What is time, does it flow, and if so what defines its direction?

I will adopt a different viewpoint than other answers. The flow of time can be used in relativity.

Some answers above and this answer to Does it actually take infinite (observer) time for someone to fall into a black hole? point to time as in the Block Universe. Space-time is an unchanging 4D block. Time does not flow.

According to this answer to Understanding emergent phenomena in the block universe. (Reworded question), for a flow of time to exist, there would have to be a physically meaningful parameter $lambda$ that would define the flow via $dt/dlambda$ on the 4D block. There is no flow. People define an affine parameter like this all the time in general relativity, but it is has no physical meaning.

In this view, I am something like a roll of film. Each frame is me at a particular time. Each frame exists independently, regardless of which frame is "now" or whether or not the movie is being played back at all. In each frame, I am in a state that feels the flow of time and remembers the past.

However, as this answer to How can a particle's position be random and uncertain in quantum mechanics if it is already pre-determined in relativity? makes it clear, this view does not mean the future is predetermined in a way that is incompatible with quantum mechanics. However quantum mechanical wave functions and measurements turn out, the resulting events can be laid out in a 4D space-time block. The block is a convenient way to view events for relativity.

Many physicists adopt the view that the Block Universe is the way the universe works. But as John Rennie said in his answer, there are other views that this is just mathematical skulduggery, and that time really does flow.

For example, general relativistic numerical simulations often use the ADM formalism to start with an initial condition and calculate future states. Space-time is divided into spatial sheets labelled by t. The 3D metric tensor for each sheet is used as a dynamical variable, along with its conjugate momentum. A Hamiltonian can be derived, and equations of motion. Future states can be calculated without knowing them in advance.

This all shows that the Block Universe is compatible with physics as we know it, in spite of its violation of common sense. But the Block Universe isn't required.

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Part of the common sense view of time is that it flows. The present is all that exists. The future hasn't happened yet. The past is over and gone. I would like to make a case that this is also compatible with relativity. The flow of time is not required by relativity, but if it exists it does the theory no harm. To do that, I will look a little deeper at what simultaneity means.

Light travels at $3 times 10^8$ m/s. We are comfortable at $3$ m/s. We find relativistic physics counter-intuitive.

Consider a world where the fastest motion is $3 times 10^{-8}$ m/s. This is about 1 m/year, the speed of a glacier. Glacier world physicists move too slowly to see effects of motion that we are comfortable with. We can learn about our conceptual difficulties by looking at what they find counter-intuitive about everyday physics.

In classical glacier world physics, each object has a fixed, intrinsic property called position. Every observer agrees on the position of a given object. Position can be used as the identity of the object.

However, precise measurements or measurements over long time intervals show that position changes with time. This leads to counter-intuitive concepts like the "failure of sameplaceity" and "velocity".

These can usually be ignored. But observers traveling at everyday velocities would see strange effects. Glacier physicists Bob and Alice both agree that they both have position $x_0$ at time $t_0$. At $t_1$, Bob says he has position $x_0$, just as one would expect. Likewise, Alice says all is normal with her. But Bob says Alice is at $x_1$.

This leads to confusion. Neither Bob nor Alice has changed their unchangeable position. And yet they are now at two different positions. Are those positions somehow the same? Is is somehow possible to occupy two positions at the same time?

We have a similar confusion over the failure of simultaneity. We think of time as absolute. A time $t_0$ uniquely identifies a slice of space-time. When $t_0$ is now, all events in that slice have their moment of existence. The state of the universe is uniquely specified by $t_0$. All observers agree on this. The flow of time is the progression of the universal state from cause to effect. This is why the flow of time makes sense.

However measurements show the speed of light is constant. This leads to the failure of simultaneity. Everyday physicists Robert and Alicia pass each other at relativistic velocity. Both choose coordinate systems that agree they pass at $(x_0,t_0)$. They look at the event Robert sees as $(x_1,t_0)$. Robert says this is having its moment of existence now. Alicia says the same event already happened or hasn't happened yet. This leads to confusion. One event happens at two different times.

The Block Universe is one way around this. If an event exists at two different times, it must not have a momentary existence. It must exist whenever it is time for it to exist in any frame of reference. This preserves the notion that all events in a slice of space-time defined by a time exist at that time. However, it destroys the notion of flowing time. A succession of events do not come into existence and disappear. The whole block of events in all of space-time just statically exists.

Another way is to let go of the notion that simultaneity is absolute. Each object follows a world line where time flows. Each event that object experiences comes into existence at its time and disappears. But there is no universal way of matching up times of separate events. The match up is as motion dependent as matching up the position of separate events.

This is a much harder conceptual stumbling block for us than anything else in special relativity. We are used to Robert seeing two events as the same place, but Alicia seeing them as different because of her motion. But Robert seeing two events at the same time and Alicia seeing them as different because of her motion violates our notion of what time is. It is hard to get used to the idea that time is not what we thought it was.

This does not change what simultaneity is in relativity. The simultaneity of two separate events measured in a particular inertial frame of reference can be inferred. Robert, sitting at $x_0$, can always rely on his clock at $x_0$ to be uniform. He sends out a pulse of light that reflects off an event at $x_1$ and returns. He records the sendoff and return times. The speed of light is constant, so the pulse spends half the time going out and half returning. The reflection is therefore simultaneous with the event Robert experienced at the time halfway between.

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The difference between the Block Universe and Flowing Time viewpoints is purely philosophical. They both use the same space-time diagrams, do the same calculations, and arrive at the same answers. One viewpoint regards the space-time diagram as a block of statically existing events. The other regards it as a collection of loosely matched up histories.

Both notions of time are different from the common sense notion. I don't know of any experiment that can distinguish these interpretations of time. Use whichever one makes you the most comfortable.

However, there is one difference. The Block Universe does not include the progression of state from early time to later time. It must be added. There is some question of how to motivate it and explain the direction. The Flow of Time doesn't explain this progression. It is just assumed.

I got bumped off my question (it being perceived as a duplicate of this one) so I'll leave an answer here as a parting shot. John Rennie mentions that time flows in a direction at the rate of one second per second. That is fine to say but there are an infinite number of variations of that rate within our Universe alone.

Even for ourselves as individuals the rate appears to change according to how our brain is functioning. Our brains have different gears (alpha, beta, gamma, delta [and maybe more] wave types) and depending on which combination of them is in process time can appear to move more slowly or quickly. When we're multi-tasking we want it to slow down and when we're bored it appears to drag.

But enough of the philosophy already - we're talking the physics of time here which is basically the physics of something that doesn't exist. I can't remember who it is, some celebrity actually but the person experiences a bit of a anomaly, a "medical" condition and they don't have a perception of time like the rest of us (whatever % of human race).

Just like some people have no appetite reflectors which tell them when they have eaten enough (people have died from this) the person I forget who cannot judge time. So are they dysfunctional or more attune to nature than some?

I don't think animals have a perception of time. I think they perceive events and can remember them (actually and strangely experiments have proved that birds can pass "memories" of specific threatening persons to their offspring and the children react accordingly to the persons without previously meeting them) - however only humans have an abstract conception of time as a "thing".

In fact in Shakespeare's Henry the IV the protagonist mocks a fool because he refers abstractly to time. The kind of moral is that people who have time to wonder about time don't really achieve much. That was written 400 years ago and now (laughing at myself) I am doing the same thing. So I guess I won't after this little answer.

Recently listening to Julian Barbour (who has given a solution to the three body problem) he kind of says the same thing. He tells us that time as we see it is a kind of leftover from events that don't match up perfectly. Like if you're supposed to meet a friend and they arrive at the meeting place after you one might begin to notice the passing of time. If you had met up simultaneously your mind might have remained fully engaged in life and time would have been unimportant.

To give a similar example a dude said he once met Bruce Lee (the martial artist) in an elevator and they began discussing Kung Fu. They got to the same floor so they tried out a few moves. The next time the dude glanced at his watch it was 7am. He got into the elevator at 10.35pm so time disappears when we don't watch it.

Similar to quantum mechanics if we go to sleep we can depend that our reliable alarm clock will tick (as John Rennie says) one second per second all night until it wakes us from unconsciousness. BUT we have no way of proving this is what actually happens. Just like an uncollapsed wavefunction what is time actually doing when we aren't watching?

How do we even prove that we are the same person who woke up that went to sleep? The Buddhists say that the ego is a function. It is temporary and sometimes is there, sometimes not. That is the actual meaning of the "if a tree falls in the forest and no-one is around, does anyone hear it?" koan. It translates as "if the ego is not there to watch is there an ego?" - well, obviously not if it is not there for a moment. It might come back though if the Physics Stack Exchange answer you worked hard on gets a lot of downvotes ha ha ha.

So on earth the nearest accurate measure of time is days. Everything else is nit-picking that humans get up to because they want a lot of things so they start calculating how to get them and that means we divide events into portions we call time.

That is Barbour's point too. We need to perceive division in order to function and we possess an innate time-dividing tool just as we possess an identity dividing tool (the ego) but as is seen in people who have anomalies this can be absent.

Even an emotion like sadness is never the same twice and as a famous philosopher said "You never jump into the same river twice." I'm not sure how to end this ramble conclusively. Perhaps I'll just say we agree to believe we all roughly know what an hour is and a second is a very convenient measurement.

No, wait. I'll end the answer with a question. How many of you despite your best efforts arrived late for an event, meeting or failed to do something that you had to do in a specific amount of time? The time was accurately fixed in your brain but life was just larger and more impersonal and beyond your control so much that the concept of time didn't work.

What do I know though? I'm just a clock. Sometimes I tick, sometimes I tock.

Physicists treat time as a coordinate which can be assigned to an event along with the three spatial coordinates, and use the term spacetime to refer to the resulting 4 dimensional mathematical representation.

Physics effectively defines time as a count of change. Specifically, a second is 9,192,631,770 successive hyperfine transitions of a caesium 133 atom.

If you think of time, therefore, as a count of change, you will see that it does not flow, it is simply a running total of the counter. It follows that time will always increase, as a count of successive instances always results in an increasing number.

What is time? Ask ten different people and you'll get ten different answers. This is how I understand it. The word time is a term describing temporal motion. Time is what we do. Time doesn't flow, we time. We don't move through the time dimension, we time through the temporal dimension. Time is a verb.

It's our motion from one moment to the next moment. Clocks measure our progression. Clocks measure how far we time. Temporal distances are measured in seconds, minutes, hours, days, years, etc. So we move through the spatial dimension and we time through the temporal dimension.

The speed at which we time is known as our temporal velocity. Timing through the temporal dimension is not something only humans do. Everything except massless particles, times.

Now, because the spatial dimension and the temporal dimension are interlinked, the faster we move, the slower we time, but due to the fact that our perceptions are determined by our temporal velocity, we never perceive time dilation locally.

Now, I may be wrong, but there is no scientific evidence which suggests that my theory is implausible. Correct me if I'm wrong, but, if you disagree, just because it doesn't match your current views, don't comment.

Alright, let me too give it a try:

• What do physicists mean by time? Let me start with a quote by Hermann Weyl:

The world simply is, it does not happen. Only to the gaze of my consciousness, crawling up along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.

When he writes "The world simply is" he is obviously not using is in the sense of "existing in time". Time (as meant in the quote) comes into existence by means of our consciousness (which divides the world into a past and a future, but which in reality "are" just there) which "crawls up" along the lifeline of our body. An analogy would be a barrel organ together with a punctuated paper music roll. We are the barrel organ (consciousness) and we move along the roll (the world that simply is), thereby producing the music that's already present in its entirety on the roll. This is also what Einstein thought:

For us, who are convinced physicists, the distinction between past, present, and future has no other meaning than that of an illusion, though a tenacious one.

I think that this is the way most physicists think about time (the coming into existence, by means of our consciousness of the movement of a world that simply is; the past, present, and future co-exist). But for certain, there are some (or many, I didn't investigate) who think time does exist objectively.

Because by the same token, we can say that it is the world that is continuously changing and not a priori "simply is". Our consciousness perceives this constantly changing world as a world in which there is time. Or, in the analogy of the barrel organ, we are not moving along the music roll but the music roll unfolds before us (without all the music already being there).

Time would then be constituted by the constantly changing configurations of all elementary particles. I think one can say that time is then a phenomenon emerging from the collective behavior of truly elementary particles for which time does not exist. How can time exist for a truly elementary particle if it always stays the same? In other words, how can it age? Aging, a good example of time, emerges from the collective behavior of elementary particles.

• How does time flow? I can't answer this because time can not flow. Which is, I think, how most physicists think. The moving elementary particles from which time emerges obviously can flow. Through space, with which time forms an absolute spacetime. Subjectively one can perceive time as "flowing" very slowly (when one has a bad time) or very fast ("time flew by very fast when we kissed"), but that's a question of words. Or look at the situation when you dream: the clock next to your bed says five minutes have passed, but in your dream, it seemed that much more time has passed. For animals, the perception of time may be very different too. But this aside. So the answer to your question is that time can't flow in the usual sense. Which means it hasn't a direction either.

• Why is there an arrow of time? First, let me emphasize that the arrow of time doesn't imply that time has a direction. The arrow doesn't point to a point in space. The arrow neither points from the past to the future, because time obviously has no direction in time. The arrow of time just means that the past is different from the future, and an arrow is a good analogy to address this. So the question you ask is in fact: "Why is the past different from the future?" According to the second law of thermodynamics, the collective behavior of all particles in the Universe is such that the particles tend to configure themselves increasingly disordered (if the initial configuration isn't in a maximal disordered state). Now, by definition, the past of the Universe finds itself in a more ordered state than the future finds itself in. You can ask yourself what would happen if the motions of all particle fields were reversed (which can be done only in the mind, but the late Stephen Hawkins thought once that this would happen if the Universe would change from expansion into contraction, a thought of which he later luckily admitted that is was a wrong one). Something to think about!

• If you want a short answer, then:
  • Physicists mean a coordinate axis mostly, when they talk about time . Also in relativity theory we are measuring time in meters!
• How does time flow?
  • I would say rather: how does it passes. In relativity theory we 'assume' that time is passing with the speed of light. We define the direction of the time by that 'definition':
    • The direction of the passage of time is the same direction where entropy grows. If you do not know what entropy is, then in a nutshell entropy is a measure of the disorganisation of the matter. So the time is passing in the direction where the disorganisation of the particles in our universe increases.

It was a short answer and it wasn't extremely accurate, but if you want a deeper explanation of these things you can read a lot of books and watch a lot of videos on youtube on this topic. I hope my answer was useful.

Of course John Rennie's answer is right, as well as John Duffield's answer. Both answers are correct mostly, and technical. But let me try to explain it in a more simpler way. So please let me try to add something by explaining it in a really simple, understandable way and by comparing the two answers where they are in agreement and where not.

1. What is time?
   By John Rennie:

   it is what physicists mean, by definition of the four vector, and that time is a coordinate in that 4 dimensional system we call spacetime.

   
   By John Duffield:

   it is the movement of the clock's pieces.

   
   Both are right. But both try to take it from another angle. Let's take them one by one, trying to unify them.

The first answer says the universe is set up so and the four vector is set up so that time as a dimension is just another dimension like the 3 space dimensions that we perceive easily. Now this is correct and it is just the measurement of spacetime. It explains that any movement (speed) in the space dimensions is affected by (and vica versa) the movement (speed) in the time dimension. If something moves at speed $c$ in space, it must not move in time (it's speed in the time dimension is 0). If something moves slower then speed $c$ in space, it must start moving in the time dimension. That is just an empirical fact, and that is how the universe and the four-vector is set up.

This also comes down to the block universe, where all the 'timeslices' already exist, and we just perceive them as we 'move' along the time dimension's coordinates.

Now there is no accepted theory on what everything consists of, and how rest mass is gained, but one theory is that massless particles (gluons) oscillate at speed c in some confinement.

In SR, time dilation is in one example explained by a photon clock, with two mirrors and a photon moving in between (like oscillating) the mirrors (some confinement). So if we take this as an analogy, we could say that time is the movement (in space) of the photon or the gluon between mirrors or confinement. So everything existing could consist of little gluon clocks.

So the first answer could lead to saying that by the movement (oscillation) of the little clocks' gluons and their speed in space (oscillating at speed $c$) affects their speed in the time dimension (speed 0, time frozen). So the four vector will show that these are not moving in time.

But as soon as we look at the higher level particles that they build up, they somehow, because of the confinement gain rest mass to the higher level particles, and those (the higher level particles) slow down in space. So they must start moving in the time dimension, and that is what their four vector will show, they will move in the time dimension with speed > 0.

So this is where the first answer connects to the second answer, it says that the movement (speed) in the space dimensions affected the movement (speed) in the time dimension.

The second answer says that time is just the movement of a clock's parts. So basically it agrees to the first answer by saying that it is all just like a photon clock and that movement in space affects movement in time.

2. Does time flow? No, both answers say no, because it is not time flowing, it is us moving along the time dimension's coordinates.

The first answer explicitly says that we are moving along the time dimension and that it is more like a block universe, where everything already exists.

The second answer says that time is just particles moving, so time itself does not flow but the particles are moving and that is what we perceive as the flow of time.

The first answer is right. that is just how the universe and the four-vector is set up. Let me show why the second answer is wrong, and to do that, I need to explain something that non of these answers mentions.

3. The case that none of these answers explain is: if everything in the universe (all the particles, energy) would not move in space (they would be stationary in space dimensions).

In this case the first answer would say that based on the four vector, we still would see that everything moves in the time dimension at speed $c$. That is just what the four vector says, and it is empirically true.

The second answer would imply that in this case, since nothing is moving in space, we do not perceive time, and there is no time, it does not exist. But that is not true. The second answer is based on a misconception. It implies that particles are moving in space and that our (who have rest mass) speed is normal, and that we would need to speed up (in the space dimensions) to the speed of light. That is not true. Everything massless in the universe is moving at speed $c$ in space (and moving at speed 0 in the time dimension) right away when it is created. Since we could use the analogy and say that everything consists of small photon (gluon) clocks, these are all moving in space at speed $c$ as they are created. If they somehow gain speed for the higher level particles that they build up (by oscillating in some confinement), then they slow those higher level particles down in space. That is the only way to move slower then speed $c$ in space, to slow down. As they slow down, by gaining mass, they (the higher level particles) start perceiving time as we do (who have rest mass), and start moving in the time dimension too.

Now, if we slow that much that we become stationary in space, that is just not special. We are then still moving at speed $c$ in the time dimension. The misconception you see is that you may think that speed 0 in space is special (for considering it's effect on the speed in the time dimension). It is not. Speed 0 in space is just the same (it's effect on the speed in the time dimension) as any other speed<$c$ in space. The only speed in space that has special effect (on the speed in the time dimension) is speed $c$. But that is the only speed that massless particles can have and if anything wants to move slower — it has to slow down by gaining mass.

So the second answer is wrong, when it says that particles are moving in space and that it is what we perceive as the flow of time. Because, even if nothing would move in space, we would still move in the time dimension at speed $c$.

How is that possible? The only solution is really the block universe, where our universe's versions are all existing in the time dimension, and we are just moving along the time dimension and realizing the actual version of our world.

But what makes us all perceive the same universe at the same time? How are we all moving along the time dimension together? The answer is: because we are all moving at the same speed (approx., flying with the Earth as it moves in space) in the same gravitational field. As soon as we move out of the Earth's gravitational field, we perceive time at a different rate (then the ones left on Earth), since we will be moving in the time dimension at a different speed because of GR. And as soon as we fly with a different speed in space, (in a fast spaceship) we perceive time differently (then the ones left on Earth) because of SR.

OK, but how are we able to move in the time dimension at different speeds in the first place? How can then somebody move away from the Earth and come back to the same point in the time dimension as the ones left on Earth? The answer is that if someone moves away from the Earth, so moves out of the gravitational field, and moves at a different speed then the ones on Earth, then that person moves away in all four dimensions. To come back to the same point in all four dimensions and meet the ones left on Earth at the same point in the time dimension too, he has to use extra energy to make up for the difference. Otherwise that person would only come back to Earth to the same space location but never to the same point in the time dimension as the ones left on Earth (since his speed in the time dimension is different). We would see that as if he would have aged less (or more).

So the first answer is right because even if something is stationary in space, it still might be in a gravitational field, and that changes it's speed in the time dimension (compared to the ones outside the gravitational field).

This is not my topic, and I am not an expert here at all, but I believe in extra circumstances in a black hole, time might even stop (their speed in the time dimension is 0) for the ones inside the black hole (from a far away observer's view), and they might seem to be stuck at a certain point in the time dimension. So there might be a special case when the speed in the space dimension is 0 and in the time dimension too (for a non-rotating black hole at the singularity). To start moving again in the time dimension, they would have to use extra energy to get out of the black hole's gravitational effect and start moving in both the space and time dimensions.

What is time?

As Einstein said, time is what clocks measure. And if you take a look at what a clock actually does, if you open up a clock and take a cold scientific look at the empirical evidence, you will see cogs turning or a crystal oscillating. You will see that the clock features some kind of regular cyclical motion along with something like gears or a counting device, and it gives some kind of cumulative display of the thing we call "the time". However all that's actually happened is that cogs etc have moved, and the big hand has moved along with the little hand. Once you appreciate this, special relativity is simplified, particularly when you think of the wave nature of matter. The maximum rate of motion is the speed of light, so macroscopic motion through space of necessity results in a reduction in the rate of local motion. See the simple inference of time dilation on Wikipedia.

Does it flow?

No it does not. Water flows. You can see it flow. But when you try to see time flowing, all you can see is cogs turning or a crystal oscillating. The notion that time flows is just a figure of speech, an abstraction. Light moves, planets move, blood moves, electrochemical signals move, cogs move. Everything moves, and you can see this motion. But you can't see time flowing. Because it doesn't. A clock is not some cosmic gas meter that measures the flow of time. Open up a gas meter and you find gas flowing through it. Open up a clock and you don't find time flowing through it.

and if so what defines its direction?

There is no time direction. That's an abstraction too. Can you point in this direction? No. Can you move in this time direction? No. Because time is just a cumulative measure of motion, you cannot move through a measure of motion, and there is no direction you can take.

What do physicists mean by time?

Most physicists mean the same as anybody else. But some will tell you it's a coordinate, which is another abstract thing, rather than something associated with motion or change.

What do physicists mean by time?

We’ll start with the easy question what do physicists mean by time.

Note that it’s easy to get mixed up between the concepts of time and the flow of time. When non-physicists talk about time they usually mean the flow of time i.e. the fact that in the human experience time flows inexorably onwards (at one second per second). We’ll get on to this, but for now we’ll ignore the question of why time flows and just address what time means to a physicist.

If you want to locate some position in space one method is to set up some axes, e.g. $x$, $y$ and $z$ axes, and you can then uniquely identify any point in space by its coordinates $(x, y, z)$.

To distinguish between events happening at the same point in space but at different times we need to specify when an event happened as well as where it happened, so we add a time coordinate $t$. Events can then be uniquely located by their spacetime coordinates $(t, x, y, z)$. To a physicist time is just a coordinate used to specify events in spacetime. In figure 1 above we have an $x$ axis stretching from $-infty$ to $infty$, a $y$ axis stretching from $-infty$ to $infty$ and a $z$ axis stretching from $-infty$ to $infty$. To these a physicist adds a $t$ axis stretching from $-infty$ to $infty$, and that’s what time is - just a coordinate.

But everyday experience tells us that time is special - certainly different from space - so what justifies the physicist’s view that time is just a coordinate? To understand this start with time in the everyday world as described by Newtonian mechanics.

Suppose I set up a coordinate system with myself at the origin, $x$ to the East, $y$ to the North and $z$ straight up. For time I’ll use my wristwatch. And suppose you do the same, but let’s say you’re in a different country from me. Our two sets of coordinates won’t match, because our East, North and up axes point in different directions.

Or suppose you are moving relative to me. Even ignoring the curvature of the Earth’s surface, our coordinates won’t match because your origin is constantly moving relative to my origin - what appears to be stationary to me is moving in your coordinates and vice versa.

So spatial coordinates are observer dependent. However time is absolute. Assuming we both use Greenwich Mean Time (or some other similar standard) we will always both agree on the time no matter where we are on Earth or however we are moving relative to each other. In Newtonian mechanics time is special for this reason, so it makes sense to consider it separately from space.

However since 1905 we have known that to properly describe the world around us we have to use special relativity, and in relativity time is not the same for all observers. Let’s go back to ordinary Newtonian mechanics for a moment, and suppose you’re moving relative to me along the $x$ axis at some speed $v$. If we draw my time $t$ and position $x$ axes and your $t’$ and $x’$ axes they’d look like:

Our two time axes point in the same direction, so we both agree on what it means to define a time axis. But now suppose you’re moving at relativistic speed $v$ and draw the same diagram.

When we include special relativity our axes no longer point in the same direction. If I draw my time axis straight up then relative to me your time axis is rotated by an angle $theta$ given by:

$$ tan(theta) = frac{v}{c} $$

So your time direction is a mixture of my time and space directions. You would see exactly the same - if you draw your time axis straight up then you’d see my time axis rotated by $-theta$. In effect we have different definitions of time, and indeed this is why we get time dilation in relativity.

The point of all this is that in relativity time is not uniquely defined. When we consider the coordinates used by different observers we find that time and space get mixed up with each other. Time is no longer distinct from space, and that’s why physicists treat it as just one of the four coordinates that together make up four dimensional spacetime.

How does time flow?

The previous section explained what physicists mean by time, but made no mention of time flowing. This is because in relativity time doesn’t flow - more precisely the flow of time doesn’t exist as a concept.

This is going to take some explaining, so let me attempt it using a simple example. Suppose I throw you a ball and you catch it. Everyday experience tells us that time flows forwards and as it does so the ball rises up from my hand then falls down to your hand. If we graph the height of the ball, $h$, against time, $t$, we’ll get something like:

In Newtonian physics this has a nice simple interpretation: time flows forwards and the height is a function of time. We can write the height as $h(t)$. But now let me draw a different graph. I’ll graph the height of the ball, $h$, against the distance, $d$, the ball travels horizontally as it moves from me to you:

This looks awfully like the previous graph, and indeed I can write the height of the ball as a function of the horizontal distance travelled, $h(d)$. But we wouldn’t say that distance $d$ flows forward and the height changes as it does so, because, well, time is different from distance. The two graphs are just different views of a four dimensional graph showing the trajectory of the ball in spacetime (I’m only going to draw three dimensions because I can’t do 4D graphs):

In the previous section I went to some lengths to explain that time is just a coordinate, like the spatial coordinates, so this graph doesn’t show time flowing any more than distance or height are flowing. The trajectory of the ball is just a line in a 4D.

In relativity we call graphs like the above world lines, where the world line is just the set of all spacetime points $(t, x, y, z)$ that the ball occupies during its trajectory. This world line is a fixed object in four dimensional spacetime - it doesn’t change with time. All that changes is the ball’s position on the world line. This is why we say that time doesn’t flow. Time is just one of the four dimensions that the world line occupies.

In fact any physical property, pressure of a gas, strength of a gravitational field, or whatever, can be written as a function in the four spacetime dimensions, $F(t, x, y, z)$. Written this way the geometrical object $F$ exists in all of space and all of time - it’s not something that evolves in time any more than it’s something that evolves in space. In principle we could have some function that represented the whole universe, $mathcal{F}(t, x, y, z)$, and this would exist for all values of $t$, $x$, $y$ and $z$. This idea (or a range of ideas like it) is called the block universe - the idea that the whole universe exists simultaneously and time doesn’t flow.

At this point I should note that many physicists, and I would guess the vast majority of non-physicists, would say this is just mathematical skulduggery and it’s nonsense to say time doesn’t flow. I’m not going to make any comment, except to say that this nicely brings us onto the last of our questions.

Why is there an arrow of time?

However mathematically convincing the idea of a block universe may be, the fact remains that our everyday experience tells us that:

1. time flows

2. it flows in one direction — forwards, and never backwards

So, how do we reconcile this with the idea of a block universe? Many physicists have expended much thought on this, and there are lots of differing views. However there is a something of a consensus that it is related to entropy. Indeed this is encapsulated in the second law of thermodynamics, which roughly speaking states that for any isolated system entropy only ever increases.

Consider some mechanism. We won’t worry exactly what it is, for example it could be something mechanical, an interstellar gas cloud or a human brain. When we talk about time flowing forward we mean that the state of the machine changes in a specific direction, e.g. a clock ticks forwards, and the second law of thermodynamics tells us that it changes in the direction of increasing entropy.

Assuming the human brain is just a mechanism, it changes in the direction of increasing entropy just like every other mechanism. But if consciousness is the result of the brain changing then it follows that any conscious being will observe mechanisms changing in the direction of increasing entropy. This isn’t so much a physical law as a correlation. Since our brains change in the same direction (of increasing entropy) as everything else that means they will necessarily observe everything to be changing in this same direction. We call this direction increasing time.

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If I’m allowed a personal opinion I would say this all seems a little trite — too good to be true — and it seems a suspiciously simple explanation for something as complicated as the universe. However I have no better suggestion to make. Indeed, I don’t think anyone has a better suggestion, or at least not one better enough to convince large swathes of the physics community.