Q: If time is relative, then how can we talk about how old the universe is?

Physicist: One of the most profound insights ever made by peoplekind is that time is relative.  This isn’t some abstract idea, mistake, or mathematical artifact.  If you have two identically functioning clocks, you can start them together, move them to different locations or along different paths, then when you physically bring them back together to compare, they will literally have registered different amounts of time.

You may be inclined to say, “sure, it’s weird… but which clock is right?”.  The existentially terrifying answer is: there is no such thing as a “correct clock”.  Every clock measures its own time and there is no such thing as a universal time.  And yet, cosmologists are always talking about the age of the universe (a mere 13.80±0.02 billion years young).  When talking about the age of the universe we’re talking about the age of everything in it.  But how can we possibly talk about the age of the universe if everything in it has its own personal time?

The short answer is: almost everything is about the same age.  The biggest time discrepancy is between things deep inside of galaxies and things well outside of galaxies, amounting to a couple parts per million (or one or two seconds per week).  Matter that has been in the middle of large galaxies since early in the universe’s history should be no more than on the order of 50,000 years younger than matter that has managed to remain in intergalactic space.  Considering that our best estimates for the age of the universe are only accurate to within 20 million years or so (0.1% relative error), a few dozen millennia here and there doesn’t make any difference.

There are two ways to get clocks to disagree: the twin paradox and gravitational time dilation.  The twin paradox a bizarre consequence of the difference between ordinary geometry and spacetime geometry.  In ordinary geometry, the shortest distance between two points is a straight line.  In spacetime geometry, the longest time between two points is a straight line.  A “straight line” in spacetime includes sitting still, so if you start with two clocks in the same place and take one on a trip that eventually brings it back to its stationary partner, then the traveling clock will have fallen behind its sedentary twin.

The Twin Paradox: The straighter the path taken between two locations, the more time is experienced.  Gravitational Time Dilation: Things farther from mass experience more time.

Assuming that the traveling clock travels at a more-or-less constant speed, you can figure out how much less time it experiences pretty easily.  If the traveling clock experiences \tau amount of time and the stationary clock experiences t amount of time, then \tau=t\sqrt{1-\left(\frac{v}{c}\right)^2} (which you’ll notice is always less than t) where v is the speed of the traveling clock and c is the speed of light.  The ratio between these two times is called “gamma”, \gamma = \frac{t}{\tau} = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}, which is a useful piece of math to be aware of.  If the traveling clock changes speed, v(t), then you’ll need calculus, \tau=\int_0^t \sqrt{1-\left(\frac{v(t)}{c}\right)^2}\,dt=\int_0^t \frac{dt}{\gamma(t)}, but there are worse things.

Gravitational time dilation is caused by the warping of spacetime caused by the presence of energy and matter (mostly matter) which is a shockingly difficult thing to figure out.  When Einstein initially wrote down the equations that describe the relationship between mass/energy and spacetime he didn’t really expect them to be solved (other than the trivial empty-space solution); it took Schwarzschild to figure out the shape of spacetime around spherical objects (which is useful, considering how much round stuff there is to be found in space).  He did such a good job that the event horizon of a black hole, the boundary beyond which nothing escapes, is known in fancy science circles as the “schwarzschild radius”.

Fortunately, for reasonable situations (not-black-hole situations), you can calculate the time dilation between different altitudes by figuring out how fast a thing would be falling if it made the trip from the top height to the bottom height.  Once you’ve got that speed, v, you plug it into \gamma and wham!, you’ve calculated the time dilation due to gravity.  If you want to figure out the total dilation between, say, the surface of the Earth and a point “infinitely far away” (far enough away that Earth can be ignored), then you use the speed something would be falling if it fell from deep space: the escape velocity.

By and large, the effect from the twin paradox is smaller than the effect from gravity, because if something is traveling faster than the local escape velocity, then it escapes.  So the velocity you plug into gamma for the twin paradox (the physical velocity) is lower than the velocity you’d plug in for gravitational dilation (the escape velocity).  If you have some stars swirling about in a galaxy, then you can be pretty sure that they’re moving well below the escape velocity.

The escape velocity from the surface of the Earth is 11km/s, which yields a gamma of \gamma = 1.0000000007.  Being really close to 1 means that the passage of time far from Earth vs. the surface of the Earth are practically the same; an extra 2 seconds per century if you’re hanging out in deep space.  The escape velocity from the core of a large galaxy (such as ours) is on the order of a thousand km/s.  That’s a gamma around \gamma = 1.00001, which accounts for that several seconds per week.

Point is, it doesn’t make too much difference where you are in the universe: time is time.

Now admittedly, there are examples of things either trapped in black holes or screaming across the universe at near the speed of light, but the good news for us (on both counts) is that such stuff is rare.  The only things that move anywhere close to the speed of light is light itself (no surprise) and occasionally individual particles of matter.  Light literally experiences zero time, so the “oldest” photons are still newborns; they have a very different notion of how old the universe is.  No one is entirely sure how much of the matter in the universe is presently tied up in black holes, but it’s generally assumed to be a small fraction of the total.

Long story short: when someone says that the universe is 13.8 billion years old, they’re talking about the age of the matter in the universe, almost all of which is effectively the same age.

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16 Responses to Q: If time is relative, then how can we talk about how old the universe is?

  1. A muon whirling around a storage ring has a big time dilation. Indeed the time it takes them to decay is much longer and w/o this effect they wouldn’t live long enough to be stored

  2. Leo says:

    Thank you very much for this, a very clear explanation!

  3. Ashok Narayan says:

    An excellent explanation. I can only add that the universal time is measured with reference to the Hubble Parameter (H). The surface along which H has the same value is considered to be a surface of the same universal time. In other words, the state of expansion of the universe determines the universal time.

  4. Douglas Franks says:

    You discussed how the clock traveling as constant speed experiences time, but the reciprocal is also true. That clock can be considered stationary and the “at rest” clock moving away, so they both experience the longer time.

    It was also discussed that the preponderance of mass in the universe has little relative motion so all experiences similar time approximated by the age of the universe. Isn’t it still legitimate to do the thought experiment of traveling with a neutrino and find a very different passage of time and a different age? It gets hard to wrap your head around.

    About the photon experiencing the null interval, is it always at the beginning of time, or is it always current at what we consider present time. Or is it at all times always??

  5. The Physicist The Physicist says:

    @Douglas Franks
    That’s why it’s important to stipulate that the clocks start together and end together. Otherwise, the rate of time is (as you point out) subjective. However, the straight-line vs. curved-path thing is objective and does lead to different results.
    Things that travel near the speed of light (relative to some frame) between two locations/events (defined by that same frame) will experience less time on the journey than things that are stationary (in that same frame). A photon traveling from here to the Moon will take about a second according to us, but zero time according to the photon. It turns out that asking “what is the photon’s point of view” is remarkably fruitless; if a photon doesn’t physically run into something, then that thing may as well not exist. While we material beings can talk about coordinates and time and travel, photons don’t. Not really.
    High speed things have a very different perspective on the universe, no more or less valid than any other, but I figured it would be more useful to consider what the vast majority of the matter in the universe sees when it considers the vast majority of the matter in the universe.

  6. To get the twin paradox one twin must accelerate differently than the other. If they are riding in ‘elevators’ they experience different ‘gravitational’ fields and the effect is not different from going near a black hole (but not too near!) and coming back out to meet up again.

  7. The Physicist The Physicist says:

    @Traruh Synred
    It’s subtle, but although there must be acceleration for a curved path, acceleration is not the cause of the twin paradox. The top diagram here is a decent explanation of why.

  8. We’ll that’s interesting.

    I remember doing a calculation (homework, I think) as an undergrad. counting ‘heart beats’ and it seemed the acceleration was critical (It was taken to be infinite, so we’d only need special relativity). There has to be some acceleration to break the symmetry, but I guess it can be anything. But indeed the difference in twin age depended on how far/long and the speed of the traveling twin.

    The approaching black hole equivalence is wrong apparently.

  9. Will a clock at the bottom of an upward accelerating elevator run differently than one at the top? Seems like it must by the equivalence principle… for a ridged elevator the clocks would always be at the same speed.

    What happens if the elevator stops?

  10. Bert Plante says:

    If infinity is a solid, the momements of the two clocks relative to infinity are different.
    If time is momement relative to the solid of infinity, of course, to each other they would
    be different.

  11. Douglas Franks says:

    It was stated that all of the matter in the universe has little variation in frame of reference, and therefore similar time.

    I have understood that out at the visual horizon of our universe galaxies are moving away from us at close to the speed of light caused by be some high bred of the expansion of space and conventional motion. If there are such high speed differentials would there not also be significant differentials in time for large structures in the universe?

  12. Neruz says:

    @Douglas Franks
    Expansion of space is not the same thing as movement for relativistic purposes.

  13. Let’s call a photon, “A”. A is outside time. “A” also crosses the event horizon of a black hole. What time did it cross that line? Did it enter a specific time when it crossed? Could it ever cross over? If so when did A cross the event horizon?

  14. David says:

    We can estimate the age of the universe, as you say, with an approximate universal time. But incidentally, wouldn’t matter moving at just below c, such as neutrinos, get further out of sync with other matter over 12 bn yrs (give them a bit of time before they get emitted), than 50,000 years? If they travel at 0.999976 c then they’re time dilated by a factor of 144, which gets them out of sync with other matter by 83 million years.

  15. Neruz says:

    Sure, but neutrinos are nearly impossible to detect as-is. So very little is known about them, IIRC we only recently figured out that they can ‘oscillate’ into different kinds (flavors I think?) of neutrinos which are even harder to detect than the normal kind.

    I have no clue if we can date neutrinos. They are a very, very strange particle.

  16. Fahim says:

    Hi, I have done a blog on an experiment that I did in freshmen year of school, relating to the refractive index. Please feel free to check it out and leave feedback. Thank you.

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