Relativity gives us no universal clock, yet cosmologists say the universe is about 13.8 billion years old. The answer lies in cosmic time: a natural way of keeping time supplied by the expanding universe itself.
Relativity took away the master clock.
Before Einstein, it was natural to imagine time as an invisible current flowing uniformly through the universe. Clocks might run badly, people might lose track of the hour, and trains might fail to respect the timetable, but beneath these local disappointments there was supposedly one true time. If two events happened simultaneously, they happened simultaneously for everyone.
Relativity tells us otherwise. Observers moving relative to one another can disagree about which distant events happened at the same time. Clocks following different paths through spacetime can accumulate different durations. Gravity changes the rate at which clocks tick. There is no clock mounted somewhere above the universe, safely outside the proceedings, displaying the correct cosmic hour.
And then cosmologists say that the universe is about 13.8 billion years old.
This sounds suspiciously like the master clock has been smuggled back into the building.
Whose 13.8 billion years are these? A clock on Earth? A clock drifting between galaxies? A clock moving at nearly the speed of light? There is no inertial reference frame for the entire expanding universe, so where does this age have any truth?
The short answer is that the age of the universe is not the reading on an arbitrary observer's clock. It is an amount of cosmic time: the proper time measured by an ideal family of observers who move with the average expansion of the universe.
That answer needs unpacking, because "ideal family of observers" is precisely the kind of phrase that can make a simple question feel as though it has been escorted into a government office.
It is sometimes said that relativity proved time is not real or not fundamental. Relativity itself does not go that far. What it removes is a single universal time that every observer must share.
Suppose two travelers leave the same event, follow different routes through spacetime, and later meet again. Their clocks need not show the same elapsed duration. One may have traveled rapidly; one may have spent time deeper in a gravitational field. Neither clock is malfunctioning. Each has measured the time along its own path.
This path-dependent duration is called proper time. It is not merely a matter of perspective in the casual sense, as though the travelers could settle the disagreement by talking it over. Proper time is what a real clock records along a particular worldline. Everyone who performs the calculation correctly will agree about what that clock recorded.
Relativity therefore gives us something more interesting than "time is subjective":
There is no single elapsed time between widely separated events until we specify the path or the rule by which distant clocks are to be compared.
For a lone astronaut, the path is the astronaut's worldline. For the universe as a whole, cosmologists need a sensible rule for comparing clocks spread across billions of light-years.
Fortunately, the universe contains a clue about how to do that.
The laws of relativity do not select a preferred state of motion. But the contents of a particular universe can.
There is no preferred direction written into the laws of mechanics, for example. Nevertheless, if you jump into a river, the water has a definite direction of flow. You can measure your motion relative to it. The river has not repealed relativity; it has simply supplied some physical material against which motion can be described.
Our universe contains its own extraordinarily large-scale reference material. Galaxies participate, on average, in the cosmic expansion. Even more importantly, space is filled with the cosmic microwave background, or CMB: ancient radiation released when the universe became transparent, roughly 380,000 years into its hot early history. Today it reaches us from every direction with a temperature of about 2.7 kelvin. ESA's overview of the CMB explains how this radiation records the state of the early universe.
If you are moving relative to this background, it appears slightly hotter in the direction you are traveling and slightly cooler behind you. If you remove that dipole pattern, you have identified the local frame in which the CMB is, on average, equally warm in every direction.
Our Solar System is not at rest in that frame. Measurements of the CMB dipole show it moving at about 369 kilometers per second relative to the cosmic background. That speed sounds dramatic until one remembers that light travels nearly 300,000 kilometers per second. Cosmically speaking, we are fidgeting in our chair.
The CMB frame is not preferred by the laws of physics. An experiment performed in a smoothly moving laboratory still obeys the same local laws. But the CMB frame is distinguished by the actual distribution of matter and radiation in our universe. It is the frame in which the large-scale cosmic material is most nearly at rest.
Now imagine observers scattered throughout the universe. Each observer remains at rest relative to the average cosmic matter nearby. They do not fly through the expanding material under their own power; they are carried along with the large-scale flow.
Cosmologists call them comoving observers.
As space expands, the physical distances between widely separated comoving observers increase, but their labels on a cosmological map remain fixed. They are like raisins in rising dough, provided we forgive the dough for having an outside, an oven, and altogether too much confidence as a model of the universe.
Each comoving observer carries a clock. In an ideal perfectly homogeneous and isotropic universe, all of these observers agree about the large-scale state of the cosmos around them:
These shared physical conditions allow their clocks to be synchronized. A particular cosmic time does not mean "whatever time the Earth clock says." It means "the epoch at which the universe has this large-scale density, temperature, and degree of expansion."
The proper time kept by this family of comoving clocks is called cosmic time.
There is still no global inertial frame. An expanding, curved spacetime cannot be covered by one enormous special-relativistic laboratory. Each comoving observer has only a local frame. But the regular large-scale structure of the universe lets cosmologists connect those local frames into a natural global description.
The universe has not given us one master clock. It has given us a way to organize many clocks.
The geometry used in standard cosmology is described by the Friedmann-Lemaître-Robertson-Walker, or FLRW, metric. In a compact form, part of it can be written as
$$ ds^2 = -c^2 dt^2 + a^2(t)d\Sigma^2. $$
The symbols look more forbidding than the idea.
By convention, cosmologists commonly set $a=1$ today. Earlier in cosmic history, $a$ was smaller. If the scale factor was $1/2$ at some earlier epoch, two sufficiently distant comoving locations were then half as far apart as they would be today, setting aside local structures held together by gravity or other forces.
For a comoving observer, the spatial coordinates do not change. That means
$$ d\Sigma = 0. $$
The metric along that observer's path therefore becomes
$$ ds^2 = -c^2dt^2. $$
Along a clock's worldline, the interval is also related to the clock's proper time $d\tau$ by
$$ ds^2 = -c^2d\tau^2. $$
Comparing the two expressions gives
$$ d\tau = dt. $$
This is the mathematical heart of the idea: for an ideal comoving observer, cosmic time is proper time. The coordinate $t$ has not been chosen merely because it makes the equations attractive. It corresponds to what that family of clocks would measure.
Knowing what cosmic time means does not tell us its present value. We still need to reconstruct how long the expansion has been underway.
The key quantity is the Hubble parameter, written $H(t)$. It measures the fractional rate at which the scale factor changes:
$$ H(t)=\frac{\dot a(t)}{a(t)}. $$
The dot over $a$ means "how quickly $a$ is changing with time." Dividing by $a$ turns that change into a fractional rate. It is similar to distinguishing "my investment gained $50" from "my investment gained five percent." The second statement tells us the rate relative to the amount already present.
Because
$$ \dot a = \frac{da}{dt}, $$
we can rewrite the definition of $H$ as
$$ H(a)=\frac{1}{a}\frac{da}{dt}. $$
Now solve for the small amount of cosmic time $dt$:
$$ dt=\frac{da}{aH(a)}. $$
This equation says that if we know the scale factor and the expansion rate at a particular stage of cosmic history, we can determine how much cosmic time the universe spent moving through a small interval $da$.
To find the total age, we add all those small intervals—from the hot early-universe limit, where the scale factor approaches zero, to today, where we have defined it to be one:
$$ t_0=\int_0^1 \frac{da}{aH(a)}. $$
The integral sign means "add continuously." It performs the same conceptual job as adding the durations of every leg of a journey, except that the legs have been made infinitesimally short.
The age therefore does not come from taking today's expansion rate and naively running it backward at a constant speed. The expansion rate has changed. Radiation dominated the early universe, matter later became more important, and dark energy dominates the recent expansion. Each component affects $H(a)$ differently. The integral accounts for the whole changing history.
No clock survived from the first instant with a warranty card and an intact battery. Cosmologists infer the age by combining observations with a model of how the universe evolves.
Several kinds of evidence constrain different parts of that model:
Different choices for the cosmic ingredients produce different functions $H(a)$ and therefore different ages. Astronomers adjust the model parameters until the predicted CMB pattern, galaxy distribution, supernova distances, and other observations agree with what telescopes actually see.
Under the standard flat $\Lambda$CDM model—the model containing ordinary matter, cold dark matter, radiation, and dark energy represented by a cosmological constant—the result is approximately 13.8 billion years. The Planck Collaboration's final cosmological analysis found a remarkably precise fit to this model. NASA's LAMBDA summary of age measurements also emphasizes that the result is an inference rather than a direct measurement and that its exact value depends somewhat on the cosmological model.
That last qualification matters. "The universe is 13.8 billion years old" does not mean that a scientist found the manufacturing date stamped beneath a galaxy. It means that the best-fitting standard model requires about 13.8 billion years of cosmic expansion to produce the universe we observe.
In the ideal FLRW universe, the answer is: every comoving location on the same cosmic-time slice.
Imagine a distant astronomer in a galaxy moving with the Hubble flow. After correcting for the galaxy's local motion and gravitational environment, that astronomer should infer the same large-scale density, CMB temperature, expansion history, and present cosmic age that we do.
There is an important catch. When we look at that galaxy, we see old light. If its light has traveled for ten billion years, we observe the galaxy as it was at a much earlier cosmic time. We can use the cosmological model to speak about what is happening "now" at its location, but we cannot see that present state yet.
This distant cosmic "now" is not forced upon every conceivable coordinate system. General relativity permits other ways to divide spacetime into space and time. The FLRW slicing is special because it follows the observed large-scale symmetries and matter flow of our universe. It turns "now" from an arbitrary coordinate convenience into a physically natural cosmological convention.
Real spacetime is also not perfectly smooth. Galaxies, clusters, voids, black holes, and planets introduce local motion and gravitational time dilation. A clock near a black hole will not keep cosmic time. Neither will a spacecraft traveling close to the speed of light relative to the Hubble flow.
Suppose such a spacecraft leaves a comoving observer, travels rapidly, and eventually reunites with another comoving observer. The spacecraft may record less proper time than the cosmic interval between departure and arrival. Its crew has not discovered that the universe is younger. They have discovered that their worldline through spacetime was different.
The distinction is simple once stated:
For a comoving observer in the ideal model, the two coincide. For an arbitrary observer, they need not.
There is one more trapdoor beneath the familiar number.
When cosmologists calculate the age, they extrapolate the standard expansion model backward toward $a=0$. In classical general relativity, this leads to the Big Bang singularity. But a singularity is usually a warning that the theory has been pushed beyond the conditions under which it can give a complete physical account.
We do not yet possess an experimentally confirmed theory of quantum gravity that describes the earliest conceivable stage. We therefore should not casually turn 13.8 billion years into a claim about an absolute beginning of reality, the birth of time from nothing, or what happened before the word "before" had completed its paperwork.
The scientifically careful statement is narrower:
About 13.8 billion years of cosmic time have elapsed between the hot, dense early-universe limit of our best-tested cosmological model and the present cosmic epoch.
That is already a tremendous claim. It does not need metaphysical overtime.
The apparent contradiction arose from imagining only two possibilities.
Either the universe possesses one universal clock, in which case relativity must be wrong; or every observer has a different time, in which case the age of the universe must be meaningless.
Nature has chosen a more interesting arrangement.
There is no clock outside the universe and no privileged inertial observer at headquarters. Instead, the large-scale contents of the universe define a family of comoving observers. Their local proper times can be joined into cosmic time because they share the same average history of expansion.
The age of the universe is true not at one special location, but across a whole cosmological epoch. It is written indirectly in the cooling of the cosmic background, the distances between galaxies, the remnants of ancient sound waves, and the oldest stars.
Relativity did take away the master clock.
But the universe, as it turns out, knows how to keep time without one.
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