Edit to add: A simple explanation of my objection to Blandford and Thorne's definition of inertial reference frame (which they use synonymously with inertial frame) is that, if I'm in free float, sitting is the seat of a spaceship, and I feel a force acting on me, that informs me that I am changing inertial frames. I don't need a latticework of clocks and rulers to make that assessment.

I also intended to add that what Blandford and Thorne are setting up, if it also is equipped with detection devices and data recorders colocated with each clock, is what I call an observer system. Such a system is sometimes simply called an observer.

The following is from Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics, by Kip S. Thorne and Roger D. Blandford. And I find it contrary to my understanding of what an inertial frame of reference is.

enter image description here

An inertial reference frame is a 3-dimensional latticework of measuring rods and clocks (Fig. 2.1) with the following properties:

  • The lattice work is purely conceptual and has arbitrarily small mass, so it does not gravitate.
  • The latticework moves freely through spacetime (i.e., no forces act on it) and is attached to gyroscopes, so it is inertially non-rotating.
  • The measuring rods form an orthogonal lattice, and the length intervals marked on them are uniform when compared to, for example, the wavelength of light emitted by some standard type of atom or molecule. Therefore, the rods form an orthonormal Cartesian coordinate system with the coordinate $x$ measured along one axis, $y$ along another, and $z$ along the third.
  • The clocks are densely packed throughout the latticework so that, ideally, there is a separate clock at every lattice point.
  • The clocks tick uniformly when compared to the period of the light emitted by some standard type of atom or molecule (i.e., they are ideal clocks).
  • The clocks are synchronized by the Einstein synchronization process: if a pulse of light, emitted by one of the clocks, bounces off a mirror attached to another and then returns, the time of bounce $t_{b}$, as measured by the clock that does the bouncing, is the average of the times of emission and reception, as measured by the emitting and receiving clock: $t_{b}=\frac{1}{2}\left(t_{e}+t_{r}\right)$.

Compare this to the original formulation of inertial system, given by Ludwig Lange in 1885 prefaced by:

Newton's absolute space is a phantom which should never be made the basis of an exact science.

From Inertia and Gravitation: The Fundamental Nature and Structure of Space-Time

Definition I. 'Inertial system' is called any coordinate system of the kind that in relation to it three points $P;P^{\prime}; P^{\prime\prime},$ projected from the same space point and then left to themselves--which, however, may not lie in one straight line--move on three arbitrary straight lines $G;G^{\prime}; G^{\prime\prime},$ (e.g., on the coordinate axes) that meet at one point.

Theorem I. In relation to an inertial system the path of an arbitrary fourth point, left to itself, is likewise rectilinear.

Definition II. 'Inertial timescale' is called any timescale in relation to which one point, left to itself (e.g., P), moves uniformly with respect to an inertial system.

Theorem II. In relation to an inertial timescale any other point, left to itself, moves uniformly in its inertial path.

To my understanding, Lange is not requiring any specific set of points (point masses), he is merely stating a minimum requirement for empirically determining an inertial system. He indicates that they may move along coordinate axes, but that is not necessary.

My concept of an inertial frame (of reference) is far more abstract than that given by Blandford and Thorne, and I consider it to be fundamental to physics. So here I am, challenging a guy with a Nobel Prize in Physics for his work in General relativity, regarding the foundations of the same.

There are many ways I might try to communicate what I mean by inertial frame. I'll use this one. An inertial frame is a family of parallel world lines filling Minkowski spacetime. It is a priority geometric entity in the same sense that point, line, plane and space are in Euclidean geometry. By "priority geometric entity," I mean that it exists prior to coordinatization.

Blandford and Thorne are, in my assessment, defining a coordinatization of an inertial reference frame. This wouldn't be worth posting about if it were not for the fact that the book's target audience is graduate students of physics, and professional physicists. Furthermore, the very emphatic thesis of the book is 'real physics is geometrical'.

Am I right to contest Blandford and Thorne's definition of inertial reference frame?

  • $\begingroup$ Proper definition of an inertial reference frame is a complicate task. It is strongly entangled with the definition of clocks, synchronization and free motion.The most delicat point being this last concept. I do not think I have an answer, but as a comment I would say that your definition should say something explicit about the clocks, without using the concept of inertial reference frame. Can you do that? $\endgroup$ Commented Nov 20, 2019 at 0:00
  • $\begingroup$ I am assuming that it is understood that the affine parameter of the worldlines is propertime. I guess I should have said "a family of timelike worldlines", in my definition. Clock and rulers are used to quantify relationships between events. Were I to provide a means of setting up a system which can be used to do quantitative physics, I would use the method that Blandforn and Thorne cribbed from Taylor and Wheeer. But that is a coordinatization of an inertial reference frame. Not a definition of one. $\endgroup$ Commented Nov 20, 2019 at 0:18
  • $\begingroup$ Ok, but how do you measure propertime? by which kind of clock? How do you know it is a proper clock? $\endgroup$ Commented Nov 20, 2019 at 0:47
  • $\begingroup$ @GiorgioP My definition (which was "off the cuff") is that of a mathematical abstraction. There is no need to give a specific means of parameterizing the worldlines. The definition presupposes sufficient experience in working with specific cases in order to take the step of abstraction. Lange's definitions are pseudo-emperical, in that they suggest experiments by which relative inertial space and inertial time may be determined. But leave room for abstraction. Blandfor and Thorne lock us into one concrete representation, leaving no room for abstraction. $\endgroup$ Commented Nov 20, 2019 at 18:25
  • $\begingroup$ Abstraction could be a good thing. However, one key property of the inertial systems is that free motion is uniform (linear increase of the dependence of the coordinates on time). That requires a careful choice of the clock. Many possible choices of the proper time would not be consistent with the description of free motion as uniform. So, I think that your "family of parallel world lines filling Minkowski spacetime" should be further specified as far as the parametrization of time is concerned. $\endgroup$ Commented Nov 20, 2019 at 22:31

2 Answers 2


You can contest the Blandford and Thorne definition as being unnecessarily specific. They do not define a general inertial reference frame, but a specific conceptual model of one. As you say, there is no reason to posit the existence of imaginary clocks etc. There is also no requirement, generally, for the spatial axes to be orthogonal or to each have the same units (although ditching those requirements will make a reference system more difficult to use).

You can also criticise it for being internally inconsistent. For example, it says that the lattice is purely conceptual, but goes on to say that it has an arbitrarily small mass-- a purely conceptual lattice would have no mass.

The statement 'so that ideally there is a clock at every lattice point' is nonsense. Why not simply state that there is a clock at every point? If you are going to define a reference system with the idea of an infinite number of imaginary clocks why allow that it might be less than ideal?

Phrases such as 'the clock that does the bouncing' are atrocious English.

Hopefully by now you get my drift. Physics is a discipline in which the loose or ill-considered use of language is a menace to comprehension. Almost every science book I have read suffers from some essential ambiguity as a result of its author taking insufficient pains to make their meaning utterly clear, and many of the questions on this forum are a lamentable symptom of that tendency.

  • $\begingroup$ All of your criticisms have merit, but my point is that an inertial reference frame is prior to any coordinate system, and certainly not a mere "crutch". Ironically much of my thinking about this topic is influenced by Taylor and Wheeler's Spacetime Physics, and Misner, Thorne and Wheeler's Gravitation. Yep, the same Thorne. $\endgroup$ Commented Dec 9, 2019 at 22:58
  • $\begingroup$ I agree. The more abstract underlying idea is a continuous space of points maintaining fixed distances between themselves, none of which is accelerating. I think you then need to overlay some system of coordinates if you want to be able to use it as a reference frame, ie to refer to specific points within it. $\endgroup$ Commented Dec 9, 2019 at 23:18
  • $\begingroup$ I like the definition I came up with when formulating my original question. That is, a family of parallel time-like world lines filling Minkowski spacetime. Another way to say the same thing is that an inertial frame is a time-like direction in (flat) spacetime. Since I had only completed one basic algebra class at the time I decided to learn the theories of relativity, I relied heavily on my intuition. Acceleration became a 4-dimentional rotation. That is, a change in the spacetime direction of the object's world line. And that is still how I think about it. $\endgroup$ Commented Dec 10, 2019 at 23:05
  • $\begingroup$ I think of acceleration in a similar way as the tilting of an object's plane of simultaneity. I find it thought provoking that the tilt is the consequence of an applied force, or, equivalently, the gain or loss of energy. $\endgroup$ Commented Dec 11, 2019 at 7:48

Definition 1:

An inertial frame is a family of parallel world lines filling Minkowski spacetime.

Definition 2 (Lange, 1885):

'Inertial system' is called any coordinate system of the kind that in relation to it three points P;P′;P′′, projected from the same space point and then left to themselves--which, however, may not lie in one straight line--move on three arbitrary straight lines G;G′;G′′, (e.g., on the coordinate axes) that meet at one point.

Definition 3 (Thorne and Blandford):

(Paraphrased, and eaving out some of the details): we have a latticework of clocks and rulers, with the clocks synchronized by Einstein synchronization.

Definitions 2 and 3 are operational definitions. This is from a philosophical school called operationalism, articulated by Percy Williams Bridgman. They spell out the operations needed to measure a thing, and this is taken to be a definition of the thing. Definition 2 is, with historical hindsight, a sloppy operational definition, because it implicitly talks about time and motion, but it assumes that time operates in a trivial way that hardly even needs to be discussed.

Definition 1 is not an operational definition. That isn't necessarily a bad thing -- not all definitions have to be operational -- but it makes it possible for it to contain hidden ambiguities. Depending on what is understood by "Minkowski spacetime," this definition might or might not imply a lot of assumptions about how the spacetime behaves, what mathematical apparatus is associated with it (a connection?), and so on.

If I wanted to nitpick about definition 1, I could complain that you haven't specified that the lines be timelike, you haven't said whether they're directed lines, and you haven't said anything about an orientation of the observer, which in general is part of what we mean by a frame of reference. If you fill in some of this, you will end up with something like the notion of a frame field.

As a matter of style, I would prefer to disentangle the notion of a reference frame from other assumptions, such as that spacetime has a bilinear metric rather than a degenerate metric as in Galilean relativity.

You seem to assume that it is automatically meaningful to talk about whether distant lines are parallel. This is actually not true in a curved spacetime. I guess this is covered by the reference to Minkowski space, but this happens in a way that sweeps a lot under the carpet. The intention of definition 3 seems to be not to sweep anything under the carpet.

Blandford and Thorne are, in my assessment, defining a coordinatization of an inertial reference frame.

Sure, but this is a very natural way to describe Minkowski space, which has the structure of a vector space with certain preferred choices of basis, and a clear one-to-one correspondence between the choice of basis and the choice of a frame of reference.

  • $\begingroup$ Note that I addressed the "timelike" issue in a comment. I am not assuming that all once parallel worldlines are always parallel in the real Univers. That's why I qualified my spacetime as Minkowski. My problem with Blandford and Thorne's definition is that it is defining a coordinate system, rather than the entity being coordinatized. If they had used inertial reference frame to mean their specialized definition, but had left inertial frame and/or inertial system to mean the more abstract concept, I would not protest. $\endgroup$ Commented Nov 20, 2019 at 0:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.