# Why is time referred to as "The" capital T 4th dimension?

Often you hear people say "time is The 4th dimension". What confuses me is that people talk about "The 4th" dimension as if it's a specific thing and I don't understand why.

What I mean by that, is that I can plot the position of a point in 3D space with the 4th dimension being just about anything. For instance, I can plot my position in space with time being the 4th dimension showing me my position at any given timepoint, but I can also plot my position in space with, say, my level of hunger being the 4th dimension where I'd be able to see where I was in space based on hungry I was.

What I'm trying to get at is, is there a reason why time is widely considered as THE 4th dimension rather than just another parameter? Is the "The" really describing an intrinsic relationship?

• "For instance, I can plot my position in space with time being the 4th dimension showing me my position at any given timepoint, but I can also plot my position in space with, say, my level of hunger being the 4th dimension where I'd be able to see where I was in space based on hungry I was." This sentence shows that you have a much better understanding of the mathematical, abstract notion of dimensions than most people. Most people don't call hunger a dimension and are happy with time being "the" fourth dimension (and they don't want a fifth)
– Stef
Commented Dec 2, 2022 at 10:19
• One problem with "the hunger dimension" is that you do not have a unique solution for the x,y,z,h state vector – there can be multiple x,y,z positions with the same h. When calculating space-time, you only have one unique position vector for each t: time doesn't loop. Commented Dec 2, 2022 at 15:48
• Isn't hunger just another unit to tell time? hunger only evolves within time between meals.
– user16500
Commented Dec 2, 2022 at 17:12
• @JacobKrall I don't want to be that guy but general relativity does allow for time loops to exist (closed timelike curves). Also, when you look at a slice of constant time, every point has the same value of $t$. Commented Dec 3, 2022 at 23:44
• @JacobKrall There is no law or constraint on dimensions that prevents duplication of values.
– Eric
Commented Dec 4, 2022 at 18:49

This is because time is the fourth dimension in the theory of General Relativity which describes gravity.

It turns out that a good way to describe the paths that objects or light take when in a gravitational field is to describe a curved four-dimensional space with coordinates x,y,z,t. A particle in space becomes a curve in this four dimensional space, and we can use general relativity to find the shape of this line, and so the position of the particle at any given time.

If the force of gravity is weak and the velocity of the particle is small then x, y, z correspond to the ordinary position of the particle in 3d space at time t, in the usual Newtonian way. If the gravity is strong, or the velocity is high, then only the General Relativity description matches observations.

Now you can, of course describe your (x,y,z,H) position (where H is hunger) however this doesn't lead to much interesting physics. It doesn't help develop a theory of hunger.

So time is the fourth dimension, because that helps us to describe reality.

• This is a wonderful answer. So time is THE 4th dimension specially in reference to Einstein's theory of GR , because in GR gravity warps time in a similar manner to the way it warps space. Am I getting it right? Commented Dec 1, 2022 at 20:17
• Yes, time is warped as well as space. The two are so interlinked that it doesn't make sense to treat the curvature of time separately from the curvature of space. Therefore 4D spacetime. Commented Dec 1, 2022 at 20:40
• And on the other hand, even (x,y,z) is specific to a subset of theories. For example Lagrangian mechanics uses a completely different coordinate system.
– jpa
Commented Dec 2, 2022 at 7:10
• I feel like part of the answer is that it's partially just convention: that is, why x,y,z,t and not t,z,x,y. Time is the 1st dimension! It could also be the second. Your location as x,t,z,y is just as valid (especially since x,y,z themselves are presumably some arbitrary frame. Which way is "x" in space?) Time being 4th is, I think, just arbitrary convention. Commented Dec 2, 2022 at 16:40
• @JamieB Actually, it's quite common in physics to put time at the start of spacetime vectors: (t, x, y, z), making it the first (or zeroth) dimension. Commented Dec 3, 2022 at 5:11

For pretty much the same reason that the first 3 are (x, y, z) and not (Hair Colour, Body Type, Salary).

It's part physics and part linguistics. As @JamesK said x,y,z,t are the most useful to describe physical phenomena, therefore these are the ones that people typically use and linguistically have become the dimensions unless otherwise specified by the speaker.

Just my 2c worth, I think there's an easier explanation. Not that I don't love the physics and maths behind this question.

Just linguistically speaking, we want to describe an object's position that's "good enough" without lacking detail or being superfluous. Thinking back many years in a book by Rudy Rucker on the 4th dimension, the "Flatland" analogies: In one-dimensional space, which is pretty much a single line, only one measurement is necessary to describe an object. Dimension, meaning exactly that, a measurement. In 2 dimensional space, x and y, and in 3 dimensional space a z attribute is required.

Think of a moving object in 3D space, like an aeroplane. Without time, it is completely impossible to satisfactorily describe its position. Even with the Hunger analogy, you'd have to say the plane was at x,y,z when I was "a little hungry" or "very hungry", but isn't hunger then just a reference to a state in time?

The point (excuse the pun) is to measure an object's position, and be able to communicate, record or reproduce this position. Without time, it's "not good enough".

Let's not go down the road of a Multiverse, because then 4 dimensions just aren't good enough :-)

• Is it perhaps Flatland and not Discworld? Commented Dec 2, 2022 at 17:39
• I'm sorry, that's possible, the 90's are so long ago 🙂 Oh, just googled it, I stand corrected.
– Andi
Commented Dec 3, 2022 at 18:30
• I've edited the answer to say Flatland Commented Dec 3, 2022 at 19:41

As mentioned in the comments, one problem with your proposal is that $$hunger = f(time)$$ is not injective.

If you are really hungry every day at 07:59, 11:59 and 19:59, specifying your spacetime position with 48°N,9°E,450m,"really hungry" would not give any information about the date or even the time of day. Your units would only be useful for a few hours at most.

But indeed, $$x,y,z,t$$ could just as well be specified as:

• $$x,t,z,y$$,
• $$t,x,y,z$$,
• $$x+y,x+z,z,t$$
• or even $$x,y,z,t + x/c$$.

The theory of relativity does, however, force us to change fundamentally our ideas of space and time. We must accept that time is not completely separate from and independent of space, but is combined with it to form an object called space-time.

It is a matter of common experience that one can describe the position of a point in space by three numbers, or coordinates. For instance, one can say that a point in a room is seven feet from one wall, three feet from another, and five feet above the floor. Or one could specify that a point was at a certain latitude and longitude and a certain height above sea level. One is free to use any three suitable coordinates, although they have only a limited range of validity. One would not specify the position of the moon in terms of miles north and miles west of Piccadilly Circus and feet above sea level. Instead, one might describe it in terms of distance from the sun, distance from the plane of the orbits of the planets, and the angle between the line joining the moon to the sun and the line joining the sun to a nearby star such as Alpha Centauri. Even these coordinates would not be of much use in describing the position of the sun in our galaxy or the position of our galaxy in the local group of galaxies. In fact, one may describe the whole universe in terms of a collection of overlapping patches. In each patch, one can use a different set of three coordinates to specify the position of a point.

An event is something that happens at a particular point in space and at a particular time. So one can specify it by four numbers or coordinates. Again, the choice of coordinates is arbitrary; one can use any three well-defined spatial coordinates and any measure of time. In relativity, there is no real distinction between the space and time coordinates, just as there is no real difference between any two space coordinates. One could choose a new set of coordinates in which, say, the first space coordinate was a combination of the old first and second space coordinates. For instance, instead of measuring the position of a point on the earth in miles north of Piccadilly and miles west of Piccadilly, one could use miles northeast of Piccadilly, and miles north-west of Piccadilly. Similarly, in relativity, one could use a new time coordinate that was the old time (in seconds) plus the distance (in light-seconds) north of Piccadilly.

Take a look at this equation from special relativity. You don't need to know what it represents; it's good enough to know that $$t$$ is time, and $$x, y, z$$ are the three spatial dimensions.

$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$

It should be clear that in this equation, the four dimensions are on equal footing, but they are not the same, because one of them has a negative sign and the rest are all positive.

The fact that they are on equal footing is why we say there are four dimensions. You can't include hunger to this equation because it doesn't make physical sense to add hunger to the other four dimensions - hunger is measured in different units, and it doesn't make sense to add hunger with time/space. You can no more add hunger to time/space than you can add the current in a wire (measured in amperes) to its mass (measured in kg).

The fact that time is nonetheless different from the three spatial dimensions is why we single time out. Time is the fourth dimension because it makes sense to group the three spatial dimensions together as the first three. If one day we discover a fourth spatial dimension, then time would become the fifth dimension.