###### By Don Lincoln, Ph.D., Fermi National Accelerator Laboratory

## Does mass increase as velocity increases? Why is it that we can’t move faster than the speed of light? Let us try to find answers to these questions using the concepts of relativity and spacetime. Also, let us try to debunk some common misconceptions using simple analogies and mathematical equations.

### Relationship between Energy, Velocity, Mass, and Relativity

It is common knowledge that if you increase the velocity of an object, its energy increases and, understandably, the converse is true as well. When energy levels are smaller, relativity also works typically as in classical physics. But, when the equations of relativity are used at higher energy levels, these predictions of classical physics fail to work.

The speed of light, denoted by ‘c’ is a universal constant and numerically translates to 186,000 miles per second or 300,000 kilometers per second. One consequence of this assumption is that massive objects cannot travel faster than the speed of light.

The typical reason for this is stated as mass increases when velocity increases. This statement seems to make sense because our intuition tells us that if we increase the velocity of an object, its energy increases and vice-versa.

A mathematical representation would be the velocity of an object increases as the square root of the energy. However, as the energy increases, the velocity in relativity is a little smaller than that in the classical theory. No matter how much more energy is given to an object, the velocity doesn’t change rapidly and never crosses the speed of light. Well, that is definitely contrary to common perception and thus, the entire idea of mass increases with an increase in velocity arises.

Albert Einstein’s familiar equation E equals mc squared provides the technical explanation of why mass increases as velocity increases. However, this equation works only when an object is stationary or has zero velocity. When an object is moving the right equation to use would be E equals gamma mc squared, where the term gamma has a velocity inside it.

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Understanding the Misconceptions of Science. Watch it now, on The Great Courses Plus.

### The Concept of Spacetime

The concept of spacetime is a little tricky or, should we say, there are misconceptions around it. The concept of space is real, can be defined, and it is possible to identify a unique point in space.

For example, you can move around in any direction or you can drive to work and come back home. But, time is different; it flows or moves constantly in one direction only and is unstoppable. It is a mental construct where you can only think of past or future events but cannot actually go there.

Going by this reasoning, it may seem that space and time are very different things, but in reality this could be a hasty conclusion.

Technically, Lorentz transforms prove that space and time cannot be disentangled as the equations just mix them up using the two points of view of the primed and the unprimed observer.

To illustrate with an analogy, someone living in the center of United States can move east/west and north/south. However, the effects of moving in the two directions are different experiences as moving east/west doesn’t change the temperature you experience very much, but moving north/south does. Yet, there is no issue in accepting that they are all components of the directions in the map.

Spacetime can be understood using this analogy, imagine the left/right motion as analogous to space, while the up/down motion is analogous to time. The other crucial point to remember is that the only speed in spacetime is the speed of light.

Learn more about untangling how Quantum Mechanics works.

### Why Can’t Objects Travel Faster than Light?

Spacetime is a crucial concept to answer why objects cannot move faster than light. We have understood that mass changing with velocity is not the reason why things can’t travel faster than light.

We will try and find answers as to why objects can’t travel faster than light using a few analogies and mathematical equations.

Let us use co-ordinate geometry to mark an arrow for velocity with end of the arrow at the origin. If the arrow can be rotated, we get different projections on the x and y axis. Taking velocity into consideration, mathematically this can be represented as v_x squared plus v_y squared equals v squared.

Hence, the object is always moving at a constant speed of v. It can move either entirely north or entirely east or toward north east at a speed that will never exceed v.

The next step would be to replace the x and y axis of the algebra above with space and time respectively. The horizontal axis for space can remain as ‘x’ while the vertical axis for time may be renamed as ‘t’.

Now, start rotating the velocity arrow under the assumption that all objects move at the speed of light in spacetime. Suppose the velocity arrow is pointing entirely in the time direction, it means that the object will move entirely through time and not through space.

Next, rotate the arrow toward the horizontal space direction and as the object is rotated closer and closer to the horizontal axis, the higher and higher is the velocity through space. At the same time, we have less and less velocity in time.

When the arrow is pointing horizontally, the velocity is entirely through space. This analogy shows that objects can move through space at velocities lower than the speed of light and they can’t move through space at velocities higher than the speed of light.

Once objects are moving through spacetime at the speed of light and when they’re moving entirely through space and not at all through time, there’s no way to get more velocity. All of the speed is through space.

This analogy also illustrates that an object moving extremely fast through space moves very slowly through time. Hence, an object moving at the speed of light through space experiences no time at all or in other words is frozen in time.

So, the real reason why we can’t move faster than the speed of light is that once we’re moving entirely through space, there’s no more speed to be gained. And this is a more accurate reason than the reasoning of changing masses.

Learn more about can you go faster than light?

### Caveats to the Above Explanation

- There could be quantitative issues if the above reasoning is dissected, however the reasoning is qualitatively correct.
- The reasoning is based on mathematics of circles, specifically t squared plus x squared equals a constant squared. This has significant consequences on calculations as relativity is really built on the mathematics of hyperbolas.

### Common Questions about Special Relativity

**Q: What are the three ways to travel at nearly the speed of light?**

Particles can be accelerated at nearly the speed of light using electromagnetic fields, magnetic explosions, and wave-particle interactions.

**Q: What is the famous equation of special relativity?**

The theory of special relativity explains how space and time are related for objects moving at a constant speed. Einstein’s famous equation E = mc2 is a part of this theory.

**Q: Why was the concept of relativistic mass introduced?**

The idea of relativistic mass of an object was introduced to simplify the concept of relativity. It was introduced to derive something similar to Einstein’s familiar equation.