# Special relativity: is "the accelerating Earth" possible as by flat-earth science

An explanation for the gravitational pull on Earth provided by some who claim we live on a flat world is that the Earth is constantly accelerating "up" at a rate of $9.81\frac{m}{s^2}$. That would mean Earth has been gaining speed since the dawn of time. It would be wildly incorrect to assume Newtonian mechanics and conclude that the speed of light has been passed long ago: $v = a\cdot t = 9.81t> c$ for any $t>\frac{c}{9.81}$.

But, for the sake of the argument, let's assume Earth is in fact accelerating upwards at a constant rate of $g$. Now, as the relativistic addition of speeds ensures one can never quite reach the speed of light: $$v_2 = \dfrac{v_1+u}{1+\dfrac{v_1u}{c^2}}$$

this must also directly apply to accelerating as well.

My intuitive knowledge of the theories of relativity is not that strong, so I must ask: Doesn't moving at velocities near $c$ mean time also stretches? If so, could it be, that even though Earth is accelerating up, we experience time slower, so the acceleration can continue at the same rate as observed by us?

A little disclaimer here: I don't actually think the Earth is flat. (what a surprise) So this is just a thought experiment for me. What do you think?

• As we'd be moving near the light speed and the closest starts are within single-digit light years, we'd be seeing these stars approaching to us in less than a decade. Unless, of course, they also magically accelerate in the same direction. It would be hard to explain the moon and other planes rotating around the sun or binary stars rotating around each other. And so many other things. Commented Sep 12, 2017 at 22:56
• up to where, exactly or even approximately? I appreciate you don't associate yourself with this argument, but it could not possibly be more vague, imo. The Earth's orbital speed around the Sun: 30 km/s (108,000 km/h, ~70,000 mph) The Sun's orbital speed around the Galaxy: ~200 km/s (720,000 km/h, 450,000 mph) (source iop). So if we don't notice this total velocity, which we don't in ordinary terms, any upward motion seems trivial.
– user167453
Commented Sep 12, 2017 at 23:05
• !!! All those specks of space dust hitting the Earth at the speed of light: we would be nuked to oblivion! This reminds me: what-if.xkcd.com/1
– user154997
Commented Sep 12, 2017 at 23:18
• Flat Earthers believe that the Sun and the Moon move in a plane about 3000 miles above the Earth. I don't know what they say about the stars, but I bet, when they say that the Earth is accelerating upward, they mean that their entire cosmos accelerates upward. Commented Sep 13, 2017 at 0:09
• The relativistic formula for constant acceleration is $v = c \tanh \left(\frac{aT}{c}\right)$, where $v$ is the speed relative to the initial rest frame, and $T$ is the proper time of the accelerating body. I derive this equation in physics.stackexchange.com/questions/343250/… but check the articles on the relativistic rocket in the Usenet physics FAQ and Wikipedia for other related equations. Commented Sep 13, 2017 at 10:57

You are correct in the sense that at velocities near c, time "stretches" or dilates. What we have to be careful about is what it means for time to dilate. I will switch over to the example of length contraction. If we had a ruler out to measure the length of an object as length contracts, the length of the ruler itself would also contract.

Coming back to time, we experience a similar situation. Here we are looking at the hypothetical case of the earth accelerating upwards at g = 9.81 m/s^2, due to some unknown combination of forces. I note here that in this hypothetical case acceleration is constant; it is speed that is approaching c. In this case we expect time to dilate. This is where we have to be careful. Time has been "stretched" rather than "slowed". This is an important distinction to make. However, as the earth continues to accelerate upwards, and approaches c ever more closely, time intervals become longer and longer. But in whose reference frame? For those standing on the earth, acceleration continues to be experienced exactly as g. For someone in space, they see the earth's acceleration as decreasing as the earth speed approaches c.

To put this into perspective, this example takes no information from real earth observations other than this particular earth object is accelerating at 9.8 m/s^2.

Just use the velocity addition equation you quoted and put in another 9.8 m/s every second. You can see that for the people doing the acceleration, it will seem like they gain 9.8 m/s every second, but an outside observer will measure a lower gain in velocity every second and the relative speed won't exceed $c$. Obviously you have to shrink your time step from 1 second to 0 and do an integral and think about GR if you want an exact result, but it's the same concept