Fastest Space Travel, Smallest Time Dilation

Note: A bug in WordPress or the LaTeX plugin used to display equations on this blog has broken the formatting at the end of this post, and I can’t figure out why or how to fix it.

You’re the captain of a medical ship responding to a distress call from the next solar system. You need to arrive as fast as possible. How fast do you go?

It sounds like a ridiculous question until you think about time dilation due to moving at relativistic speeds. If you want to skip straight to the answer, it’s ≈0.707c. If you travel any slower, it takes longer to get there (duh).. but the interesting part is that if you travel any faster, you also arrive later. While the trip time for you is purely based on how fast you move, the arrival time is based on how fast you move relative to your destination.

Welcome to relativity. A simple equation relates how much time passes between an observer at rest (your destination) and an observer in motion (you):

t = t0 / \sqrt{1 – v^2 / c^2}

t is the time at the destination, while t0 is the time that passes for you. You need to know how fast to go, so I’ve rearranged this equation to solve for a velocity based on known times:

v = \sqrt{ c^2t^2-c^2t_0 } / t

Of course, we don’t know what those times are either. It’s a bit of a conundrum (which kept me from solving this problem for over 2 years). This next part? I solved it at 3 am a few days ago, and then forgot what logical leap got me here. I have no idea why this works, I just know that it does:

v \in [0,1) \\
t_0 = 1 / v
t = t_0 / \sqrt{ 1 – t_0^2 }

The first line defines the valid range of velocities (anything from stationary to lightspeed, except for lightspeed itself). The second and third lines confuse the hell out of me. No idea how I figured out that your time is 1 / v. I know it has something to do with using multiples of c as velocity values so that I can replace occurrences of c with 1. It also has something to do with assuming 1 second passes for the stationary observer (your destination, t), allowing it to also be replaced with 1 as well.

Somehow these assumptions led to an equation with only t0 on the right side, and t on the left – what needs to be minimized! Again, we need to solve for velocity, so we substitute t0 with 1 / v:

t = v \sqrt{1 – v^2}

This is the part where I cheated and had Wolfram Alpha find the minimum for me:


And as you can see, we have a fairly interesting shape, and a minimum of 2 at 1 / √2, which is ≈0.707. Remember that this equation is using multiples of c, so that’s ≈211,985 km/s. This means that the fastest you can possibly arrive is however long it takes you at this speed, doubled, plus however long it takes you to accelerate and decelerate on either end of the trip.

If you travel as fast as possible, your arrival time is only double your trip time.

Of course, because of the acceleration and deceleration times, you will never hit exactly 2x time dilation over the total trip – but you don’t want to anyhow, as the only way to hit that value would be to go even faster – and guarantee you arrive even later than you otherwise would.

Fluoridated Water & Toothpaste Are Safe

Is fluoride bad for you? No. (Updated 2023-04-19. See footnotes for update history.)

I am not a medical professional, this article is the result of personal research, and it should not be considered medical advice. Additionally, this article is written for American audiences who are concerned with fluoride in drinking water. Unfortunately, fluoride exposure is far more common in less privileged countries (especially India and China)1, 15. In these countries, it is not from drinking water.

I’m going to get straight to the point: The worst possibility for the average person is exceedingly rare joint pain and brittle bones. This is called skeletal fluorosis1, and is caused by the body absorbing too much fluoride into bone tissue, which causes it to be brittle. It is difficult to consume enough fluoride to cause skeletal fluorosis – unless you’re eating toothpaste.

For children/teens (6-17)2, there is an increased risk of dental fluorosis, which is discoloration of teeth, and very rarely structural damage can also occur (in less than 0.25% of the population3). This is because children and teens are more likely to swallow toothpaste, and more likely to use too much toothpaste. See “How To Use Toothpaste” for more information, and please do read it because most people use too much toothpaste and don’t know it (including me before researching this article).

My Bias & My Sources’ Bias

I choose to use the worst possible case as presented from available data, because that level of caution tells me if danger is even a possibility. As a result, most of the sources I am using are from organizations against the use of fluoride in drinking water4, 6, 15, 17, and likely includes conspiracy theorists.

Ironically, the worst case data presented is that fluoride is not dangerous to the average person, and the benefits of fluoridated water outweigh the risks. This is despite several of my sources not distinguishing between dental and skeletal fluorosis4, 6, 15. Dental fluorosis is almost entirely harmless, while skeletal fluorosis can be crippling.

How Much Fluoride is Dangerous?

This is a difficult question to answer, and the average person does not need to worry about it. If you have poor kidney function, you should probably talk to a doctor about it. It is also more likely to affect you if you have a calcium or vitamin D deficiency5.

Most studies I looked at are of populations with medical problems that leave them susceptible to fluorosis. The lowest consumption I can possibly link to the average person developing any kind or amount of fluorosis is 10mg/day every day for 6 months4, 15, and this estimate is based on the most rapid onset fluorosis study I could find combined with a more reasonable estimate for the magnitude of consumption required6.

In order to consume 10mg/day of fluoride, one would need to drink 8 liters of highly fluoridated water7, 14, or eat 4 liters of raisins8, 17, or eat around 3-5 times as much maximally fluoridated toothpaste as should be used per day9And you would have to do these every day for 6 months or longer.

How Much Fluoride Am I Consuming? A Worst Case

Dental fluorosis occurs while teeth grow, and is the most common risk of fluoride. I set out to figure out the worst case consumption for a child around age 6, but these numbers can be easily extended to older or younger people.

Children 4-11 years old consume 2-2.2mg/day fluoride, of which 0.2-0.3mg is from toothpaste10. Assuming the worst case in these factors, this means children are getting 1.9mg/day from non-toothpaste sources, and spitting out 2.4mg/day from their toothpaste. But this is not the worst case. This estimate means children are drinking less than a liter of water per day and no food with fluoride in it, which is definitely not true.

A child 4-8 years old should be consuming 1.18 liters of water per day11, and is definitely getting fluoride from food as well8, 17. To make the worst case, I’m going to add 1.18 liters of maximally fluoridated water (1.42mg), a liter of the most fluoridated food (raisins, 2.34mg), and assume they’re using twice as much toothpaste as they should, using maximally fluoridated adult toothpaste, brushing twice a day, and swallowing it all. That’s 6mg/day from toothpaste alone!

Altogether? 9.76mg/day of fluoride, which is not good, but amazingly, is still below the worst threshold I could possibly link to fluorosis (10mg/day).

The most important take-away from this is: Don’t swallow toothpaste. Don’t use too much toothpaste.

How To Use Toothpaste

This is actually fairly simple, but please consult with a dentist, especially for young children. A “pea-sized” dollop is all you need. Children are at the highest risk of having dental problems, and so the right balance for them is most difficult.

If one has a hard time spitting or rinsing out toothpaste, consider finding a toothpaste with lower fluoride concentrations. This is more common in children’s toothpaste, but there is significant overlap in fluoride concentrations between adult and child toothpaste12.

(I was under the impression that children’s toothpaste should have lower fluoride than adult toothpaste, but lower concentrations are associated with worse dental outcomes, while higher concentrations are more likely to only cause mild fluorosis. Additionally, fluorosis matters far less for a child’s first set of teeth – and its presence or lack thereof can be used to guide changes in their fluoride use.)

Sources, References, Footnotes

  1. Wikipedia states this with a dead link to an article from UNICEF. However, multiple sources of similar information can be found via search. In the interest of time, I did not research more details.
  2. I am combining data from many sources to state the range of “children” at increased risk. Different sources give different ranges, and are primarily focused on pre-teen ages. Dental fluorosis can occur any time teeth are growing, and thus this includes teens.
  3. Prevalence and Severity of Dental Fluorosis in the United States, 1999-2004. (CDC)
  4. Fluoride Action Network discusses and links to many studies. However, they do not distinguish between dental and skeletal fluorosis – instead claiming that all fluorosis cases are dangerous and crippling. Additionally, they do not clarify which studies are of a sub-population (such as the many studies on kidney disease and kidney failure), and they do not clarify the difference between very rare occurrences and reasonable risks to the average person.
  5. Fluoride does not replace calcium in enamel (for teeth) and bone, but a lack of calcium increases how much fluoride is absorbed by enamel and bone.
  6. Estimated Thresholds For Fluorosis. Warning: This page appears to be misleading, as it is part of the Fluoride Action Network4, and intermixes studies across a wide range of locations & causes without clearly distinguishing between types of fluorosis.
  7. Water has been fluoridated to 0.7-1.2mg/L. At the maximum concentration, one would need to consume 8.33 liters per day to receive 10mg/day, which is not really possible14.
  8. FoodData Central. (USDA) (Originally, I used a different source17, but they are not reliable.)
  9. Adult toothpaste contains 1,000-1,500mg/L fluoride, adults should use no more than a “pea-sized” amount of toothpaste (approximately 1mL)citation needed, which means that swallowing adult toothpaste gives 1-1.5mg fluoride per swallow.
  10. Fluoride – Health Professional Fact Sheet. (NIH)
  11.’s page about hydration.
  12. Study highlights fluoridated toothpaste use in children.
  13. 1 ppm is equivalent to 1 mg/L. For consistency, I used mg/L within this article.
  14. Drinking more than 1 L/hr of water for more than 2 hours leads to water toxicity. It is likely one could die from water toxicity before they could consume enough fluoride to cause harm.
  15. Most studies “linking fluoride to negative health effects” are in India and China and based on high levels of contamination of multiple materials. These do not distinguish between lead, arsenic, and other toxins more likely to be the cause of these issues. Note: I did not do this analysis myself, and am relying on Professor Dave.
  16. In 2015, the US Department of Health & Human Services recognized that fluoride should be limited to 0.7mg/L for an optimal balance between dental protection and risk of causing dental fluorosis.
  17. Largest List of Foods That Contain Fluoride. Warning: This website is promoting pseudoscientific claims. Specifically, they claim to be able to remove fluoride from one’s body in 28 days, which is impossible due to how it bonds with bone and how slowly bone is reformed.

2022-01-20: Additional sources, footnotes; minor corrections/edits.
2022-02/03-??: Title changed to reduce likelihood of misunderstanding/spread of false information.
2023-04-19: Grammatical corrections, better word choice.
2023-08-17: Title was still too potentially misleading, changed again.

A Luke-Warm Political Hot-Take

Back in 2016, I published a blog post1 about the major election results for my state and country (California, USA). While I probably agree with most of my feelings at the time, I know for a fact that I made some bad choices with my votes, and an invalid hot take. I want to revisit this, and respond to it with a little more knowledge and hindsight.

Hilary Clinton vs Donald Trump

The voting system in this country is a joke. The only countries less democratic are those without any voting, and maybe “votes” with only one candidate.. which includes portions of the US. This is a large topic, but one of the easiest problems to attack is the presence of the electoral college. The popular vote doesn’t actually matter, what matters is a select group of 507 representatives. Most of the time, they vote according to what the populace votes for, but not accurately. Fortunately, the National Popular Vote Interstate Compact intends to fix this.

The more immediate problem is who got elected that time. Trump is a moron, and caused a lot of harm.. and that’s even before COVID-19 became a problem. He is directly responsible for the deaths of hundreds of thousands of Americans, and partially responsible for millions worldwide. Of course, every American president has a body count2 3, but Trump is a special level of stupid, careless, and evil.

That said, I’m still not sure he was the wrong person out of the two choices available4. Hilary wanted to go to war with Russia, a decision that could easily be civilization-ending when promoted by a competent politician. Of course, Russia interferes with all US elections, and in this case assisted the winner.

Flavors of Bad

I don’t know much about the particular candidates that ran for Senate and House of Representatives’ seats, but the Republican Party won the majority, and have a history of helping the wealthy grow their wealth at the expense of worldwide suffering. That said, I’m not holding the Democratic Party up as some kind of hero. They are responsible for the majority of the harm that Republicans have not caused, and have a history of pretending to be held up by Republicans when nothing of the sort has occurred.

Californian Propositions

I’m going to skip over propositions I can’t have a valid opinion on, as I lack knowledge on the specifics, with the exception of stating that on the surface 51-57, 59-65, 67 seem like progress. Of these, the following passed: 51, 52, 54-57, 59, 63-65. (Full results.)

58 (English-only schooling) and 66 (limiting death penalty appeals) are objectively bad propositions that passed, discriminating against non-English-speakers and increasing support for state-sanctioned murder. 62 (death penalty removal) might’ve been a good step towards prisoners’ rights – but did not pass. Of course, US prisons are somewhere between a death camp and slavery on average5.

I previously agreed with 58. At that time, I did not realize how much of a discriminatory it is to allow English-only schools. On the surface, it sounds fine, the majority is English-speaking, and should not have to be hampered by provisions for non-English-speakers in areas where there aren’t any non-English-speakers, right? The problem is these areas don’t really exist, and this just hides them farther under the rug6.

Many of us believed that 67 (single-use grocery bag ban) was a positive. I was led to believe that the benefits were grossly outweighed by unintended costs and increases in plastic usage to make reusable bags. For example, it would take a “reusable” bag thousands of uses to make up the difference in environmental impact of its creation vs a single-use bag. This is a feat that is not likely to be achieved. In fact, this post was written poo-pooing this proposition until I went to gather sources.

Turns out, it’s a lot more complicated!7 While the negative facts I’d heard are true, the result of California’s ban is the majority of purchases are being made without any bag use. Turns out, they just didn’t need them for the most part. (It is important to specify that these results are not generalizeable to any population, however.)

I am somewhat conflicted about 63 (background checks for purchasing ammo, prohibition of large-capacity magazines). My gut tells me this is a good thing, because gun control is a good thing. However, I have heard evidence that suggests that these kind of half-assed measures are easily worked around and serve no actual prevention of gun-related harm. At the time I went all-in on my gut feeling, instead of doing more research on the topic. At this time, I have not gone back to check sources to find out if this measure is actually helping.

There is much more to be said, but that also requires research. Please comment if you would like to see that done on these or other political topics.

  1. The original will likely go down as part of a site redesign and holds no value as content, so I’ve created an unlisted paste of it.
  2. That list is obviously outdated, as Trump easily makes the top 5. And that’s without even counting foreign deaths.
  3. Another example: Sanctions, which directly lead to somewhere between thousands and millions of deaths, and much more significant suffering.
  4. History shows that Americans rarely have a 3rd option in presidential elections.
  5. This article on modern concentration camps primarily focuses on more recent events, but offers a taste of what I am referring to.
  6. Languages spoken in California.
  7. ABC News’ article on plastic bag bans. I cannot reliably determine the net effect of California’s prop 67, but it seems to be edging into positivity.