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Distant galaxies are said to be moving away from the Milky Way (and us) at speeds approaching the speed of light. Since Special Relativity tells us that any object moving away from us at a velocity of near the speed of light will increase in mass in our observations, i.e. it will appear to be much more massive than its rest mass, approaching infinity as the speed approaches the speed of light. If we observe very large masses (like galaxies) moving away from us at near light speed, wouldn't their rest masses be tiny? So why do we not think these distant galaxies are actually just specks of matter in terms of their actual rest masses? And if that is the case then are we not just looking at what may be sub-atomic particles moving away from us at near light speed? ...The expanding universe explanation is the standard one. The balloon example is perhaps suggestive but not correct. If you occupied one point on the surface of a balloon, and you had a friend next to you and the balloon expanded, your friend would move away from you at a relative velocity v. Einstein's equations would obviously still apply. I see no easy way to explain why when I look at measurements of relative velocity between distant recessive galaxies, those velocities are somehow not what are measured and quoted but something very different because the universe is expanding. Velocity is defined (for Einstein too) as distance divided by the time to travel that distance. And we are talking about objects with mass; not a reflective beam from a searchlight or something like that. Even if the universe expands, velocity is still what it is. And if it is not what it is, then why do astrophysicists state that distant galaxies are moving away from us at nearly light speed? I guess I would like to see the intuitive argument with equations included which must be there but no one knows how to do it. I suppose it took a while before time dilation could be explained on the back of an envelope too.

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Hi Don. Your question seems to be nice. Though formatting is not a big issue, atleast you could've some mercy on us. Please consider formatting coz, some users may get bored by reading your question. Man, it is a huge story-like thing. Err... Sorry for that :-) – Waffle's Crazy Peanut Dec 28 '12 at 3:19
Why do you think the mass of distant galaxies would be small? Is it because you think we've measured their mass and found it to be similar to our galaxy, while neglecting to correct for relativity? Because I'm fairly certain that isn't true. The reason we think distant objects are galaxies is not because we've measured their mass, but simply because they look like galaxies... – Nathaniel Dec 29 '12 at 5:52
Nathaniel, an atomic particle blown up might 'look' like a galaxy. Also you are much more trusting than I. You cannot just assume that people have corrected for relativity. They might have ignored that by using the expanding universe hand waving argument. Do you remember when several years ago Nasa crashed a rocket because they were confused between meters and feet?!? No real scientist should assume other people know what they are doing. Anyway, I answered my own question by plugging in numbers in my answer below. Actually doing the calculations is sometimes the best solution. – Don West Dec 29 '12 at 19:15

Let us not forget that in addition to far-away galaxies at high redshift we can observe nearby galaxies (and many at every distance in between). Those observations of nearby galaxies can resolve individual stars which are clearly the same as stars in our own galaxy and obey physics consistent with the physics that powers our very own Sun.

Further, the mass of galaxies is measured by observing their velocity curves (indeed this was the source of the first part of the missing mass puzzle).

In short, no: there is no chance that we have made an error of orders of magnitude on the mass of these objects.

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It is not true that Special Relativity tells us that "any object moving away from us at a velocity of near the speed of light will increase in mass in our observations". This statement depends on what observation you make, ie if you measure the relativistic mass or the rest mass. For cosmological observations the relativistic mass is not usually an interesting quantity. There are various ways to estimate the mass of galaxies, one is for example gravitational lensing, which has the merit that it also takes into account dark matter. Or you can look at the angular velocities of stars orbiting the center which depend on the total mass (also takes into account dark matter). Either way, what you get is the total gravitational mass (energy) in the rest frame of the galaxy, ie its rest mass.

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No matter how you measure a mass in relative motion, relativity still applies. And the direction of movement (i.e. away) from us does not come into the equations. So your opening statement is incorrect. – Don West Dec 27 '12 at 18:45

What special relativity tells us is that a body moving with a velocity $v$ which has a rest mass $m_0$ will have an observed mass of $$m = \gamma m_0$$ where $$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

This still holds true in astrophysical situations. For example, the apparent brightness of objects called quasars is much larger than it's "real" brightness due to a very similar effect.

But what is actually happening is that space itself is expanding. No doubt you know that the farther an object is from us, the faster it is moving away from us. But the velocity with which it is moving away from us isn't due to the velocity possessed by the object itself but rather the velocity due to the expansion of space.

Think of two dots on a balloon. If you blow into the balloon, the two dots will move away from each other at some speed. But this isn't because the dots themselves have a velocity, it's because the space in which they are in (the balloon) is itself expanding. The situation with the universe isn't exactly analogous, but it's somewhat similar.

Therefore since the bodies themselves don't possess a relativistic velocity, what we actually measure is the rest mass of the object.

P.S - Lots of astrophysical papers do talk about this quantity called "peculiar velocity", which is infact the velocity possessed by the object itself. This peculiar velocity is the velocity over and above the velocity of the object due to the expansion of space. How the difference between the two velocities etc., are measured and calculated are beyond your question (and this answer), so I'm not going to get into it here, but I thought it was worth mentioning that astrophysical object do have their own velocities (on the order of a $100$ to $1000$ $km$/$s$).

EDIT: Special relativity applies only to inertial frames of reference with respect to us. The expansion of space is an accelerating and hence a non-inertial frame of reference. So one will have to use general relativity equations (which I'm not too comfortable with).

However it is useful to to remember that the increase in mass of a relativistic object comes from the mass-energy equivalence. The total energy of a body with momentum $p$ is given by $$E = \sqrt{p^2c^2 + m^2 c^4}$$

Consider you and a friend standing at two points in space. If the space is expanding with some speed, you and your friend will be moving away from each other, and therefore light from him will be redshifted with respect to you. But he isn't gaining energy by moving away from you, both of you are just standing there. Therefore the mass that you measure will be the same as his rest mass, since $p = 0$ in the above equation, and $\gamma = 1$.

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I couldn't add enough words for my comment so I edited my question instead. The expanding universe answer is the standard one but without a bit more work on the intuition and equation side, it doesn't explain anything - it just states it. I explained time dilation to my 15 year old son on the back of an envelope just from the geometry of relative motion. That is possible because Einstein's work is fundamentally intuitive and simple. Special and General relativity (unlike quantum mechanics) can be understood with basic algebra and calculus. That is what I am looking for. – Don West Dec 27 '12 at 18:55
I've edited my answer to try and explain why the expansion of the universe doesn't result in an increased mass. I hope it helps! – Kitchi Dec 27 '12 at 20:32
Thanks Kitchi. I'm still not clear on this but you have made me think that I need to use General Relativity which does sometimes change the intuition a bit. I have to think about this. By the way, One way to get to the equations of general relativity which is very simple is to assume the equations of Special Relativity hold at every moment in time and therefore one need only take derivatives with respect to v to obtain the equations of General relativity. – Don West Dec 27 '12 at 22:50

The usual answer (as noted above) comes from an empirically consistent standard cosmology for which space itself expands. Evidence includes 1. the cosmic microwave background with its origin near the early recombination epoch, redshifted by the expansion to the current temperature of 2.7 K 2. the baryonic nature of luminous matter as indicated by spectral data 3. the agreement of WMAP, BAO, supernovae and many other tests for distance measures to high redshift

If you were correct we would have difficulty explaining the details of astrophysical phenomena observed at the lower redshifts, such as star formation regions and active galactic nuclei. Don't forget that observations begin with our own solar system and Milky Way, which appears not that different to other spiral galaxies.

Having said that there is a sense in which you are correct, because a high redshift galaxy could well be another form of a molecule at rest in the laboratory, since matter potentially contains all other matter by the laws of QFT. Even the standard cosmology (which is well open to debate these days) would agree that there are string theory dualities between very distinct scales and there is no reason that we should not apply these ideas to our view of the cosmos.

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If your argument was right, we would be so confused when handling accelerated electrons in laboratories, confusing them for galaxies.

Even based on my very limited experience with relativity, the very basic formulas for energy and momentum account for this. Hence, there is a difference between rest mass ($m$), and observed mass ($\gamma m$). So rest assured, if these formulas were correct, galaxies should be huge.

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"If your argument was right, we would be so confused when handling accelerated neutrons in laboratories, confusing them for galaxies" Do they really accelerate neutrons? Meh, I suppose they do in a sense when they knock them out of targets :-) – twistor59 Dec 27 '12 at 7:44
@CrazyBuddy I think you mean protons/electrons/etc. - very hard to accelerate electrically neutral things with electromagnetic fields ;) – Chris White Dec 27 '12 at 11:29
Though the velocities experienced by particles in cyclotrons, etc., are near the speed of light; the Lorentz transform still works exactly and we would not be confused about seeing galaxies. It depends on 'how close to the speed of light' and the rest mass of the particles in question. – Don West Dec 27 '12 at 18:49
up vote 0 down vote accepted

I thought that since I brought this up and I didn't get any good explanations, I'd answer it myself. Plugging in some numbers helps. I used this for light speed = 299,792.458 km/sec. Hubble constant = 72 km/sec/megaparsec. The Hubble constant is necessarily an approximation because nobody knows exactly what it is and they also don't know if it is a constant through time. The Hubble constant is used to calculate the speed of distant galaxies due to the expansion of the universe. So based on some simple calculations using the above numbers, I come up with this: If you go out 13,573,936,292 light years, the universe is expanding at 0.99999999994 of the speed of light. I might point out that this is about 203 feet per hour slower than the speed of light so it is pretty close. Using the Lorentz Transform (relativity) we can figure out the effect on a galactic mass moving at that speed away from us. Suppose we observe a galaxy about the size of the Milky way or Andromeda moving away from us at this speed. The rest mass of the Milky way or Andromeda is roughly 700 billion solar masses. Give or take. Applying the Lorentz transform to this observation, the rest mass of the distant receding galaxy would be about 7,493,550 solar masses. So even at speeds this close to the speed of light, this is still a large mass. It is not a tiny particle as I thought it might be. Therefore we don't even have to use the hand waving argument about the expansion of the universe doesn't really count in relativistic calculations. I guess the real theoretical issue and question would be: what happens at the singularity? That is right at the outer edge of the expansion, objects are traveling exactly at the speed of light, and here relativity breaks down. But nobody knows the answer to that. The interesting thing about plugging in the numbers is that when you are really close to light speed, there are no contradictions. And my calculations are really close because 203 feet per hour is a lot slower than I walk. I could crawl faster than that.

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It would seem that your argument is invalidated by the accepted thinking (e.g. that some galaxies are receding faster than the speed of light. That would seem to suggest that expansion doesn't generate any kind of relativistic motion between galaxies in the first place. – Michael Nov 15 '14 at 15:19

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