Observations show that the expansion of the universe is accelerating.^{1} What causes this acceleration is an unsolved problem of physics; dark energy is only an ad hoc solution that was fitted to the observational data. The argument below shows that general relativity (GR) already provides the answer, making dark energy superfluous. Note that the argument assumes that space itself is not expanding, as explained at Expanding Space Obviated.

See the equations of special relativity (SR) for a relativistic rocket. Let a rod with a proper length of ten light years float in intergalactic space. Let a rocket decelerate relative to the rod and alongside it, first passing a beacon fixed at one end of it and finally coming to rest relative to it at its other end. Let the trip take just one year as the crew measures, as the equations allow. Then the crew observes the beacon recede to a distance of ten light years in one year, at which point the rocket is at rest relative to both the beacon and the rod.^{2, 3} The equations can be used to show that the crew measures the beacon pass the rocket at a speed of 99.98% of *c*, the speed of light.^{4}

The only way the crew can observe the beacon recede to a distance of more than one light year in one year, when its initial speed as they measure is less than *c*, is if it accelerates away from them as they measure (otherwise it would recede to a distance of *less* than one light year in one year). The rod is initially foreshortened as the crew measures due to special relativistic length contraction. As the rocket decelerates relative to the part of the rod passing by, the rod becomes less foreshortened as the crew measures; it expands in length. At a relativistic speed *v* (a speed close to *c*) relative to the part of the rod passing by, the length expansion of the rod outbalances the deceleration to make the beacon accelerate away from the crew as they measure. The difference between ten light years of recession and less than one light year of recession (the latter corresponding to their one-year trip at a speed less than *c*) is due to a *relative* (depends on the observer) expansion of the rod, and of the space containing the rod.

Remove the rod and let the rocket’s degree of deceleration vary among multiple tests. (But keep in mind that the rocket decelerates relative to the part of the now-removed rod passing by, not necessarily the beacon, which may accelerate away, even if I say “decelerates relative to the beacon” for lack of a better tangible thing to refer to.) The relativistic rocket equations can be used to show that, regardless of the degree of the rocket’s deceleration, when the speed *v* is relativistic the beacon accelerates away from the crew as they measure. To see this, for any acceleration *a*, plot the distance *d* / *γ* (the distance the crew measures) as a function of the time *T* (the elapsed time the crew measures) until the speed *v* is relativistic. Rotate the chart a half turn, so it depicts the opposite scenario, that the rocket decelerates to rest relative to the beacon, initially passing it at a relativistic speed.

Let a relativistic rocket decelerate past a floating beacon and come to rest with respect to it. As the crew measures, a curve of their distance from the beacon over time, when their initial speed relative to the beacon is low (top), has a tail when it is relativistic (middle and—closer to As measured by the crew of a relativistic rocket that passes a beacon while decelerating relative to it, their distance from the beacon as a function of time, from when they pass the beacon to when they come to rest relative to it. c—bottom). Along the tail the beacon accelerates away from the crew as they measure. The degree of the rocket’s deceleration is immaterial; these charts are producible regardless of the deceleration given. |

According to the equivalence principle of GR, during their trip the crew experiences the equivalent of a uniform gravitational field, like that experienced locally by someone standing on the Earth’s surface (an “Earth observer”). From the crew’s perspective the beacon moves as if it was thrown upward in a uniform gravitational field. Then the equations of motion for a locally uniform gravitational field are the relativistic rocket equations, replacing the relevant equations of Newton’s. (The top chart above plots an everyday activity on Earth, such as tossing up a ball.) **Let an Earth observer launch a particle upward at a relativistic speed. The particle initially accelerates away from the observer as that observer measures, the middle and bottom charts show**. Having employed only the equivalence principle and published equations of SR, locally, we now know the solution to the “keys accelerate toward the ceiling” mystery described at National Geographic News, for keys thrown upward at relativistic speed:

“When I throw my keys up in the air, the gravity of the Earth makes them slow down and return to me,” said Mario Livio, a theoretical physicist at the Space Telescope Science Institute (STScI) in Baltimore, Maryland.

But the study, along with an independent work released later the same year, showed that the expansion rate is actually speeding up.

This observation, Livio said, is as if “the keys suddenly went straight up toward the ceiling.”

(The keys in this analogy are placeholders for the high-redshift supernovae that compose the observation leading to the idea of dark energy. The only keys we’ve observed to accelerate toward the ceiling are those moving away from us at relativistic speed, if only when we drop the idea of expanding space.)

Consider an idealized case where the Earth is the sole gravitationally-significant body in the universe. Let the crew of a rocket that is hovering above the Earth at any distance from it (and therefore feeling acceleration from a thrusting engine, however slight) locally measure a particle *P* that recedes directly away from the Earth with high redshift, starting from when it passes right by the rocket at a relativistic speed. Again the relativistic rocket equations predict that *P* initially accelerates away from the crew as they measure. (Read carefully; these are *local* measurements made by the rocket’s crew, confined to a sufficiently small space and duration, that therefore can be validly predicted by those equations of SR.)

The crew can communicate those measurements to an Earth observer, who converts it to his or her own measurements (what he or she would measure about *P* while looking through a telescope) using the relative differences in the rate of local clocks and length of local standard rods that are predicted by GR between the observers. This conversion is tantamount to merely recalibrating each axis of a chart like those above, which leaves the shape of the curve unchanged and therefore results in an unchanged conclusion regarding acceleration or deceleration. For example, no matter how each axis of the middle chart above is recalibrated (to, say, change the time scale from 0 to *N* seconds on the rocket’s clock, to 0 to *Nx* seconds on the Earth observer’s clock, where *x* is the rate at which the Earth’s observer’s clock runs, expressed as a fraction of the rate at which the rocket’s clock runs), “acceleration away” is still indicated. Therefore the Earth observer would also measure that *P* accelerates away, notwithstanding the curved spacetime between the observer and *P*. **In other words, acceleration away is predicted by GR (across curved spacetime) as well as by SR (across flat spacetime).**^{5}

High-redshift supernovae accelerating away from us compose the observation of accelerating cosmic expansion, and are proxies for *P*. Then there is no need for dark energy to explain that observation. Furthermore, this thought experiment solves the horizon problem by showing that exponentially rapid cosmic expansion is possible in principle, without need for the inflaton field the inflation theory proposes.^{6, 7} Gravity alone does the trick. As an extreme example, the proposed new metric for Schwarzschild geometry (or the Schwarzschild metric) could be used to show that *in principle* an observer could measure a free object recede at a rate of ten light years per year, increasing within a second to twenty light years per year or any higher rate.^{8} Meanwhile, causal contact between the object and the observer would be retained, just as it is between the beacon and the rocket’s crew, or between *P* and the Earth observer.

Recall that the argument above assumes that space itself isn’t expanding. When the expanding space paradigm is discarded in concordance with Occam’s razor, it can now be seen that three major problems of physics vanish, with no new assumptions or theories needed (including the inflation theory): the flatness and horizon problems, and the nature of dark energy. Note that observations purporting to confirm the expanding space paradigm are explained by the argument above. Space can *measurably* expand in a *relative* way (it depends on the observer). Free objects can recede from an observer at speeds greater than *c* without violating SR or GR, in space that isn’t expanding in an absolute way (a way that stretches or breaks sufficiently large objects). Even if cosmologists keep the superfluous notion of space itself expanding (absolute expansion), they need to add the notion of relative expansion, assuming they agree with GR.

It isn’t necessary to feel acceleration, as the Earth observer does, to observe relative expansion or acceleration away. It’s enough to be at a fixed distance from a center of gravitational attraction; e.g. freely orbiting. That’s because gravity, not non-inertial acceleration per se, causes the deceleration on which both effects depend. The space between any observer and any receding decelerating free object expands (in a relative way) as the observer measures. If the object recedes sufficiently fast (i.e. if its redshift is sufficiently high) then the relative expansion of space causes the object to accelerate away as the observer measures. It’s not just the Earth’s gravity that makes the high-redshift supernovae accelerate away as an Earth observer measures. Also contributing (at least) is the gravity imparted by the whole Milky Way.

**References and Notes**

1. *Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant*, A. G. Riess et al., *Astronomical Journal*, vol. 116, pg. 1009 (September 1998), http://arxiv.org/abs/astro-ph/9805201.

2. Herein, as in most relativity texts, we ignore the travel time of light that prevents a remote event from being seen until after it has occurred. In principle a network of observers can be set up in locally inertial frames that momentarily co-move with the rocket, so that events can be detected by equipment right next to them, and the records subsequently compiled and analyzed.

3. Though the crew observes the beacon recede to a distance of ten light years in one year, a recession rate greater than *c*, this doesn’t mean that Einstein’s speed of light limit is violated, for that limit applies only to the speed of objects moving in spacetime as measured with respect to locally inertial frames, whereas the crew’s frame is non-inertial. In any locally inertial frame that momentarily co-moves with the rocket, the rod and the beacon’s speed is less than *c*. This “greater than *c* recession rate” also isn’t new information: The Relativistic Rocket site reports that a rocket accelerating / decelerating at 1 Earth gravity can travel from Earth to the Andromeda galaxy, 2 million light years away and arriving at low speed, in 28 years on the crew’s clock. Then the rocket’s crew would observe a beacon floating at the midpoint between the galaxies recede 1 million light years in the 14 years after they pass it.

4. Ideally this speed is measured in a locally inertial frame that momentarily co-moves with the rocket, since the rocket is decelerating.

5. A simpler way to see this: Flat spacetime is curved spacetime with zero curvature. Then, when acceleration away is predicted for flat spacetime it is concomitantly predicted for curved spacetime.

6. See *The Horizon Problem* at http://archive.ncsa.illinois.edu/Cyberia/Cosmos/HorizonProblem.html.

7. See *WMAP Cosmology 101: Inflationary Universe* at http://map.gsfc.nasa.gov/m_uni/uni_101inflation.html.

8. The relativistic rocket equations can predict that scenario, for an observer on the surface of a sufficiently large / massive object. (Note the hockey-stick shape of the bottommost chart above.) Then it’s predictable as well by the proposed new metric for Schwarzschild geometry. The relativistic rocket equations are special relativistic equations of motion for a locally uniform gravitational field (as deduced above), and the metric approximates SR locally, and the size of “local” is unlimited in principle. This isn’t to say that the metric couldn’t predict that scenario for a non-uniform gravitational field too.