Time is a Fourth Dimension

Super fact 58 : In relativity, time is considered the fourth dimension, inseparable from the three spatial dimensions to form a four-dimensional continuum called spacetime. Adding time as a fourth dimension, not (x, y, z), but (x, y, z, t), results in spacetime measurements called spacetime intervals that all observers can agree on.

Before relativity the distance between two points was the same for all observers. The distance between points is calculated using the Pythagorean theorem: (d^{2}=x^{2}+y^{2}+z^{2}). You calculate the distance between two end points in a coordinate system using Pythagoras theorem because the points make right angled triangles along the x-axis, y-axis and z-axis. See the picture below.

The image shows the formula for Pythagoras theorem in two and three dimensions and Pythagoras theorem applied to the distance between two points.
Pythagoras theorem in two and three dimensions which also apply to the distance between two points. The points are indicated in red.

Let say you add another coordinate system (x’, y’, z’). The new coordinate system could be translated and rotated compared to the first one. The values of (x, y, z) and (x’, y’, z’) would be different and yet the distance between point-1 and point-2 would be the same. Well as long as you don’t change units, like using meters in one coordinate system and feet in the other. The distance between the points would be a so-called invariant. Now imagine that you forgot to include one coordinate in Pythagoras theorem, for example, y and y’ or x and x’, then your calculation for the distance would be different for the two coordinate systems. We need all coordinates, or all dimensions. See the picture below.

The picture shows two different coordinate systems. One is rotated and translated compared to the other. There are also two points and the distance between them is indicated. The formula for Pythagoras theorem is shown for both coordinate systems.
Pythagoras theorem is used to calculate the distance between two points from two different coordinate systems, with different coordinate values for the points. You still have the same distance for both coordinate systems. The points are indicated in red.

 In relativity the length of objects, as well as the time between events is relative and varies from observer to observer. In other words, distance and time varies from coordinate system to coordinate system. However, if you add time to the three space dimensions and calculate the distance between events using the Pythagorean theorem for intervals (between two events): or  (s^{2}= x^{2}+y^{2}+z^{2} – t^{2}) or (where the ‘t’ represents time in appropriate units), then the difference between different observers vanish. The interval is the same for all observers. It is a so-called invariant. The formula for the spacetime interval comes in a few different forms. One for distance like intervals (space distance bigger than time) (s^{2}= x^{2}+y^{2}+z^{2} – t^{2}), and one for time like intervals (time is bigger than the space distance) (s^{2}= t^{2} – (x^{2}+y^{2}+z^{2})). There is also one that includes the imaginary number (s^{2}= x^{2}+y^{2}+z^{2} + (it)^{2}). See below.

The image shows three formulas for the spacetime interval Euclidian: “(s^{2}=x^{2}+y^{2}+z^{2}+(it)^{2}”. For Time like intervals, the standard form: “(s^{2} = t^{2} – (x^{2}+y^{2}+z^{2}))”. For distance like intervals: “(s^{2} = ((x^{2}+y^{2}+z^{2}) – t^{2}))”.
The three formulas for the spacetime interval above all assume that the unit used for time is the time it takes light in vacuum to travel the distance unit used. If that is meters, it would be the time it takes light to travel one meter. The top formula is the Euclidian form of spacetime. It contains only the ‘+’ operator at the expense of adding the imaginary number (square root of -1) in front of the time coordinate. The second form is typically used with time like intervals and considered the standard form. The third form is used when the distance between two events is larger than the time distance, or distance like intervals.

The interval concept was developed, not by Einstein, but by Hermann Minkowski (a few years after special relativity) and is often referred to as Minkowski space. Time is like a space coordinate but the opposite signs in the equation make it different. Based on articles I found it appears that the opposite signs (minus vs. plus) means that you cannot move “backwards” in time as you can in a space dimension.

I admit that this is a very abstract super fact, but it basically means that if you add time as an extra coordinate to the three space coordinates x, y, z you get something, the spacetime interval, that everyone regardless of speed, orientation, etc., agrees on, despite relativistic length contraction and despite time dilation and non-simultaneity.

Time Expressed in Appropriate Units

I would also like to explain what I mean by (where the ‘t’ represents time in appropriate units), as I stated in the above. For physical formulas to work they need to be expressed in consistent units. For example, you can’t use kilometers for the coordinate x, and miles for coordinate y, not without adding a constant to adjust for it. For the formula (s^{2}=x^{2}+y^{2}+z^{2}-t^{2}) to work you need to express time in a unit that corresponds the time light travels in one meter if x, y and z are expressed in meters. If you express x, y, and z in meters and express time in seconds you must adjust the formula with the constant c = 299,792,458, the speed of light in meters per second, so you get (s^{2}=x^{2}+y^{2}+z^{2}-(ct)^{2}). See the picture below.

The image shows the formulas for the spacetime interval with the constant representing the speed of light in vacuum “(s^{2}=x^{2}+y^{2}+z^{2}+(ict)^{2}”, “s^{2}= (ct)^{2} – (x^{2}+y^{2}+z^{2})” and “(s^{2}=x^{2}+y^{2}+z^{2}-(ct)^{2}”.
If you measure the space coordinates in meters and the time in seconds you must adjust the units to match by inserting the speed of light in vacuum c = 299,792,458. The three forms of the space interval now have the constant c attached to the time coordinate.

Time Like Space Intervals

The formula for time like intervals is typically used for the situation where the time component is larger than the space component, which also means that it is possible to physically travel between the two events forming the space interval. As you can guess, that is a pretty normal situation. Let’s say you are watching TV and having a pizza. Your sofa is your coordinate system. You turn on the TV and 100 seconds later you move 2 meters to get a slice of pizza. Let’s calculate the spacetime distance between those two events.

The space component is easy, that’s 2 meters. However, if we express time in the time it takes light (in vacuum) to travel one meter we get 100 times 299,792,458. If you express time in seconds, you adjust it using the constant c = 299,792,458, and again you multiply 100 with 299,792,458, which is 29,979,245,800. So, the distance in time is almost 15 billion times larger. You really did not move far in space, but you moved very far in time. Now ask yourself. Are you spending your time well?

The Minus in Front of the Time Coordinate

There is one obvious difference between time and the space coordinates. In a coordinate system you can walk forward, along let’s say, the x-axis and then walk back the same way. You can walk back and forth as many times as you want, no problem, but you cannot do that with time. Time may be a space-time coordinate, but it is different from the other three coordinates in that way, and that’s where the opposite signs in the formula for the space-time interval comes in. This is beyond the scope of this super fact blog post, but you can read more about this here and here.

Other Relativity Related Superfacts

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Author: thomasstigwikman

My name is Thomas Wikman. I am a software/robotics engineer with a background in physics. I am currently retired. I took early retirement. I am a dog lover, and especially a Leonberger lover, a home brewer, craft beer enthusiast, I’m learning French, and I am an avid reader. I live in Dallas, Texas, but I am originally from Sweden. I am married to Claudia, and we have three children. I have two blogs. The first feature the crazy adventures of our Leonberger Le Bronco von der Löwenhöhle as well as information on Leonbergers. The second blog, superfactful, feature information and facts I think are very interesting. With this blog I would like to create a list of facts that are accepted as true among the experts of the field and yet disputed amongst the public or highly surprising. These facts are special and in lieu of a better word I call them super-facts.

59 thoughts on “Time is a Fourth Dimension”

    1. Thank you Ada. I realized after posting that I need to clarify and maybe add a couple of things. For example, the last picture shows two formulas for the spacetime interval. They are really the same except the first one (proper distance) you use when the space distance is larger than the time, and the second one (proper time) when the time “distance” is larger than the space distance. I am not explaining this. I’ll fix in a few days.

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    1. Thank you Robbie. I should say that after I posted I realized I need to update a couple of things, so if he sees it before I do he will probably correct me. For example, the last picture shows two formulas for the spacetime interval. They are really the same except one is the negative of the other. The first one (proper distance) you use when the space distance is larger than the time, and the second one (proper time) when the time “distance” is larger than the space distance. I am not explaining this, so I need to do that. I’ll fix in a few days.

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            1. Thank you so much Pooja. Unfortunately, there are formula(s), but you could just note that while relativity makes space distances and times relative to each observer, if you add time in together with the space coordinates you get something (spacetime interva) that is the same for all. It is as if time is just another space coordinate, well almost. Like a space coordinate you can’t walk backwards in.

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  9. Fascinating to read about, although I failed grade 12 math and it’s a bit over my head. 😉 Until this post, the only things I knew about the “space-time continuum” came from the “Back To The Future” movie series.

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    1. Thank you so much Debbie. I guess the take away can be that adding time as another dimension to the three space dimensions was for a compelling reason and not just on the whim. Another thing I brought up, which there seemed to be more than one opinion on, so that’s not part of the super fact, is that the minus sign, or rather the different sign between the time component and the three space dimensions, make it so that you cannot travel back in time. However, I need to improve this super fact post. I was not clear. One thing is that there are two ways to write the spacetime interval, that are each others negative, and one you use when the time component is bigger (proper time) and one when the distance (x,y,z) is bigger proper distance. I kind of messed up explaining that. Anyway, I remember back to the future. It was a fun movie.

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      1. Too bad! Going back in time is a fantasy that many of us have had. 🙂 And I never would have known you messed anything up, until you said that.

        Back To The Future was a series of three movies, two of which went back in time (1985-1955, 1985-1955-1885) and one into the future (1985-2015). The real 2015 was nothing like how the movie depicted it. 😄

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            1. Thank you so much Debbie. It isn’t enough evidence to add that to the super fact but it looks to me like the physics Professor who said that the minus sign prevented physical objects from returning back along the time axis, was the one who knew what he was talking about. What he said made perfect sense.

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  10. This is amazing, Thomas, and a bit confusing. This morning, I listened to Dr. Roman Yampolskiy talk about AI. Scary stuff, but the phenomenon has a connection to your post. I wish I understood it better.

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    1. It is difficult to explain. Harder than I thought. However, suffice it to say that time as another dimension added to our three visible space dimensions (x,y,z) is not added on a whim but has physical reasons. I should say that I am about to update this post. About AI, I agree, we have to be very careful with how we use it, or it could become very dangerous. I don’t know Dr. Roman Yampolskiy but I refer to the book Superintelligence by Nick Bostrom.

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    1. Well you have a lot of it. The last 10 seconds you’ve had 3 billion units, if you count a unit is the time it takes for light to travel one meter. What did you do with your 3 billion? In your life you have almost 800 quadrillion units. But I’ll understand your concern. They go by fast, at the speed of light in fact.

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  16. Your clarity in merging abstract physics with vivid examples makes relativity feel accessible, almost like storytelling. It invites not only scientists but dreamers too. I hope your work continues sparking curiosity across generations, inspiring both learning and wonder about the mystery of time.

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