Radiation paradoxes 2: Space and inertial frames

In trying to understand and resolve the paradox I wrote about in the first post in this series, I will be taking a somewhat circuitous route in order to lay some important groundwork before we can directly confront the paradox.

We can start the journey by looking at one of the most fundamental concepts in physics, that of the nature of space. On the surface, space seems like a very straightforward concept. It is seen as a kind of container in which everything in the universe exists. But difficulties arise when one asks questions such as whether space can be viewed as something positive, a tangible entity that has its own properties that can be detected, or whether it is viewed as something negative, that signifies the absence of matter in a region. Another way of posing the distinction is asking whether, if one can conceive of removing all the matter and energy in the universe, what would we be left with? Just ’empty space’? In the absence of matter, would such a thing have any meaning at all?

To address this question, we can start by examining the nature of the motion of matter through space. That brings us to the issue of what are called inertial frames that are intimately connected with the nature of space. Many of the laws of physics are believed to be valid only in inertial frames but it is not easy to identify those frames and part of the difficult lies in the fact that the nature of space is hard to pin down.

A ‘frame’ is short for ‘reference frame’ and refers to any system an observer uses to measure the motion of objects. The surface of the Earth is often used as a reference frame because it is the one we live in and are most comfortable with but you could just as easily use a moving car or train or boat or plane and measure the motion of any object with respect to it. When we get to relativity, it is necessary to be more specific about what a frame consists of and it is usually described as a system of standardized rulers and clocks that are spread out over all of space and move with the observer. An ‘event’ is any occurrence that all observers will agree about (such as my picking up a cup or a snap of my fingers) and is uniquely specified by the ruler and clock readings at the location of that event, since only one event can occur at any place at any time.

Different observers that are in motion with respect to one another will have their own sets of rulers and clocks that move with them. A key point is that while all observers will see the same events, we cannot assume that the location and time of any given event as measured by the rulers and clocks of one observer will have the same values when measured by the rulers and clocks of another observer who is moving relative to the first. So while events are unique, the space-time parameters that identify them will be different, their differences depending on whether the two frames are moving with low relative constant speeds (in which case we can use Galilean transformations), or at constant speeds that are appreciable when compared to the speed of light (in which case we must use Lorentz transformations), or they have a relative acceleration (in which case we must use general relativity). Those distinctions will not concern us in this series of essays until very much later.

Most people are familiar with some formulation of the law of inertia, also known as Newton’s first law of motion, that says that an object will continue in its state of rest or constant speed in a straight line unless acted upon by a net external force. (The technical term ‘velocity’ is used denote the combination of both speed and direction.) An inertial frame is one is which that law is seen to hold. (One can read more about the law of inertia at https://www.britannica.com/science/law-of-inertia)

But identifying which frame is inertial and which is not is tricky because a subtle circular reasoning can enter, leading to a sort of chicken-and-egg problem. An inertial frame can be identified as one in which the law of inertia is seen to hold for an object. But the law of inertia only holds in an inertial frame. In other words, we can say that we are in an inertial frame if we observe that an object that is subjected to a zero net external force is at rest or moving with constant velocity. But the problem is how would we know that there is no net force on the body? We know this if it is at rest or moves with constant velocity. But that is true only if we are in an inertial frame. That is the circular logic. This subtlety is usually elided when teaching physics, especially at the introductory level. (This is justifiable since deep subtleties can be mystifying and off-putting to novice learners, but we need to come to grips with it in this series of posts.)

Can one eliminate the circularity? Isaac Newton tried to do so by saying that space is something fixed and an inertial frame was a frame that was at rest with respect to that space or moving with a uniform velocity through it. Newton’s idea of space can be used to try and identify an inertial frame intuitively. I recall one of my professors in college saying that in an inertial frame you ‘feel at home’ and can ‘drink a cup of tea’ with no problems. (This was in Sri Lanka where tea metaphors are common.) That homespun image was meant to evoke the fact that a person in an inertial frame can tell if they are accelerating or not. If you are in a car that is at rest or traveling at constant speed in a straight line (i.e., with constant velocity), you can drink a cup of tea without spilling. But if the car changes its speed by braking or accelerating or changing direction (i.e., if it changes its velocity), then the tea sloshes around and can spill. We can know that the car’s velocity is changing purely by what happens within the car without even looking out the window. Changes in velocity always indicate accelerations and this means that the frame is not inertial. Thus we can feel whether we are in an inertial frame or not. So when we are at rest on the Earth or moving with constant velocity horizontally with respect to the Earth’s surface, we can feel that we are in an inertial frame.

But that is not strictly true because the surface of the Earth is not an inertial frame. The Earth is spinning on its axis and is also orbiting the Sun, and rotational motion always introduces accelerations. One can add another layer to the rotational motion by noting that our Solar system lies in a spiral arm of the Milky Way galaxy and that arm too is rotating about its center. And then we can also add the motion of the galaxy as a whole. The reason we do not sense these accelerations is because they are small, too small to detect in our everyday lives. The acceleration due to the spinning of the Earth about its axis is 0.034 m/s2 and that due to its orbital motion about the Sun is 1.2×10-5 m/s2 which are small compared to everyday accelerations like that due to gravity on the surface of the Earth, which is 9.8 m/s2. Such small accelerations can only be detected with sensitive measuring instruments. If you have a frame that is at rest or traveling horizontally on the surface of the Earth with constant velocity, then for all practical intents and purposes, that acts approximately as an inertial frame since the effects of the Earth’s spinning about its own axis and its orbital motions around the Sun are so small.

The smallness of those non-inertial effects is both fortunate and unfortunate. It is fortunate because we believe that Newton’s laws of motion and the laws of electromagnetism codified in Maxwell’s equations are strictly valid only in inertial frames. They could be (and were) discovered and formulated in the frame of the Earth, even though it is not an inertial frame, because the non-inertial effects are so small. If the non-inertial effects had been significant, then the simple forms that these laws take in inertial frames would have been much harder to discern. On the other hand, it is unfortunate because it likely prevented us from getting to grips with the true character of inertial frames for a long time.

The small effects due the Earth’s spinning can be detected using careful measurements (Newton did experiments to demonstrate them as I will discuss later) and they also show up in large scale systems like the atmosphere where they affect air flows, an effect that we commonly ascribe to ‘Coriolis forces’. This is one of the so-called ‘fictitious forces’ or ‘inertial forces’ that are introduced to explain the deviations from the behavior expected in inertial frames using Newton’s laws and ‘real’ forces, i.e., forces whose sources we can identify such as attached ropes and springs and gravity. The fictitious Coriolis force is due to the Earth’s rotation that makes it a non-inertial frame and it causes air currents to appear to veer away from straight line motion. Another fictitious force is the so-called ‘centrifugal force’ that is used to explain why, when we are in a car and take a sharp turn, we feel as if we are being ‘pulled’ outwards. For the people inside the car, it appears as if a mysterious force has suddenly kicked in.

It is the presence of these small fictitious forces that alerts us that we who are stationary on the surface of the Earth, despite appearances to the contrary, are not in an inertial frame.

(To be continued.)


  1. consciousness razor says

    The acceleration due to the spinning of the Earth about its axis is 0.034 m/s² and that due to its orbital motion about the Sun is 1.2×10-5 m/s² which are small compared to everyday accelerations like that due to gravity on the surface of the Earth, which is 9.8 m/s². Such small accelerations can only be detected with sensitive measuring instruments.

    Well … those are needed in the “can feel it spilling your tea” sense, where those kinds of instruments (not you or me) are the things which are sensitive enough to “feel” it.

    But of course, we can certainly detect that Earth is rotating and so on (if that’s all you’re aiming to do) via astronomical observations which need not be very sensitive at all. I mean, pretty much all you need to do is to watch the Sun and other stars “moving” across the sky every day. Obviously, that by itself wouldn’t spit out a number for the acceleration using fancy equipment and such; but you can at least conclude that, whatever the exact figure might be, it definitely isn’t zero.

  2. Rob Grigjanis says

    Even if the Earth were in deep space far from any other bodies, and not spinning, an object sitting on its surface is not in an inertial frame. If you let go of your cup of tea, it accelerates downwards, from your point of view.

    You are in an inertial frame if a nearby uncharged object with initial relative zero velocity stays where it is. And that’s certainly not a frame in which you could comfortably sip your tea from a cup.

    That said, there are a lot of experiments done on the surface of the Earth for which our frame of reference approximates an inertial one. For example, experiments in which motion of particles is constrained to a plane perpendicular to gravity. But even for the LHC, minor adjustments have to made to account for deflection of beams by gravity.

  3. Mano Singham says

    cr @#1,

    The question is whether you can sense accelerations purely within your frame, without reference to any external objects like stars.

  4. JM says

    Guessing where you are going with this I think the answer that you are going to arrive at is that all inertial frames are approximations. Space itself is expanding and that means that any two different points are accelerating away from each other by some very small amount. This means that no real physical frame (in this universe at least) is perfectly inertial. Inertial frames are idealized approximation for situations where all those factors like universal expansion, galactic rotations and such are too small to matter.
    Given that it’s just an approximation we can select a frame based on what makes the math simplest. Happily that matches our monkey brain assumption that the earth is a fixed base for normal terrestrial problems.

  5. Rob Grigjanis says

    JM @4:

    all inertial frames are approximations.

    Yes, of course. Everything in physics is an approximation. Unless you count thought experiments!

  6. Mano Singham says

    JM @#4,

    Actually, I am not even going into the effects of expanding space . This issue is complicated enough without bringing that in!

    In fact, expanding space may not matter in this case since the observer’s frame will expand along with it. What is significant is the local motion relative to the comoving frame

  7. billseymour says

    I just had a (probably not so bright) idea:  could there be some meta-frame in which spacetime as we know it exists?

    I’m almost totally ignorant of physics, so I’m just blathering here; but could that idea be used as an explanation of FTL travel in some science fiction story?

  8. Mano Singham says

    billseymour @#7,

    I am not sure what you mean by saying “could there be some meta-frame in which spacetime as we know it exists?”

    Space-time as we know it does exist, by which we mean that we can specify events uniquely by using four space-time coordinates.

    Can you clarify?

  9. dictyostelium says

    People respond with “inertial frames” about gravity and radiation due to gravitational acceleration, _because_ that should be equivalent to the frame that co-accelerates. Which shouldn’t have radiation.

    I think the question here is probably pretty much exactly equivalent as the discrepancy between Newtonian mechanics and Electrodynamics. These have different frames of reference; one has the Galilean and the other Lorentzian.

    Instead we have electromagnetics, and general relativity, trying to do an accelerating frame and noticing they disagree. (because they matter-a-fact do)

    There is no way to “interpret” Electromagnetic in inertial frames so that the Galilean transformations work.(other than as approximation) And similarly here, i don’t think for GR and electrodynamics you can..

    Personally, i expect no radiation from falling charged objects. Since in the frame accelerating along, for them nothing is happening. We do have models for GR+EM together.. Including for charged black holes. Models which can be applied straight to a charged sphere.

    Although afaik experiments and observations we can do have very little hold on GR+EM models? I doubt a simple charged object with earth gravity, assuming i am wrong and it actually radiated, would produce measurable radiation. Seems like someone would have done the experiment, already, if it did..

    Hmm, now i think about it though, what if the charge isn’t allowed to fall, it’d be accelerated upward constantly to stay in position and radiate?… So maybe i am wrong somehow.. Still, as i said, there basically are models for GR+EM, which presumably say something about this?

  10. flex says

    The fact that the earth is not an inertial reference frame led me to thinking that time travel, if it wasn’t impossible for other reasons, would be far more difficult than jumping into a DeLorean and getting to 88MPH.

    If you did manage to back in time, or jump forward for that matter, you would need to account precisely for the accelerations from the rotating earth, the orbiting earth, and even the motions of the solar system and galaxy. They are small accelerations, too minor for us to personally detect, but even a minor mistake in correcting for them would leave the time traveler in trouble. An error of a few feet could put the time traveler halfway into a wall, a larger error could miss the earth entirely. Then, are there possible unpredictable fluctuations which limit the precision of the correction factors? How precise could a possible time traveler be? As an example, if Planck’s Constant applied (and I don’t know if it would), and any calculation which relied on a precision which exceeded Plank’s Constant was unpredictable, that might limit the time travel to 30 days, or 30 minutes, or even the 13 seconds (within that inertial reference frame) used by the movie Galaxy Quest.

    I don’t know if that thought is correct. I proposed it on a forum a couple decades ago and was told that, “No, this would not be a concern for a time-traveler.” But I don’t know if that response was correct, and I don’t know physics well enough to know if my idea has any validity, or it is complete crap.

  11. consciousness razor says

    The question is whether you can sense accelerations purely within your frame, without reference to any external objects like stars.

    I get that. I was just responding to the statement that it’s only detectable with very sensitive equipment, which didn’t stipulate anything about which objects you’re allowed to reference and which ones are off-limits.

    That does seem like a somewhat fraught subject though. Here in this room, with the blinds closed on the window, I can still see where the daylight hits the floor and how that position changes over the course of the day, over the year, etc.

    So … am I “referencing external objects” then? Implicitly, perhaps, but in a much more direct sense, it’s about local events that are only happening in the room where I’m typing this. Besides, I shouldn’t have to be in a closed system (or pretend as if it were one) in order to talk about this reference frame, since its status as an inertial or non-inertial frame simply isn’t about whether any physical systems in it are open or closed. And anyway, like I said, I don’t literally need to see anything that’s going on outside. It’s all available in my immediate vicinity (spatially and temporally), just like it is when you observe the behaviors of gyroscopes or whatever sorts of instruments are supposed to be fair game because they’re giving you only “local” information.

  12. Rob Grigjanis says

    dictyostelium @9: I started composing an answer to your post, but it occurred to me that there are many people smarter than me, who have been thinking about this for a lot longer. My bolding;

    The familiar relativity of kinetic energy corresponds to the symmetry between different frames of reference, which is to say, we can always find a system of inertial coordinates with respect to which any given object (at a given instant) has zero kinetic energy. Our consideration of charged particles in a gravitational field seems to suggest similarly that we can always find a system of coordinates (at least locally) with respect to which a charged particle (at a given instant) does not radiate -- even though the particle may be radiating at that instant with respect to some other system of coordinates.

    It’s also possible to question whether the equations of electrodynamics really do imply that an accelerating charge necessarily radiates. Surprisingly, this is still an open question for the classical theory. The difficulty is in knowing how to correctly account for the influence of a charged particle on itself.


    Feynman thought that, while a charged particle undergoing cyclic acceleration did radiate, one undergoing linear acceleration did not radiate. Many very clever people disagree vehemently with the latter.

  13. billseymour says

    Mano @8:  oh dear, I’m not sure I know what I mean. 😎

    I’m wondering about the expansion of space-time; but now I reread your comment @6 and see that that’s outside the scope of this discussion.  I should just shut up and wait for the rest in your series of posts. 😎

  14. GerrardOfTitanServer says

    The question is whether you can sense accelerations purely within your frame, without reference to any external objects like stars.

    I also wrote this for part 3. I’m very, very confused. Isn’t the point that you can detect whether you are undergoing acceleration or not via experiments that you can do in a completely closed box with no access to the outside world, e.g. an accelerometer?

    The answer seems to be: Of course you can, to the limits of the precision of your accelerometer.

  15. GerrardOfTitanServer says

    As for the opening problem, here’s my half-assing based on a mere physics undergrad education which wasn’t even enough for a minor.

    Take the charged particle, the uncharged particle, and observer #1, and release them at the same time (according to observer #1) from the same height above Earth’s ground in a vacuum. Observer #1 is in an inertia frame. So are the two particles. It seems to me that they all must hit the ground at the same time according to observer #1. I would be really surprised otherwise.

    Take observer #2, standing on the ground of the Earth. They are in a non-inertial frame. They are undergoing constant acceleration. So, all bets are off, and I have no idea what they see. Seems like a very complicated question of general relativity which is way above my pay grade.

    A more interesting setup to me is concerning two observers who observe the particles, both in inertia frames, but in different inertia frames. The complete novice that is me doesn’t see how you can avoid Feynman’s conclusion, because otherwise one observer can see the two particles collide in finite time but the other observer seems them remain at constant distance for eternity -- but again I’m way above my pay grade.

  16. Rob Grigjanis says

    Gerrard @15:

    concerning two observers who observe the particles, both in inertia frames, but in different inertia frames

    You mean the two particles which were released at the same time? There is no frame, inertial or otherwise, in which they will collide, unless they go down a very big hole and converge at the centre of the Earth (in thought-experiment world, anyway). And I don’t see what that has to do with Feynman’s conclusion.

  17. GerrardOfTitanServer says

    Sorry Rob,
    Let me clarify. I’m an idiot. I briefly confused “charged particle at constant velocity causes radiation” to “charged particle at constant acceleration causes radiation”. Please ignore that third paragraph.

  18. GerrardOfTitanServer says

    Let me reformulate what I was thinking.

    Two test particles, one charged, and observer #1, at 0 meters, 1 meter, and 2 meters on the x-axis, all at rest relative to each other in an inertial frame, far from anything else. Throw on observer #2, constantly accelerating. If you position observer #2 just right, and if charged particles radiate away (kinetic) energy from constant acceleration, then observer #2 should eventually see the two test particles collide in finite time. However, the other observer sees the two test particles remain at constant finite distance forever.

    That seems to be a simple formulation of the paradox. So, without knowing anything, I’ll side with Feynman as the “obvious solution”.

    Having said that, the real proper answer must involve lots of fun (general?) relativity. Yay.

  19. GerrardOfTitanServer says

    I hope I beat Rob to a correction.
    I’m still an idiot. I was assuming equivalence of acceleration, approx, but that’s not true. Observer #2 might see the test particles behave as though they were accelerating, but the test particles are not. Observer #2 can measure this absolutely. Velocity is relative. Acceleration is not.

  20. Rob Grigjanis says

    Gerrard @18: The OP has the particles released from the same height, but OK, we’ll do it your way. It’s actually a neat reframing, since it asks “collision or no collision?”.

    First, let’s get Feynman out of the way. His claim was that a charged particle undergoing linear acceleration with respect to an inertial frame does not radiate. In your scenario, the particles have constant velocity with respect to any inertial frame (I’m assuming the domain of the experiment allows us to ignore spacetime curvature).

    Observer #1 will certainly not see radiation, and in fact will see the two particles remain at rest. That (and the equivalent consideration in the OP’s scenario) is what leads me to suspect that the accelerated observer will not see radiation from the charged particle; if they did, that would seem to imply that the charged particle was losing energy, and how could that be unless it was losing kinetic energy relative to the uncharged particle? Indeed, if you had two accelerated observers with opposite accelerations, one would then see the particles converge, while the other would see them diverge.

    But the literature paints a somewhat more complicated picture. Maybe the accelerated observer could see radiation without it affecting the charged particle’s motion.

  21. GerrardOfTitanServer says

    Thanks much Rob!
    I’m sorry for being an idiot.

    (I’m assuming the domain of the experiment allows us to ignore spacetime curvature).


    is what leads me to suspect that the accelerated observer will not see radiation from the charged particle


    But the literature paints a somewhat more complicated picture. Maybe the accelerated observer could see radiation without it affecting the charged particle’s motion.

    I need to read more. Maybe I’ll understand something of it.

  22. John Morales says

    Gerrard, I think perhaps you’d be better off boning up on your tensor calculus and stuff.

  23. Rob Grigjanis says

    John @22: Gerrard’s picture of the particles being on the same axis as the acceleration of observer #2 was a positive contribution to the discussion, in the opinion of someone who knows his tensor calculus and stuff.

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