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In classical (Newtonian) mechanics we only ever seem to consider 3-dimensional space, with physical quantities being represented by 3-vectors. My understanding for this is because in classical (Newtonian) mechanics time is absolute and hence a universal quantity, identically measured in each inertial reference frame. As such, time and space are completely independent of one another and time simply serves to parametrise paths of particles, systems, etc. through 3-dimensional space.

I understand that the concept of space-time was nothing new when special relativity came along and that one could use this notion from the outset (even in classical mechanics).

My confusion, however, is why time is most often simply treated as a parameter in classical mechanics, with all physical systems being described in a 3-dimensional space, whereas, in special relativity it is elevated to a coordinate such that physical systems are described on a 4-dimensional background?


This is my understanding so far:

The motivation for time being considered a coordinate in special relativity (as I understand it) is due to the fact that it is no longer an absolute quantity, but dependent on the (inertial) reference frame of a given observer. As such, it transforms non-trivial between different inertial coordinate frames, becoming a mixture of space and time coordinates relative to another inertial frame. In this sense, space and time become inextricably linked and must therefore be considered as forming a 4-dimensional continuum, so-called space-time.

Having watched Professor Shankar's lectures on special relativity, it is implied that time is not really treated as a coordinate in classical mechanics as he states that the reason why we consider time as a coordinate (and part of a space-time continuum) is precisely due to the fact that it transforms under a linear coordinate transformation (in analogue to the behaviour of spatial coordinates under coordinate transformations), unlike in classical mechanics where it is absolute and hence is trivially transformed under coordinate transformations.

This has left me confused as to how I should interpret the usage of time in classical mechanics, and whether or not I have understood correctly the reasoning for why it becomes part of a 4-dimensional continuum in relativity?!


If someone could enlighten me on this subject it'd be much appreciated.

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  • $\begingroup$ Time is not special in NM since it is absolute; no matter which frame you are in, you have the same time as mine in another frame. It is invariant in Galilean Transformation which further implies time-interval is also invariant in the same transformation. This is what not happens in SR. $\endgroup$
    – user36790
    May 3, 2016 at 13:30
  • $\begingroup$ Time is not a coordinate, but considered as absolute in Newtonian mechanics (or say Galilean relativity). So, there, it's just a parameter that has only a meaning in explaining motion. But special relativity deal time and space in equal footing. The principle of relativity states that time is not absolute, but relative to different observers. Otherwise, as Kip Thorne says, time is "personal" in SR. The 4th coordinate of time is not "purely" time coordinate. It represents a translation coordinate along time. Overall the object moves through space time at the speed of light. $\endgroup$
    – UKH
    May 3, 2016 at 13:31
  • $\begingroup$ @Unnikrishnan I'm just wondering why it can be considered as simply a parameter in Newtonian mechanics (NM)? I mean, is it simply because it is universal and absolute in NM and as such simply serves to parametrise 3D space? I can see why in SR it must be treated as an additional dimension, since time is not absolute in SR and is intrinsically linked to space, with the two interwoven into a 4-dimensional space-time, but I find it hard to explain it fully. $\endgroup$
    – user35305
    May 3, 2016 at 13:42
  • $\begingroup$ Newtonian mechanics is true if you are doing experiments in inertial frame and in a very weak gravitational field and also for low speed objects. This is not the case when something moves very close to c. Then the time measured between two events will not be the same and depends on the velocity of the observer's frame of reference. Time is absolute means any observers see same time interval between two events. This is true since Newtonian mechanics is applicable at low speed inertial frames. That's why time has just a parameter importance in NM. The SR is nothing but an extension to it $\endgroup$
    – UKH
    May 3, 2016 at 13:50

2 Answers 2

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It depends what exactly you mean by "coordinate". If your Lagrangian/Hamiltonian is time-independent, then you may consider time to be purely a parameter parametrizing e.g. the integral curves of the vector field associated to the Hamiltonian on phase space. If your Lagrangian/Hamiltonian is time-dependent, you should indeed properly consider your theory on configuration space together with time, i.e. you may consider time a coordinate of the extended space $Q\times\mathbb{R}$. However, it is not a coordinate in the sense that there were any choice to it - classical time is absolute, and the split of your total space into $Q$ and the time $\mathbb{R}$ is fixed.

In classical mechanics, you never have need or even motivation to consider transformations that mix time and space, so time is a "parameter" in the sense that it does not mix with the actual generalized coordinates you use to describe your system. Statements that explicitly include a dependency on time are invariant under the symmetry groups of classical mechanics, there is nothing "bad" about having explicit time dependence.

In contrast, the Lorentz transformations of special relativity clearly mix space and time - the classically fixed splitting into a "spatial part" and a "time part" is not unique anymore, and depends on your chosen frame. This means you now need to properly consider time as a coordinate, and should ideally phrase statements in an invariant way that doesn't rely on a particular choice of time axis, since otherwise you cannot be sure that a statement is true independent of frame.

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  • $\begingroup$ So is the point that, in classical mechanics, time is completely independent of space. It is an absolute (frame invariant) universal quantity and as such simply serves to parametrise paths in 3D space? Is the reason why it becomes a coordinate in SR then simply due to this mixing of space and time under Lorentz transformations, such that they are no longer independent quantities, but part of a 4-dimensional continuum? .... $\endgroup$
    – user35305
    May 3, 2016 at 14:11
  • $\begingroup$ ... Can proper time then be seen as the analogue of time in classical mechanics, since it is a Lorentz invariant quantity, it simply serves to parametrise timelike curves? What I find quite confusing though is that space and time only become mixed if the two inertial frames in consideration are moving at a uniform velocity with respect to one another, so why does one need to consider the "base" coordinate space to be 4-dimensional when it is only in this case that time is frame-dependent (it isn't under rotations or translations)? ... $\endgroup$
    – user35305
    May 3, 2016 at 14:15
  • $\begingroup$ ... Or is it that as time is frame-dependent in at least one case (i.e. under boosts) we must consider the base coordinate space to be 4-dimensional as otherwise we do not cover the general case of Lorentz transformations? $\endgroup$
    – user35305
    May 3, 2016 at 14:15
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I think this issue is best clarified by closely looking at the way time is mixed into coordinate frame transformations in Classical Mechanics as opposed to Relativistic Mechanics.

Let's take the case of an observer, Alice, moving at velocity $v$ in the positive $x$ direction away from her friend Bob. Both Alice and Bob are looking at an object situated at $x_{B}$ where the subscript is there to signify a point in Bob's coordinate system. Assuming Alice and Bob started out at the same position and started their clocks together, in classical mechanics we would have:

$$\begin{align} x_A&=x_B-vt_B\\ t_A&=t_B \end{align}$$

Where now $x_A$ is the coordinate of the same point in Alice's system. Note that the time coordinates are the same since both Alice and Bob experience time moving at the same rate. Notice that here time is just used as a tool to label different points in the evolution of both systems and does not depend on the dynamics of the system. Intuitively speaking, the first equation is Alice's coordinate as calculated by Bob.

In special relativity however, Lorentz transformations come into play and we get:

$$\begin{align} x_A&=\gamma(x_B-vt_B)\\ t_A&=\gamma (t_B-\frac{v x_B}{c^2})\\ \gamma&=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \geq 1 \end{align}$$

So now the time coordinate is dependent on the dynamics of the system (i.e. on the velocity and position). It can no longer be used as a label to denote the evolution of the system from the point of view of both Alice and Bob.

So you have understood correctly why time becomes linked with the spatial coordinates in special relativity. In classical mechanics though, time is merely a label to parametrise the evolution of the system.

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  • $\begingroup$ Thanks for your answer. So I've understood why time becomes a coordinate and interwoven with space into a 4-dimensional space-time correctly then? Is time in classical mechanics simply a parameter then because it is universal and independent of the motion of an observer and so simply parametrises the occurrence of events in 3-dimensional space? In SR is it then that, since time is an observer dependent quantity, each coordinate frame must consider both the time and spatial position of an object to specify its location? $\endgroup$
    – user35305
    May 3, 2016 at 14:01
  • $\begingroup$ Yes to your first question. Yes to your second question as well. And yes to your last question. Some comments, in SR (and also GR) rather than using the word location to specify a specific point in the space the word "event" is used as this takes into account the fact that time is also being specified. $\endgroup$
    – A.Far
    May 3, 2016 at 16:59
  • $\begingroup$ Also note that in the equations above if the velocity is much smaller than the speed of light, then the usual classical mechanical equations are retrieved. So the best way to understand what is going on is to think of time as a coordinate as in SR all the time since this is the physically accurate picture. $\endgroup$
    – A.Far
    May 3, 2016 at 17:04
  • $\begingroup$ OK, thanks for your help. Could one say that in classical mechanics, a physical system can be described by 3 spatial coordinates along with a parameter, time, which specifies at what point in the evolution of the system one is considering. Since time is absolute, space and time are completely independent hence why we can separate them in this way (as a 3 dimensional space along with a time parameter). In special relativity however, the Lorentz transformations reveal that space and time are no longer independent and must be considered as a single 4-dimensional entity, spacetime.. $\endgroup$
    – user35305
    May 3, 2016 at 18:54
  • $\begingroup$ ... Hence, now time is one coordinate specifying one dimension of a 4-dimensional spacetime. It is thus a dimension since it under a coordinate transformation from one frane to another it becomes a linear combination of the coordinates in the new frame relative to the old frame in analogue to a rotation in 3D space in which the rotated coordinates are linear combinations of the "old" (un-rotated) coordinates. $\endgroup$
    – user35305
    May 3, 2016 at 19:01

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