even a vector field can’t capture all of them. For that, you need a tensor field, like Einstein’s theory of how mass, matter, and more curve spacetime.

(One book I like that goes into the gory details about the difference between scalars, vectors, and how they allow us to derive different features of our real world is called Div, Grad, Curl, and All That; if you struggled with advanced math in college, this can help make some complex ideas more clear.)

A scalar field is just a field that has a value — or amount — assigned to it and nothing else. If you want to know anything else, even as simple as which way something points, a scalar simply won’t do. There may be extra scalars floating around out there in the form of fields or quanta we have yet to meet, but as far as we know, there isn’t one that’s part of Einstein’s theory. Finding out why is one challenge string theory has yet to surmount.

Send in your Ask Ethan questions to startswithabang at gmail dot com! (And yes, I know that startswithabang is more than 2 syllables!)

Starts With A Bang is written by Ethan Siegel, Ph.D., author of Beyond The Galaxy, and Treknology: The Science of Star Trek from Tricorders to Warp Drive.

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Ask Ethan: What Is A Scalar Field? ›

A scalar field is just a field that has a value — or amount — assigned to it and nothing else.

What is a scalar field in simple words? ›

In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (with units).

What is an example of a scalar in real life? ›

Examples of scalars are volume, density, speed, energy, mass, and time. Other quantities, such as force and velocity, have both magnitude and direction and are called vectors.

What is a scalar field and vector field? ›

A scalar field is when a scalar is assigned to each point in a space region. A scalar field is, for example, the temperature at a given location on the planet. A vector field is a set of vectors assigned to each point in a space region.

What is an example of a scalar and a vector field? ›

Examples of scalar quantities include mass, electric charge, temperature, distance, etc. A vector, on the other hand, is an entity that is characterized by a magnitude and a direction. Examples of vector quantities are displacement, velocity, magnetic field, etc.

What is scalar for dummies? ›

A scalar is a quantity that is fully described by a magnitude only. It is described by just a single number. Some examples of scalar quantities include speed, volume, mass, temperature, power, energy, and time.

What is an example of a scalar field? ›

A scalar field is a name we give to a function defined in some sort of space. Thus, in ordinary three dimensional space the following are examples of scalar fields: sin xyz, cos z, x2 + y2 + z2. A linear field is one of the form ax + by + cz + d for some constants a, b, c and d.

What are 5 examples of scalar? ›

Some examples of scalar quantities include:
  • temperature – eg 10 degrees Celsius (°C)
  • mass – eg 5 kilograms (kg)
  • energy – eg 2,000 joules (J)
  • distance – eg 19 metres (m)
  • speed – eg 8 metres per second (m/s)
  • density – eg 1,500 kilograms per metre cubed (kg/m 3)

How is scalar used in everyday life? ›

Scalar quantities, are the measurements that strictly refer to the magnitude of the medium. The motion of an aircraft can be described by vectors pertaining to displacement, direction, acceleration and velocity. Eg speed , strength . The examples of scalar magnitudes they are present in everyday life.

Why is scalar important? ›

Scalars and Vectors

Even though scalars just provide information about magnitude, they are very important and used all the time in physics. In contrast, vectors are used to describe quantities with both magnitude and direction.

What is scalar field in cosmology? ›

Scalar fields are likely to play an important role in cosmology, already being crucial in both particle physics and quantum field theory. In the early Universe they may describe the inflaton field responsible for inflation or be linked to the spontaneous breaking of symmetries which give particles their mass.

What is the symbol for scalar field? ›

We often write the symbol that represents a scalar field as V ( r → ) where the position vector not only reminds us that the scalar field may vary from point to point in space, but also give us a coordinate independent symbol to describe the point at which we are evaluating the field.

Is a scalar field just a function? ›

A scalar field is a function that gives us a single value of some variable for every point in space. As an example, the image in Figure 1.2.

How do you find the scalar field? ›

A scalar field is a function of spatial coordinates giving a single, scalar value at every point (x, y, z). 2. The gradient of a scalar field φ grad φ is defined by: ∇ ϕ = ∂ ϕ ∂ x i + ∂ ϕ ∂ y j + ∂ ϕ ∂ z k = ( ∂ ϕ ∂ x , ∂ ϕ ∂ y , ∂ ϕ ∂ z ) .

What are 20 examples of scalar? ›

Scalar quantities: mass length time speed temperature electric current Vector quantities: force velocity acceleration displacement magnetic induction
  • mass.
  • length.
  • time.
  • speed.
  • temperature.
  • electric current.

What is a real world example of a vector field? ›

Examples of vector fields: Gravitational force field, electrostatic force field, fluid flowing to a river, wind is blowing.

What is the literal meaning of scalar? ›

1. : a real number rather than a vector. : a quantity (such as mass or time) that has a magnitude describable by a real number and no direction.

What is the difference between vector and scalar? ›

Mathematicians and scientists call a quantity which depends on direction a vector quantity. A quantity which does not depend on direction is called a scalar quantity. Vector quantities have two characteristics, a magnitude and a direction. Scalar quantities have only a magnitude.

Is time a scalar or a vector? ›

For example, displacement, velocity, and acceleration are vector quantities, while speed (the magnitude of velocity), time, and mass are scalars.

Is magnetic field a scalar field? ›

Just like a vector quantity, a magnetic field is described with both magnitude and direction. Thus the magnetic field is vector quantity.

Is electric field a scalar? ›

Explanation: Electric fields are not scalars.

Is the Higgs field a scalar field? ›

The Higgs field is a scalar field, with two neutral and two electrically charged components that form a complex doublet of the weak isospin SU(2) symmetry. Unlike any other known quantum field, it has a "Mexican hat-shaped" potential.

What are scalar objects? ›

Scalar objects are used for singular variables that are not part of a table or an array.

What is scalar function? ›

A scalar function is a function that returns one value per invocation; in most cases, you can think of this as returning one value per row. This contrasts with Aggregate Functions, which return one value per group of rows. Category. Description. Bitwise Expression Functions.

What is scalar values? ›

Definition: A scalar valued function is a function that takes one or more values but returns a single value. f(x,y,z) = x2+2yz5 is an example of a scalar valued function.

What are the three types of scalar quantities? ›

Some common scalar quantities are distance, speed, mass, and time.

What are the two things that all scalars must have? ›

Scalars are quantities that only have magnitude (or size), while vectors have both magnitude and direction. Explore some examples of scalars and vectors, including distance, displacement, speed, and velocity.

Is stress a scalar quantity? ›

Stress is a vector quantity.

What is scalar in deep learning? ›

Scalars, vectors, and matrices are fundamental structures of linear algebra, and understanding them is integral to unlock the concepts of deep learning. A scalar is a singular quantity like a number. A vector is an array of numbers (scalar values). A matrix is a grid of information with rows and columns.

Is dark energy a scalar field? ›

Unlike vacuum energy, which must be spatially uniform, scalar-field dark energy can clump, providing a possible new observational feature, but in most cases is only expected to do so on the largest observable scales today (see Section 10.2. 1). ~ 10-60, means that the scalar field potential must be extremely flat.

Is dark matter a scalar field? ›

The dark matter can be modeled as a scalar field using two fitted parameters, mass and self-interaction. In this picture the dark matter consists of an ultralight particle with a mass of ~1022 eV when there is no self-interaction. If there is a self-interaction a wider mass range is allowed.

Is gravity a scalar field? ›

Newtonian gravitation is a scalar theory in which the gravitational interaction is described by a gravitational scalar field or potential \varPhi , that satisfies a Poisson equation. It is complemented by Newton's second law of mechanics for the trajectories of particles moving in the gravitational potential \varPhi .

What is another word for scalar? ›

Words used to describe amounts or quantities. sheer. underlying. quantifiable.

What is the simple definition of gradient of scalar field? ›

The Gradient of a Scalar Field

We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar field as the gradient of that scalar field.

What is the difference between a scalar quantity and a scalar field? ›

The difference between a scalar and a scalar field is that the former is one single value of the latter. The scalar field exists in all points of space and at any moment of time while the scalar is its value at a certain location at a certain time.

Which of the following best describes a scalar? ›

Correct answer:

Scalar quantities are defined by a magnitude with no applicable direction. In contrast, vector quantities must have both magnitude and direction of action.

Why is it called scalar? ›

The word scalar derives from the Latin word scalaris, an adjectival form of scala (Latin for "ladder"), from which the English word scale also comes.

What is a scalar function? ›

A scalar function is a function that returns one value per invocation; in most cases, you can think of this as returning one value per row. This contrasts with Aggregate Functions, which return one value per group of rows. Category. Description. Bitwise Expression Functions.

What is a scalar field which is independent of time called? ›

A scalar field which is independent of time is called a stationary or steady-state scalar field. u = Φ(x, y, z, t) . Vector field.

What is the rate of change of a scalar field? ›

The greatest rate of change for a scalar field with respect to position is represented by its gradient, which is always parallel to the field surface at that point. The gradient of a scalar field is also known as the directional derivative of a scalar field since it is always directed along the normal direction.

Why are vectors important in everyday life? ›

Vectors are utilized in the storage of locations, directions, and velocities in video games. The position vector tells us how far away the object is, the velocity vector tells us how long time it will take or how much force we need to apply, and the direction vector tells us how we should apply that force.

What is a real life example of scalar and vector? ›

For example, consider the forces acting on a boat crossing a river. The boat's motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. Scalar quantities, are the measurements that strictly refer to the magnitude of the medium.

Is time a vector or a scalar? ›

For example, displacement, velocity, and acceleration are vector quantities, while speed (the magnitude of velocity), time, and mass are scalars.

Is speed a vector or scalar? ›

By definition, speed is the scalar magnitude of a velocity vector.

Is energy a scalar or a vector? ›

Energy = Work= F*s, here both the force and the time is vector quantity. And we know that the product of two vector quantities give the scalar quantity. So we can say that energy is a scalar quantity.


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